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IRD 101: QUANTITATIVE SKILLS I
MOI UNIVERSITY
IRD 101: QUANTITATIVE SKILLS I
BY:
S.I. NG'ANG'A
NG’ANG’A S. I. 15TH DEC 2009
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IRD 101: QUANTITATIVE SKILLS I
QUANTITATIVE SKILLS DEPARTMENT
COURSES OUTLINE FOR IRD 101 - QUANTITATIVE SKILLS I
1ST SEMESTER: 16 WEEKS
1. NUMBER SYSTEM: (2 HOURS)
1.1 Sets of Numbers
1.2 Properties of Real Numbers.
1.3 Fractions and their properties
2.
BASIC SET THEORY: (3 HOURS)
2.1
Definition of sets. A collection of District Objects e.g. all salty Lakes in Africa
2.2
Symbols in sets UNCXES
2.3
Operation on sets. '
2.4
Application of set theory to problem solving
3. COMPUTATION SKILLS: (6 HOURS)
4.1 Exponents and Logarithms
•
Definition of Exponents, base, mantissa characteristics, logarithm
•
Laws of Exponents and logarithms
•
Use of logarithms in computation.
4.2
Use of calculators and computers. (General, principles)
4.
EQUATIONS: (5 HOURS)
4.1
Equation as a Function
4.2
Formulation of simple equations
4.3
Systems of Equations
•
Graphic representation
•
Simultaneous equations .and their solutions: (two and three unknowns)
•
Use of matrices to solve simultaneous equations.
5.
GRAPHS: (6 HOURS)
5.1 Principles of Graph constructions
5.2 Types of Graphs and their uses.
5.3 Construction of the Lorenz curve, z-curves, Semi-log
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6 FREQUENCY DISTRIBUTION: (12 HOURS)
6.1
Methods of Data collection,
6.2
Frequency Tables, Polygons and curves
6.3
Measures of Central Tendency
- Mode, mean and median (mention others too)
6.4
Measures of Dispersion
Range, Standard Deviation, Quartile Deviation, Variance.
6.5
Bivariate Data
7.
TIME SERIES: (8 HOURS)
7.1
Definition of time series concepts
7.2
Examples of time series
7.3
Moving averages
7.4
Estimation of trend,
- Use of scatter diagrams.
REFERENCE BOOKS
1. Gupta S.P: Statistical Methods Enlarged Edition, 1983
2. Carolyne Dinwiddy: Elementary Mathematics for Economists
3. Marray Spiegel: Probability and Statistics Fifth Edition
4. Robert L. Childress: Calculus for Business and Economics
5. D.N. Elhance: Fundamentals of Statistics
6. W. Swokowski: Functions and Graphs
7. G.L. Thirkettle: Business Statistics and Statistical methods
8. Clare Moris: Quantitative approaches in business studies
9. Sabah Al-hadad & Scott: College Algebra with Applications
10. Gustafson & Peter Frisk: Algebra for College Students
11. Van Doorne: Elementary Statistics
12. Core Texts that Students are advised to buy
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IRD 101: QUANTITATIVE SKILLS I
TABLE OF CONTENT
Contents
1.0 NUMBERS................................................................................................................ 6
1.1 SET OF NUMBERS ........................................................................................................ 6
1.2 Properties ......................................................................................................................... 8
1.3 Arithmetic of real numbers ............................................................................................ 10
1.4 Fractions and their properties......................................................................................... 11
1.5 Algebraic Fractions ........................................................................................................ 12
1.6Revision questions .......................................................................................................... 14
2.0 BASIC SET THEORY ............................................................................................ 15
2.1 Introduction .................................................................................................................... 15
2.2 Types of sets .................................................................................................................. 15
2.3 Set Concept and Their Symbols ..................................................................................... 15
2.4 Finite and Infinite Sets ................................................................................................... 18
2.5 Complement of a Set ...................................................................................................... 18
2.7 Product of Set ................................................................................................................. 19
2.8 Venn diagram ................................................................................................................. 19
2.9 Basic Set Operation........................................................................................................ 21
2.10 Application of Sets ....................................................................................................... 24
2.11Revision questions ........................................................................................................ 28
3.0 COMPUTATION SKILLS ..................................................................................... 31
3.1 Exponents and Logarithms ............................................................................................ 31
3.2Definition: ....................................................................................................................... 31
3.3 Logarithms ..................................................................................................................... 32
3.3.1 Laws Of Logarithms ............................................................................................... 33
4.0 EQUATIONS .......................................................................................................... 37
4.1 Introduction .................................................................................................................... 37
4.2 Solutions of Equations ................................................................................................... 37
4.2.1 Categories of equation/types of equations .............................................................. 38
4.2.2 Problems leading to quadratic equations: ............................................................... 40
4.3 MATRICES ................................................................................................................... 46
4.3.1 Introduction ............................................................................................................. 46
4.3.2 Types of Matrices ................................................................................................... 47
4.3.3 Addition and Subtraction of Matrices ..................................................................... 53
4.3.4 Multiplication of matrices by a real number ........................................................... 53
4.3.5 Multiplication of Matrices ...................................................................................... 54
4.3.6 Determinants ........................................................................................................... 55
4.3.7 MINORS ................................................................................................................. 57
4.3.8 Cofactor Matrix ........................................................................................................... 58
4.3.9 Adjoint Matrix ........................................................................................................ 62
4.3.10 Inverse of a matrix ................................................................................................ 63
4.3.11 Solutions of Linear Simultaneous Equation by Matrix Algebra ........................... 65
4.3.12 Solution of simultaneous equation by inverse method ............................................. 67
4.3.13Revision Questions ................................................................................................ 69
5.0 GRAPHS: (DATA PRESENTATION) .................................................................. 71
5.1 Introduction .................................................................................................................... 71
5.2Frequency distribution .................................................................................................... 71
5.3 Cumulative Frequency Distribution .......................................................................... 72
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5.4 Ogive ......................................................................................................................... 73
5.5 Relative frequency distribution ................................................................................. 76
5.6 Histograms and bar charts ......................................................................................... 77
5.7 Frequency polygon .................................................................................................... 77
5.8 Graphs ....................................................................................................................... 78
5.9Pie-Charts ........................................................................................................................ 79
5.8Tables .............................................................................................................................. 79
5.10Other Diagrams ............................................................................................................. 81
5.11 SPECIAL TYPES OF GRAPHS ................................................................................. 83
5.11.1 Z Charts ..................................................................................................................... 83
5.11.2 Scatter Graphs ....................................................................................................... 86
5.11.3 Semi - logarithmic graphs: .................................................................................... 88
5.12Revision questions ........................................................................................................ 97
6.1Sampling and sampling design ..................................................................................... 101
6.1.1 Sampling ............................................................................................................... 101
6.1.2 Sample Examination Questions -Sampling .......................................................... 108
6.2 Methods of Data collection .......................................................................................... 111
6.3 DATA ANALYISIS .................................................................................................... 118
6.3.1Introduction ............................................................................................................ 118
6.3.2 Qualitative data analysis ....................................................................................... 118
6.3.3 Quantitative data analysis ..................................................................................... 122
6.3.4 Descriptive statistics ............................................................................................. 122
6.4Measures of central tendency........................................................................................ 122
6.5 Measures of Dispersion................................................................................................ 126
6.6 Skewness and Peakedness............................................................................................ 132
6.6.1 Skewness ............................................................................................................... 132
6.6.2 Peakedness (kurtosis) ............................................................................................ 134
6.7 Bivariate Data .............................................................................................................. 134
6.8 Revision Questions ...................................................................................................... 139
7. 0
TIME SERIES: (8 HOURS) ......................................................................... 144
7.1 Definition of Time series graphs.................................................................................. 144
7.2 Components of a time series ........................................................................................ 146
7.3 Method of semi averages ............................................................................................. 150
7.4 Method of least squares: .............................................................................................. 156
7.4Revision question .......................................................................................................... 163
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1.0 NUMBERS
1.1 SET OF NUMBERS
This is a group or combinations that are used in mathematics. We can group all numbers in
any of the following category:
(i)
Natural numbers
(ii)
Prime numbers
(iii)
Composite numbers
(iv)
Whole numbers
(v)
Integers
(vi)
Rational numbers
(vii)
Irrational numbers
(i) Natural numbers (N)
These are the numbers we normally use in counting. They are counting numbers ie
1,2,3,4 etc. these numbers constitute the set of natural numbers, N, defined as:
N = (1, 2, 3 ….)
Any subject of the set of natural numbers can be drawn on a coordinate line. The first
step would be to draw the natural number line and then plot the set on the N- line.
If a person was asked to count a number of hens, dogs, cows, students, one
would definitively start by counting 1,2,3,4 etc. These numbers come into ones mind
most naturally when counting anything thus called natural numbers.
(ii) Prime numbers (P)
These is any natural number greater than one that is divisible without remainder only
by it self and one ie 2, 3,5,7,11,13,17,17,23, etc.
(iii) Composite number (C1)
These are natural numbers greater than one that is not a prime number. It can be
divided by other numbers without a remainder besides one and itself, ie
4,6,8,8,9,10,12,etc.
(iv) Whole Numbers (W)
When zero is added to the set of natural numbers, the set N is transformed into the set
of whole numbers, W, defined as
W = (0, 1, 2, 3…..).
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(v) Integers
The set of integers is an extension of W by the incorporation of negative numbers.
Hence they are a set of all negative and positive whole numbers including the Zero ie
-5,-4,-3,-2,-1, 0, 1, 2,3,4,5. Zero is neutral, being neither positive nor negative.
Any subject of I can be plotted on the coordinate line. The procedure for plotting
subjects of I is illustrated in the example below.
Plot the following set:
P= (-3, 0, 2)
-5
-4
.
-3
-2
-1
.
0
1
.
2
3
4
5
Rational numbers Vs irrational numbers. (Q).
A rational number is a number of the form
a
b
in which a and b are integers with no common
factor ( if there is a common factor, it should be cancelled) eg
2
4
= ½ where b is not
supposed to be 0 ie b≠ 0 but b can be 1 and other numbers a can be larger than b eg
5
3
Irrational numbers
Irrational numbers are the opposite of rational numbers. the set of irrational numbers ,
is the set of all those numbers which cannot be expressed as a ratio of the integers. Π,
2 and 3, are examples of irrational numbers.
A simple way of disguising rational from irrational numbers with decimals is to study
their decimals. The decimals of rational numbers are periodic or repeating decimals,
whereas irrational numbers have non-periodic or non repeating decimals.
22
7
Is a rational number which has always been used as an approximation of the
irrational number
The decimal of
π.
π are non-periodic and are given below.
π = 3.14159265358….
However the decimals of 22/7 are periodic with a periodicity of 6.
22
7
= 3.14285714285714…
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Real numbers (r)
Between any two rational numbers we have at least one irrational number, and,
conversely, between any two irrational numbers there is at least one rational number.
Hence the irrational numbers fill in the gaps between rational numbers and vice versa.
This process results in a continuum numbers constituting the set of real numbers.
Thus, A set of all rational numbers.
A real number can be represented as decimals eg – 1/6 = - 0.166….., ½ = 0.5, 1/3 =
0.33…, 2 = 1.4142,
π = 3.141…
However, some real numbers may not necessarily be written in the decimal points eg natural
numbers and integers which also belong to the set, ie 3, 5, -1, -2, etc.
e.g. the subset -3 ≤ x < 2of R is shown below as a continuous line.
-5
-4
.
-3
-1
-2
0
1
2
3
4
5
R-Line
Another way of visualizing a set of real numbers is that every real number is used as a
co-ordinate for appoint on the number line. Therefore there is 1:1 correspondent
between the set of real numbers and the number line.
1.2 Properties
1. Equality property
If x, y & z are real numbers and x=y then we can say that:
x + z= y + z
x–z=y–z
xz=yz
x/z = y/z
if z ≠o
2. Reflexive property
If a is any real number, then a = a. any real number is equal to itself.
3. The symmetric property
If a, b, are real numbers and if a = b, b = a
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4. The transitive property
If a, b, and c are real numbers and if a = b and b = c then a = c.
If one number is equal to a second and if the second number is equal to the third then
the first number is equal to the number.
5. The substitution property
If a and b are real numbers and a = b then b can be substituted for a in any
mathematical expression to obtain an equivalent expression.
Examples:
1. x -3 = x -3 Reflexive
2. if 5x = 3y then 3y = 5x – Symmetric
3. if 6x = 10 and 3y = 10 then 6x = 3y (Transitive)
4. x + 4 = x y and x = 2 then 2 + 4 = 2y (substitution)
6. The closure property
If a and b are real numbers then a + b is a real number,
a – b is real no.
a X b is Real No
a/b is real no. provided b ± 0
Clause property guarantees that the sum, difference, product and quotient of any 2
real numbers are a real number, provided there if NO division by Zero (0).
7. Associative property
If a, b and c are real No.s, then (a + b) + c = a +(b +c), and (a b)c = a(bc). This
property permits us to group or associate the numbers in a sum or product in any way
that we wish.
Example:
(4 + 5) + 6 = 4 + (5 + 6 ) = 15
(2.3) .4 = 2. (3.4) = 24
8. Commutative property
If a and b are real numbers, then a + b = b + a and also a b + b a. These property
permits that addition and multiplication of any 2 real numbers to be done is either
order gives the same answer.
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9. The distributive property of multiplication and addition
If a, b and c are real numbers then a (b +c) = a b +a c.
10. Identity elements and diverse elements
(i)
Additive elements
0 is the addictive identity elements because by adding 0 to any real number, the
number remains the same e.g. a + 0 = 0 + a = a
(ii)
1 is the multiplicative identity elements since a 1 = 1 a = a where a in the case of
(i) and vice versa.
Since a + (-a) = (-a) + (+a) = 0
1
a
Is called the reciprocal of the multiplicative inverse of a. Also a is the reciprocal of
multiplicative inverse of
i.e.
1
a
=
1
a
1
a
provided a ± 0
a= 1
NB: The reciprocal of 0 does not exist because there is No number that can be multiplied by 0
to get 1.
1.3 Arithmetic of real numbers
If 2 real numbers have like signs, their sum is found by adding their common sign i.e.
a + b = (a) + (b) = + (a + b)
a-b = (a) + (-b)
If two real numbers have unlike signs their sum is found by subtracting their absolute values.
The smaller from the larger and using the sign of the number with greater absolute value.
Example:
x – y = x + (-y)
5- 10 = -2
The product or the quotient of the real numbers with unlike signs is the –ve of the product or
quotient of their absolute values.
2X4=8
2 X -4 =-8
8
2
=-4
Order of operations
If an expression does not contain grouping symbols then,;
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(i)
Evaluate any exponential expression like xy
(ii)
Do all multiplication and division as they are encounter working from the left to
the right.
(iii)
Do all additions and subtractions as they are encounter working from left to right.
If an expression contains grouping symbols use the above rules to perform the
calculation within each pair of grouping symbols from the inner most pair.
Example:
2x2 + (x +1)2 + 4
when x = 1
2 (1)2 + (1 + 1)2 + 4
2 + 4 +4 = 10
1.4 Fractions and their properties
Properties:
a
1. Assume the following fractions b and c d , & d 0 and if b 0 then, we conclude that
a
b
c
=
if ad = b c and this property is property of equality.
d
Example:
7
9 =
49
63
9 49 = 7 63 because the product are equal then the fraction are equal.
a
2. If a is a real number then
1
= a and if a ≠ 0 then
a
a =1
Example:
6
6
= 1,
6
=6
1
3. Fraction are multiplied and divided according to the following definitions:
(i)
a
b
a
(ii) b ÷
c
c
d =
ac ac
=
provided b≠ 0& d≠ 0
bd bd
a
d
= b c =
d
ad
provided d 0, b 0, c 0
bc
Example:
1
1
3
3
5
÷
÷
4
4
2
5
5
=
1.2
=
3.5
4
1
2
1 .5
5
= 5 5 = 20 =
7÷ 5 =
3.4
7.5
¼
= 12 35
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4. Scaling factor
a
If b≠ 0 and R≠0 then, b =
ka
bk =
k
k
÷
a
b
=
a
b
Example:
4
2
2 .2
10 = 2.5 =
2
2 ÷ 5=
2
5
This property can also be used to build fractions by inserting common factors in both
numerator and denominator.
Example:
2
Write 5 with a denominator of 30.
Common factor = 6.
2
is
5
2
6
5
÷ 6 =
12
30 .
5. Signs
a
b =
a
b =
a a a
= - = b
b b
6. Fractions are added ands subtracted according to the following definitions:If b ≠ 0 then;
a
3
c
b + d =
4
+ 47 =
ac
b
3.7 4.5 21 20 41 6
1 35
35
35
35
Show that;
a
b
+
c
d
=
ad bc
provided that b≠0, d ≠ 0
bd
1.5 Algebraic Fractions
The rule governing the use of Algebraic fractions are identical to those used in ordinary
fraction.
1. Simplification of algebraic equations
Fractions may be simplified by removing a common factor from both numerator and
denominator.
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Example:
18bx 2
Common factor = 9b x
63byx
=
2x
7y
2. Adding and subtracting of algebraic expressions
Fractions have to have a common denominator before they can be added or
subtracted.
Example:
x y x y
Common denominator 6
2
3
3 (x + y) +2 (x – y) = 3x + 3y + 2x – 2y
6
6
=5x–y
6
3a -2b – 3b – a common denominator is ab2
b2
Ab
[
3a -2b – 3b – a
ab2
b2
ab
]
3ab – 2b2 – 3ab + a2
Ab2
-2b2 +a2 = a2 –2 b2
Ab2
ab2
3. Multiplication and division of fractions:
Example:
x2 – 1
x2 – 2x
3x – 6
4x + 4
Factoring and simplifying, we have
(x +1) (x – 1) X 3(x -2)
x(x – 2)
4 (x + 1)
= 3 (x – 1)
4x
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Assign.
ab
1
A2-b2
÷
=
ab
a –b
(a -b) (a+b)
a –b
1
= ab
a+b
Simplifications of complex fractions
Example:
a2 – b2 ÷ a + b = a2 – b2
a
3
b
3
a+b
= (a –b) X 3 = 3(a –b )
a
b
QUIZ: Change a- b to an equal factor whose denominator is d-c
c –d
1.6Revision questions
1. State whether each of the following sets is finite or infinite and justify your answer.
i. {x:x is a rational number}
2 mks
ii. {y:y is a country in the word}
2 mks
iii. {z:z is a student in a Kenyan university}
2 mks
2. List the members of the set
Q={r:r€T=3r+1 for r=0,1,2,3}
What is n (Q)?
3mks
3. a) State whether each of the following is finite or infinite and in each case justify your
answer.
(i.) A=[x:x is a whole number]
2mks
(ii.) B=[x:4<x<20; x is a rational number]
2mks
b) Simplify completely and ten find the value of b in each case if a=29
i.) 7{a+[4+5(b-3a)]}=35
3mks
ii.) 4[2a+3[5-2(a-b)]}=124
3mks
4. State whether each of the following is a discrete or a continuous variable
i. The number of students in both private and public universities of
Kenya
1mk
ii. The capacity of the Moi university water tank
1mk
iii. The speed of rotation of the earth on its axis
1mk
iv. The temperature of a coolant
1mk
1 x 1
2
x 4
5. (i) Simplify 2
(2marks)
x2
2
(ii) Solve for x
1
(3marks)
2 x 1 x 2 4 0
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2.0 BASIC SET THEORY
2.1 Introduction
A set is a fundamental concept in all branches of mathematics.
DEFINITION: A set is any well defined list, collection, or class of objects. An object in set
can be anything i.e. numbers, people, letters, rivers, mountains etc. these objects are called
the elements or numbers of the set.
Set notations
Sets are usually denoted by capital letters i.e. A,B,C, D etc. the elements or members in set
are usually represented by lower case letters i.e. a, b ,c, d etc.
2.2 Types of sets
1. Numerative sets
2. Discriptive sets.
1. Numerative sets:
If we define a particular set by actually listing its ,member e.g. let A consist of the numbers
1,3,7 and 10, then we write a set as A = (1,3,7,8,10). Numerative i.e., the elements are
separated by, comas and closed in brackets ( ). This is a Tabular form of a set.
2. Discriptive sets
If we define a particular set by stating properties which its elements must satisfy eg let B be
the set of all even numbers, then we use a letter usually x to represent an arbitrary element
and we write.
B = (x/x is even), which reads as B is the set of numbers x such that x is even. We call
this the set builder form of set.
B = (x: x is even)
NB/: The vertical line or 2 dots(:) is read as that
2.3 Set Concept and Their Symbols
1. Sets of sets
Sometimes it will happen that the object of a set are sets themselves e.g. the set of all subjects
of A. it is also known as family of sets or class of sets.
The symbol used are the script letters e.g. Β, etc
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1.
Universal set U or Σ
The family of all the subset of any set (S) is called the power set of S. we denote the power
set of S a2 2s
Let M = {a, b}
Then 2M = { (a, b), (a), (b), φ}
Let T = { 4,7,8}
2T = {(4,7,8), (4,7) (4,8)(7,8) (4) (7) (8), φ}
If a set is finite say S has n elements then the power set of S can be shown to have 2n
elements. This is one reason why the class of subjects of S is called the power set of S and is
denoted by 2s.
4. Disjoint set
If sets A and B have no elements in common i.e. if no element of A is in B and no element of
B is in A then, we say A and B are disjoint.
Example:
Let A ={1,3,7,8}
B = { 2,4,7,9} then A and B are not disjoint.
Since 7 is in both sets.
Q 2:
Let A be the +ve and B be –ve numbers. Then A and B are disjoint set since no number is
both –ve and +ve.
5. Comparability sets.
Two sets A and B are said to be comparable if ACB or BCA i.e. if one of the sets is a subject
of the other set. However, two sets A and B are said to be not comparable if A ± B or B ± A.
NB:
If A is not comparable to B then there is an element in A which is not in B and also there is
an element in B which is not in A.
Example:
Let: A = { a,b}
B { a,b,c}
A is comparable to B since A is a subject of B but we cannot say B is comparable to A
because B is not a subject of A.
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R = {a,b)
C = { b,c,d}
R and C are not comparable since a is not in C i.e. R ± C, C± R.
6. Subsets
If every element in a set A is also a member of a set B then A is a subset of B if x is a
member of A. it implies that x is an element of A and B i.e. { xEA= xEB}
We denote this relationship by writing ACB which can also be read as A is contained in B.
Example 1.:
The set C is given by elements C = {1,3,5}
D = {5,4,3,2,1} since each element 1,3,5 belonging to C also belongs to D.
If E = {2,4,6} and F = {6,2,4}, since each element 2,4,6 belonging to E also to F
NB: let G = {x1 X is even } i.e.
G = {2,4,6,8…}
F = { x 1x is a positive power of 2}
I.e. F = { 2,4,8,16…..}
Then F is a subset or contained of G.
Definition:
Two sets A and B are equal i.e. A = B iff ACB and BCA. If ACB then we can also write B
A. if A is not a subset of B.
Conclusion:
1. The null set is considered to be subset of every set.
2. If A is not a subset of B, then there is at least one element in A that is not a member of
B.
Proper Subsets
Since every set A is a subset of itself then we call B a proper subset of A if
(i)
B is a subset of A i.e. BCA
(ii)
B is not equal to A i.e. B ≠ A
In some books B is a subset of A denoted by BCA = BCA and B is proper subset of A is
denoted by BCA.
Null set (ф)
Empty set/null set is a set that contain no elements. Such a set is void or empty and we denote
it by the symbol ф.
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Example:
Let B = {x1x2 =4} and is defined as odd
Then, B = { }
Equality of sets
Set A = set B if they both have the same members i.e. if every element which belongs to A
also belongs to B and if every element which belongs to B also belongs to A we denote by A
= B.
Example:
Let A = {1,2,3,4}
B = {3,1,4,2}
A = B or { 1,2,3,4,2} = {3,1,4,2}, because all members belonging to A belongs to B.
NB: repetition is not recognized. A set does not change if its element are repeated.
Example 3:
E= {x1x2 – 3x = -2}
E = {2,1},
G = {1,2,2,1}
Therefore E = F = G
2.4 Finite and Infinite Sets
Sets can be finite or infinite. A set is finite if it consists of a specific number of different
elements i.e. if in counting the different members of the set the counting process come to an
end otherwise a set is infinite.
Example: Let M = {days of the week} finite
N = { 2,4,6,8…} N is infinite
P = { x1x is a river on the earth} therefore P is finite although it
may be difficult to count the number of rivers in the the earth, P
is still a finite set.
2.5 Complement of a Set
If A is any set which is a subject of a universal set then the complement of A normally
written as A1 or Ac is defined as all those elements that are not contained in A but are
contained in U or E.
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Example:
E = {1,2,3,4,5,6,7,8,9}
A = {2,3,4,8}
Ac or A1 = {1,5,6,7,9}
2.6 Overlapping Sets
If sets A and B have same elements but these are not subsets of another set then, these are
called overlapping sets. E.g.
A ={1,2,3,4}, B = {3,4,5,6,7) = A¢ B
3 and 4 are common elements then they are overlapping set.
2.7 Product of Set
If A and B are any two sets, then the product of A and B denoted by A X B consist of all
ordered pairs (a,b) where a is an element of A and b an element of B.
Hence A X B = { (a,): aEA, bEB}
The product of a set with itself is A X A= A2
Example:
Let A = {1,2,3} and B = {a, b}
Then A X B = {1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}
The concept of product set is extended to any finite number of sets in a natural way. The
product set of the sets A1, A2, A3…., Am is the set of all ordered in triples i.e. a1, a2,
a3,……… am where a:E A; for each is;
Example: Let M = {Tom, Mark, Eric}
W= {Andrew, Betty}, Find M X W
MXW = {(Tom, Audrey), (Tom, Betty), (Mark, Audrey), (Mark, Betty), (Eric, Audrey),
(Eric, Betty}
If we let A = {1,2,3}, B = {2,4} and C = {3,4,5}
Find A X B X C
2.8 Venn diagram
It is a simple pictorial representation of a set. We represent a set by a simple plane area
usually bounded by a circle.
Example:
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ACB
A≠B
Suppose A and B are not comparable
Example:
Let A = {a, b, c, d} and B= {c, d, e, f}
Show in a Venn diagram.
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2.9 Basic Set Operation
In the set theory, we define the operation UNION INTERSECTION & DIFFERENCE i.e. we
assign new sets to pair of sets A & B
1. UNION
The union of 2 sets A & B is the set of ALL elements which belong to A and B or both. The
union of two sets A and B is denoted by AUB read ‘A Union B’. The union of two sets A and
B i.e. AUB is shown by means of Venn diagram by the shaded region or area in the following
diagrams.
AUB is shaded. Suppose P = {a, b, c, d} & Q= {b, d, f, g} then PUQ = {a, b, c, d, f, g}
Example:
Let ℓ be the set of positive real numbers and M be set –ve real numbers. what is ℓ UM
= the set of all real numbers except 0.
Thus the union of AUB = {x1xEB}. We can conclude directly from the definition of A and B
that AUB and BUA are the same set ie AUB =BUA.
Similarly we conclude that both sets A and B are always subsets of AUB ie
AC (AUB)
BC(AUB)
NB: in some books + is used instead of U and is called the theoretic sum which reads A+ B ie
“A plus B’.
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2. INTERSECTION
Intersection of two sets A and B is the sets of elements which are common to A and B ie
those elements which belongs to A and also belong to B. the intersection of A and B is
denoted by AnB which is read ‘A intersection B’. the intersection of two sets A and B ie An
B is shown by means of Venn diagram by the shaded region that is common to both A and B.
Example: If we let P = {2,4,6,…..} i.e. multiple of 2
And Q = {3,6,9……} multiple of 3.
Then PnQ = {6, 12,18,24,30 ……..}
Example: if we let L = {a, b, c, d} & M ={f, b, d, g,}
Then ℓn M = {b, d}, hence intersection of two sets A and B can also be defined as AnB =
{x1xEA and xEB}. This we can conclude directly from the delimitation of the intersection of
two sets that is AnB = BNA. Similarly we also conclude that each of the sets A and B as a
subset i.e.
(AnB) CA
(AnB) CB
In the same way it sets A and B have no elements in common ie A and B are disjoint then the
intersection of A and B is null set i.e. AnB = ф
DIFFERENCE
The difference of two sets A and B is the set of elements which belong to A but which do not
belong to B. the difference of two sets A and B is denoted by A –B and is read as A
difference B or A minus B. the difference of two sets A and B is also sometimes denoted by
A/B or A2B read as A given B.
The difference of two sets A and B ie A – B is shown by Venn diagram by the shaded area/
region in A which is not part of B.
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Example:
Let P = {a,b,c,d} and Q = {b,d,f,g}
Then P –Q or
or P/Q = {a,c} or Q-P ={f,g}
Example:
Let L be set of real numbers and M be the set of rational numbers. Then L – M consist of the
irrational numbers thus the difference of two sets A and B can also be defined as:
A – B = {x1xEA and x ≠ B}.
Thus we conclude that set A contains A – B as a subset i.e. (A – B) CA and the sets A –B,
AnB and B –A are mutually disjoint i.e. the intersection of any two of the sets is the NULL
SET.
COMPLEMENT
Given any two sets, A and B, then we can get Ac and Bc
Example: let A {a, b, c, d} and B= {c, d, e, f}
Then,
Bc = {a,b} and Ac = {e, f}
In a Venn diagram:
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Ac is shaded
Facts about sets which follow directly from the definition of the complement of the set.
1. (a) The Union of any set and its complement A1 is the universal set i.e. AUA1 = E
(U).
(b) Set A and its complement i.e. An A1 is disjoint i.e. AnA1 = ф
2. The complement of the universal set is the null set and vice versa i.e. U1 = ф and ф =
U.
3. The complement of the complement of the set A is the set itself i.e. (A1)1 = A.
4. The difference of A and B equal to the intersection of A and complement of B ie A – B =
An B1.
We also follow directly from the definition that A – B = {x1xEA, xEA} =
{x/xEA,xEA,XEB1} = AnB1
Example:
Construct Venn diagrams to represent the following sets:
(i)
(AUB) nC1
(ii)
{(AnB)nC1} U{AnB)UC}.
2.10 Application of Sets
In a school with 94 first year studying maths, biology and chemistry. Equal number of
students were doing only two subjects. The number taking maths, biology and chemistry was
40,35 and 38 respectively. Seven students were doing maths and biology.
(i)
Draw a Venn diagram to represent the information above (3mks)
(ii)
Find the number of students doing all the courses (3mks)
(iii)
The number that was doing only maths, biology and chemistry (3 mks)
(iv)
The number doing biology and chemistry.
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Solution
Let maths (M), Bio (B), chem. (C)
n(M) = 40, n (B) = 35 n(C) =38
n(MnB) = 7
Let equal number be x doing only 2 subjects i.e. n(MnB1) = n(MnC1) =x
Maths only
40 – (7+x ) = 33-x
Biology only
35 – (7+x) = 28-x
Chemistry only
38-(7+x) = 31-x
40 +28 –x +31 - x≠ 94
99 – x = 94 = -x = 94 -99 =-5
Hence x = 5
(ii) No of students doing ALL the three courses = 2 ie 7-5 =2
(iii) Doing only maths = 28
Biology = 23
Chemistry = 26
(iv) No. of students doing Biology and Chemistry = 7. ie 5 +2 = 7
Example 2.
Given n(E) = 84
n(AnB) = 4
n(AuBuC)1 = 3
n(AnC)= n(BnC) = 7
n(A) = 30, n(B) = 40, n(C) =28.
(i)
Draw a Venn diagram to show this information (3mks)
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(ii)
Find the number of elements
n(AnBnC) (2mks)
n(AnB)nC1 (2mks)
n(A1nC1) (2mks)
n(AuB)1nC (2mks)
let n(AnBnC) =x
Hence 30 +14+x 7- x+ 29 + x3 = 84
83 +x = 84 = x = 84 -83 = 1
n(AnBnC) =1
n(AnB)nC1 = 3
n(A1nC1) = 30 + 3 = 33
n(AuB)1nC = 15
Example 3
in a café with Average of 440 customers a week, it was found that like chicken, 150 beef and
200 Githeri. It was also found that same number of customers liked both chicken Githeri, one
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third of the same number liked chicken and beef and only a sixth of those liking Githeri and
beef liked all the three foods. Find the number of customers liking
(i)
Chicken only(3 mks)
(ii)
Beef only (3mks)
(iii)
The No. of customers who liked all the three foods (3mks)
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2.11Revision questions
1. In the school of business and economics, lecturers Kamau, Kiprono, Wekesa and Munyao
have masters’ degrees, with Kamau and Munyao also having Doctorate degrees. Kamau,
Otieno, Wekesa, Nyevu, Ekeru and Okware are members of institute of certified public
accountants of Kenya (ICPAK) with Nyevu and Ekeru having masters’ degree. Identify set A
as those lecturers with masters’ degree; set B as those who are ICPAK members and set C as
doctorate holders.
a.) Specify the elements of AB and C
6mks
b.) Draw a diagram representing sets A,B and C together with their known elements
5mks
c.) What special relationship exists between set A and C?
2mks
d.) Specify the elements of the following sets and for each set, state in words what is
being conveyed?
i.) A n B
ii.) C u B
and
iii.) C n B
3mks each
e.) What would be suitable universal set for the scenario?
3mks
2. a) In a class of 17 students it was found that some were Blood A,B and O. the number
of students with Blood group A were 9. The following additional information was also
available;
n(AnBnO)=n(A n B O’)
n(B’UA’) 11
n(A’ n B’)=n(A’n O’)=n(B’ n O’)
n(AnOnB’)=2
Given also that:
AB+ I in the region (A n B n O)
O+ is in the region BnOnA’
A+ is in the region AnOnB’
Required
Draw a Venn diagram illustrating the information and find the numbers of students
who were blood group:
5mks
+
i.) AB
3mks
ii) A+
3mks
+
iii) O
3mks
b) The total number of students Registered in a department of Kileti University for
three courses A, B, C was 16,500. the lowest enrolled course had 6000 less than the
highest and 3,500 less than the second highest. How many students registered for each
of the three courses?
6mks
3. Given the following sets that n(=)ﯕ7, n(A’) =4, n(AnB)=1, n(B)=3 Find:
i.) n(A)
2mks
ii.) n(B’uA)
2mks
state whether it is correct or not to rite and why?
iii.) Aeﯕ
iv.) A’cﯕ
v.) (AnB)eA
6mks
4. A survey in a tertiary examination that was taken by 130 students revealed the number who
failed as shown in the table below. Taking E, K and H denote English, Kiswahili and History
respectively. Respond to the questions, which follow;
Subject
E
K
H
EH
KH
EH
EKH
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No of
students
who
failed
60
54
42
38
34
32
27
a.)
i.) Illustrate the information using Venn diagram
4mks
find the number of students who:
ii. Passed in all the three subjects
2mks
iii. Passed in English but failed Swahili
2mks
iv. Passed at least one subject
2mks
v. Failed at least one course
2mks
vi. Failed in two subjects
2mks
vii. Passed in History
2mks
viii. Passed English or Swahili
2mks
b.) using set notation symbolically represent the information in a.) above from question ii.) to
vii.)
5. a) Distinguish between the following terms as used in set theory:
i.)
Equivalent sets and equal sets
2mks
ii.)
Disjointed sets and sub sets
2mks
b.) The main daily newspapers in a country are: the National, The New Era and the
Citizen. The management of one of the dailies was concerned about the sales volume
of their papers. In a survey of 100 families conducted in the country, the numbers that
read the various newspapers were found to be as follows:
Name of the newspaper
The citizen
The citizen and New era
The new era
Citizen and National
The national
New era and National
All the three papers
Number of readers
28
8
30
10
42
5
3
Required
i.) Present this information in a Venn diagram
4mks
ii.) determine the number of families who did not read any of the three
newspapers
1mk
iii.) calculate the number of families that read only one of the newspapers
3mks
6. a) In a market survey by a beverage manufacturer, it was found that all the people
interviewed drank Milo or coffee. Half of the people drink Milo only, two drink both Milo
and coffee and seven drink coffee only.
i.) Illustrate this information in a Venn diagram
3mks
ii.) Determine how many people were interviewed
3mks
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b) A random sample of 400 university students found the following habits: 130 wore
sunglasses, 135 wore short trousers and 125 wore caps. If 35 wore sunglasses and
short trousers, 40 wore short trousers and caps, 45 wore caps and sunglasses and 126
did not wear any of the three items.
i.) using a Venn diagram, determine how many students wore all three items 10mks
ii.) Find out how many students wore any combination of the two items 4mks
iii.) Calculate how many students wore only one of the items
4mks
7. a) There are 54 students in Mgecon College. 30 of them take mathematics; 26 take
economics and 21 take geography. The following additional information is also provided to
you.
13 students take maths and economics
12 students takes maths and geography
11 students take geography and economics
4 students take maths and geography only
Required
i. Write the above information in a set notation
4mks
ii. Present the above information in the form of a Venn diagram 4mks
iii. How many students take all the three subjects?
2mks
iv. How many students take none of the three subjects?
2mks
v. How many of the students take two subjects only?
2mks
vi. How many students take one subject only?
2mks
b.) Given that A={t,u,v}list all the subsets of A
2mks
8. a) Using a Venn diagram, illustrate the following sets
(i) (A B) C '
(2marks)
(ii) ( A B) C'
(3marks)
( A B) C
b) In a village in Nyawara District, three mobile telephony Networks exist. It has been
established that the adult residents of the village numbering 500 all access the mobile
telephone services by use of Safaricom, Zain or Orange. The majority (300) use Safaricom,
150 uses both Safaricom and Zain only while 200 use Orange. The same number of
customers uses Zain only as do Orange only. A half of that number use both Safaricom and
Orange, while a third of that number uses Zain and Orange.
Determine the number of residents who use;
(i) Safaricom only
(ii) Orange only
(iii) Zain only
(iv) All the three networks
(v) Safaricom and Zain only
(vi) Safaricom and Orange only
(vii) Zain and Orange only
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(2marks)
(2marks)
(2marks)
(2mark)
(1marks)
(2marks)
(2marks)
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3.0 COMPUTATION SKILLS
3.1 Exponents and Logarithms
3.2Definition:
Exponents, base, matrix, characteristics, logarithms standard forms. A number written with
one digit to left of the decimal point and multiplied by 10 raised to some power is said to be
written in standard form.
5837 = 5.837 X 103
0.0415 = 4.15 X 10 -2
When a number is written in standard form the first factor is the mantissa and the second
factor is called the exponent.
Thus 5.8 X 103 has a mantissa of 5.8 and exponent of 103
2000 = 2X2X2X2X5X5X5 = 24 X53
2 and 5 are bases whereas 4 and 3 are indices.
When an index is an integer it is called a power, hence 24 is called 2 power 4
Special names may be used when the indices are 2 and 3. they are called squared and cubed
respectively.
NB: when no index is shown then the power is 1.
3.2 Law of Exponents or Indices
1. When multiplying two or more numbers have the same base the indices are add thus
am X an = a m+n
Let a = 3 32 X 34 = 3 2+4 = 36
2. When a number is divided by a number having the same base the indices are subtracted.
am ÷ an = am/an = a m-n
35 ÷ 32 = 35/ 32 = 35-2 = 33
3. When a number which is raised to a power is raised further to another power the indices
are multiplied e.g.
(am)n = amn
(35)2 = 35X2 = 310
4. A number has an index of zero (0) its value is 1
a0=1
30 =1
5. A number raised to –ve power is the reciprocal of that number raised to +ve power.
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a-n = 1/an
3-4 = 1/34
Similarly ½-3 = 23.
6.
When a number is raised to a fraction power the denominator of the fraction is root of
the number and the numerator is the power.
82/3 = ( 38)2 = 22 = 4
251/2 = ( 25) 1= ≠ 5
Similarly 27 -2/3 = 1/ (3 27)2 = 1/32 = 1/9
In general,
Am/n = nam
Example:
a3b2c4 = a2bc3
abc
x2 y3 + xy2 =
x2y3 + xy2 = xy2 +y
xy
xy
xy
x2y
= x2y
xy2 – x y
=
Quiz: simplify (Mn2)3
= x
x y(y-1)
=
(M1/2n1/4)4
y -1
M3n6
(M1/2)4(M1/4)4
= M3n6
= Mn5
M2n1
(x2y1/2) (x 3y2)
(x5y3) 3/2
3.3 Logarithms
A logarithm of a number is the power to which a base has to be raised to be equal to the
number.
Y= ax
= x = logay
Log3a = x = log3a
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3x = 9
3x = 32 = x =2
Hence log39 = 2
Log168 = x= log168 = 16x = 8
(24)x =23
4x = 3= x =3/4
Hence log168 = ¾
Example 2:
Log2y = 3
23 = y = 8
(ii) Logarithms having a base of L are called hyperbolic or napierian or natural logarithms.
Napierian logarithms of x = logex or more commonly lnx (natural log of x)
Ln 8.61 = 2.1529…
Ln 62179 =
Ln 0.149 = -9
The change of the base rule:
The change of base rule for logarithms states that:
Logay =
logby
Logba
Let t = logay = at = y
Taking the logs to base b, we get
Logbat = logby
T logba = logby
= t = logby
Logba
3.3.1 Laws Of Logarithms
1. Multiplication
Log (A X B) = log A + log B
2. Division
Log (A/B) = log A – log B
3. Power
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Log An = nlogA
Example:
Log 64 = log 128 + log 32
= 6 log 2 – 7 log 2+ 5 log 2 = 4 log 2
2x = 3 (taking log2 to base 10)
Log 2x = log3 = x log2 = log3 = x= log 3 = 0.474 = 158
Log 2
0.3010
X3.2 = 41.15 = 3.2 log x = log 41.15
= log x = log 41.15
3.2
Using logarithms, evaluate
1295 X 1.2
4.8 32
No.
Log
1295= 1.29 X 102
3.1123
1.2 = 1.2 X 100
0.0792
3.1915
48. 32 = 4.832 X 101
1.6841
1.5074 = 3.216 X 101
Example:
1.
2.873
50.49 X 0.217
2. 3 0.7214 X 20.57
69.8
3. 2.935 X 0.07652
32.74
4. Show that log t x = 1/logxt
5. Calculate 3721/3 X 0.56
457
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6. Solve for x 23x = 5x+2
7. Show that logeb logbe = 1
Represent symmetric difference
A
B, we are looking for elements that are only in A and only in B. eg A = {
a,b,c}, B= {c,d,e}, then A
A
B = {a,b,d,e}.
B is shaded
(i)
Show that log1618 = log23 (3mks)
(ii)
2loge (a-b) -2logea = log e(1 – 2b/a + b2/a2)
Solve for x in the following equations
(i)
(1/2 log316 -1/3 log527)(log34 – ½ log59) = x
(ii)
Log2x = log2e + log25
(iii)
Given that log102 = 0.3010 & log 103 = 0.4771
Find log321
A log of a number is the power/ exponent to which the base is raised to get the same number.
(i) Express these notions in 2 equivalnet expression.
(ii) Solve for x = Log10 (x2 +2x) = 0.9037
(iii) Given that X is logb T, y = logbR, and z = log1 9
Show that = logaRT = x+y
Log Rx = xy
Log + = 1/z
Show that log38 = log83
Solve t if 1nt +1n9 +3n3
Solve 3 (x+1) = 120
Evaluate logaa-1/-1
Log2 (x+4) = log2x
Solve for x if loga (x2 + 2x) = 0.9031
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Solve for t if Nt +N9 = 3 N 3
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4.0 EQUATIONS
4.1 Introduction
An equation is an expression with an equal sign. In equations, unlike in function, none of the
variables in the expression is designated as the dependent variable or the independent
variable although the variables are explicitly related.
Example:
3x + 4y = 13
-
Equations can be classified into two main groups:
1. Linear equation
2. non – linear equation
-
The two expressions below constitute examples of linear equations in the variable x.
x +13 = 15
7x + 6 = 0
-
Non –linear equations in the variable x are equations in which x appears in the second
or higher degree.
5x2 + 3x + 7
2x3 + 4x2 + 3x + 8 = 0
4.2 Solutions of Equations
To solve an equation involving a variable is to find the value or values of the variable for
which the equation holds. These values are called the roots of the equation and the set of
these values is referred to as the solution set.
Equations
An equation is a mathematical sentence/expression or an open statement containing one or
more variables. It has two sides (LHS & RHS), like a balance that they are equated by an
equal sign ‘=’ e.g. 4x + 8y = 25
Given the equation 4x + 8y = 25
i.
i x and y constitute the variables of the equations which are found by solving the
equation. They are also known as unknowns and the values to these
unknowns/variables are called solutions or roots of the equation.
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ii.
4, 8 and 25 are known as constants/parameters. They are fixed figures shown on the
left hand side of the unknown as separately.
iii.
4 and 8 are coefficients known on the lists of the unknown. They denote how many
times any specific unknown has been added.
Given this type of equation 2x2 + 8x – 20 = 0, then the 2x2 has an index power 2. it
iv.
shows how many times x have been simplified by itself.
4.2.1 Categories of equation/types of equations
i)
Linear or simple equations
ii)
Quadratic equations
iii)
Simultaneous equations
Linear equations
That which has unknown and the index of the unknown is one e.g. 4x – 10 = 0 : x is raised to
one i.e. x1 e.g.
Solve the equation:
2(4x – 2) = 3 (x +2)
8x – 4 = 3x + 6
8x – 3x = 6 +4
5x = 10
x=2
Solve the following
i) 2x = 10
5
ii) x + 5 = 12
3
2
v) 3x = x + 9
4
4
4
5
vi) 3 + 3 = 4
x
iii) x = 3x – 2
9
iv) 8 = 15
7
x
vii) x +3 – x – 1 = 1
16
4
8
Quadratic equations
These are equations formed where the highest index/exponent of an unknown is 2 e.g.
X2 + 3x + 4 = 0
The standard for of a quadratic equation is ax2 + bx + c = 0
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i)
By factorization
ii)
By formula
i) By Factorization
The part ‘bx’ is divided into two parts in such a way that b x b = a x c e.g.
Solve the equation 4x2 – x -3 = 0
Solution
4x2 – x – 3 = 0 look for two Nos. whose product would be -12 and same would be -1
4x2 – 4x + 3x – 3 = 0
4x(x – 1) + 3(x – 1) = 0
(4x +3) (x -1) = 0
Either 4x + 3 = 0 or x – 1 = 0
4x = -3 or x – 1
X= -3 and x=1
4
Check: b x b = a x c
-4 x +3 = 4x – 3
12 = 12
ii) By formula
Quadratic equations are solved using the following formula be
X = -b +
b 4ac
2a
Example:
Find the roots of the following equations.
(a) x2 + 5x – 4 = 0
(b) 5x2 – 3x = 4
Solution
X2 + 5x – 4 = 0
a = 1, b = 5, c = 4
Hence; substituting in the formulae:
X = -5 +
5 2 ( 4 x 2 x 4)
= 0.70 or 5.70
2 x1
ii) 5x2 – 3x = 4
a = 5, b= -3, c = -4
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=3+
(4 x5x 4)
2x5
X= 3 +
(9 80)
10
X = 1.24 or x = - 0.64
4.2.2 Problems leading to quadratic equations:
i) The length of a room is 4m longer than the width and the floor area is 92m2; find the length
and the breadth.
Solution
Length = (x + 4) m
Breadth = x
Floor area = Lx W = 96 i.e. x(x +4) = 96
X2 + 4x = 96 = x2 + 4x – 96 = 0
(x +12) (x – 8) = 0
Either x = -12 or x = +8
So take x = +8, since the breadth of the room cannot be negative.
ii) The sum of two digits is 10 and the sum of their squares is 58. find the digits
iii) If the average speed of a bus is reduced by 20Km/h, the time for the journey of 240Km is
by 1 hour. Find the average speed of the bus.
Simultaneous equations
These are equations whose numbers of unknown are two or more. If the numbers of the
unknown are two then the number of simultaneous equations must be 2. if the number of the
unknown are three then the number of simultaneous equations must be 3 e.g.
4x + 3y = 7
3x – 2y = 9
There are three methods of solving simultaneous equations, namely:i.
Elimination
ii.
Substitution
iii.
Graphical
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Elimination method
Solve the following equations
4x + 3y = 7
3x – 2y = 9
Here one of the unknowns has to be eliminated. We eliminate ‘Y, it would be
2x (4x + 3y = 7)
3x (3x – 2y = 9)
8x + 6y = 14
+
9x – 6y = 27
17x + 0 = 41
17x = 41
X = 41
2 7
17
and Y = 3 x 41 – 2y = 9
17
17
123 – 2y = 9
17
- 2y = 9 - 123
1
17
2y = 123 – 153
17
2y = -30
17
y = -30 2
17 1
y = -30 x 1
17
2
y = -30
34
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y = -15
17
Substitution method
Given 4x + 3y = 7 …………………………….i
3x – 2y = 9 ……………………………ii
We can take the equation ‘i’ where we express x in terms of y, hence,
4x = 7 – 3y
x = 7 – 3y
4
Then, substitute this value of x into equation ii
3(7 – 3y) – 2y = 9
4
21 – y – 2y = 9
4
- 17y = 36 – 21
y = -15
17
By the value of y into 1
4x + 3 -15
=7
17
4x – 45
=7
17
4x = 7 + 45
17
4x = 119 + 45
4x = 164
17
17
X = 164 4 = 164 x 1 = 41 = 2 1/17
17
1
17
4
12
Graphical Method
i) Solutions of Linear Simultaneous Equations
Suppose you have prior mentioned equations and you are required to find their roots over
ranges
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x+ y = 5
x- y =2
If x=0 to x if the following procedure is applied.
i.
Let x + y = 5 be labeled I and given the values of x, get the values of y.
ii.
Let x – y =2 be labeled ‘ii’ and given the values of x, get the respective values of y.
iii.
draw a Cartesian system with x values moving
iv.
Plot each of the equation in the system. Point of interaction forms the solution for
the equation.
v.
In our case above, x = 3.5: y = 1.5: These values satisfy both equations
Y simultaneously.
5
4
x–y=2
3
P(3.5, 1.5)
2
0
1
0
0 1
-1 0
2
0
3
0
X
0
4
0
-2
x +y = 5
Question 1
Graphically solve the equations
3.14x – 2.78y = 5.71
2.88x + 7.34y = 8.93
Over a range x = 0 to x = 5
Solution
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x = 2.1
i.e. (2.1, 0.4)
y = 0.45
ii) Solutions of quadratic equations
Suppose you have the quadratic equation 3x2 + 2x – 2 = 0 and you are required to find the
solution graphically.
a) You must know that 3x2 + 2x – 2 = 0, at two points where the curve cuts the straight
line y=0 which is also the axis. At these points, y=3x2 + 2x – 2 =0
b) You must know that given the equation 3x2 + 3x – 2 = y over a range say x= -2 to
x=1, the two points where the curve cuts the x – axis forms the roots of the equation,
namely; - 1.2 & 0.55.
c) Procedure
-2
X=
-1.5
3x2=
Adding
12 6.75
-1
-0.5
0
0.5
1
3
0.75
0
0.75
3
2x=
-4
-3
-2
-1
0
1
2
-2=
-2
-2
-2
-2
-2
-2
-2
Y=
6
1.75
-1
-2.25
-2
-0.25
3
Y
5
y= 3x2 + 2x – 2
(-1.2, 0.55)
4
3
2
0
1
0
-1
-0.5
0
-10
(0.55)
0.5
1
X
0
-2
-3
Question
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Solve the equation 2(x2 + 1) = 5x by graphical method over the range x=0 to x=3 i.e.
solutions line between x=0 to x=3
Answer: 0.5 and 2.0 (being the roots of the equation y=2x2 – 5x + 2 = 0
iii) Solutions of Linear and quadratic equations simultaneously.
Suppose you are given a linear and quadratic equation and you are needed to solve them
simultaneously e.g. y=2x2 – 5x + 2 and y=2x – 3 (straight line) over a range of x= 0 to x= 3
Solution procedure
i) Graph each of the equation on the same set of axes.
ii) Note their point of intersection
iii) Where the two graphs intersect give the solutions to the simultaneous equation
y=2x2 – 5x + 2 and y= 2x – 3. These points are (2.5, 2) and (1, -1)
y = 2x2 – 5x + 2
y
3
2
y= 2x - 3
(2.5, 2)
1
0
0.5
1
1.5
2
2.5
x
1
2
(1, -1)
3
NB: 0.5 and 2.0 are the roots of the equation 2x2 – 5x + 2 = 0
Suppose you have this equation
X1 + 2x2 + 3x3 = 3
2x1 +- 4x2 + 5x3 = 4
3x1 + 5x2 + 6x3 = 8
How would you find the values of X1, X2 and X3 (Hint use the substitution method)
Answer: X1 = 7, X2 = 5 and X3 = 2
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4.3 MATRICES
4.3.1 Introduction
DEFINITION: It is a rectangular array/order of numbers called elements and it is represented
by writing down the elements and enclosing them in brackets.
Thus, Matrix algebra sometimes known as Linear algebra provides us.
1. With a concise method of writing system of linear equations.
2. With techniques for determining the existence of solutions to the system.
3. With a method of determining the solutions to the system.
Example: Consider the inventory of three farmers represented by the following matrix
F1
F2
F3
2
0
1
Bags of fertilizer
120
30
75
Bags of wheat
30
11
25
Bags of corn
The matrix shows that the Farmer 1 has an inventory of: 2bags of Fertilizers: 120 bags of
wheat: and 30 bags of corn. The figures have been determined by reading down column 1,
which belongs to farmer 1.
Reading across row 2, the wheat row, we find farmer (F1) has 120 bags of wheat; farmer 2
(F2) has 30 bags of wheat and farmer 3 (F3) has 75 bags of wheat.
Thus, in matrix position and magnitude of each of the numbers in the matrix is of
considerable importance. E.g.
The column of farmer 2 and the third row, the entry is 11 bags of corn. The position of
number 11 is important because that specific location is reserved for the bags of corns
belonging to farmer two. The magnitude of the number is important since it specifies to us
the number of bags of corn belonging to farmers two.
Capital letters are used to designate a matrix and the number in the matrix referred to as
elements of the matrix are designated with small letter wit subscripts e.g.
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A=
a11
a12
a13
a14
a21
a22
a23
a24
a31
a32
a33
a34
or
A = a15
Whereby i = The row in which element ‘a’ is found
1, 2, 3
ii = The column in which element ‘a’ is found
1, 2, 3, 4
The size of the Matrix is determined by the number of rows and columns the matrix has. In
our above example, the matrix has 3 rows and 4 columns and is said to be a matrix of order 3
by 4 written 3 x 4 matrix. The number of rows and columns of a matrix also constitute the
dimensions of the matrix.
The row dimension of our above example is 3 and the column dimension is 4
4.3.2 Types of Matrices
1.
Equal matrices
Are those matrices that are identical. That is given two matrices A and B, they will be said to
be equal i.e. A=B if and only if they have the same number of rows, columns and elements in
the corresponding location e.g.
A=
1
4
7
2
5
8
3
6
9
B=
1
4
7
2
5
8
3
6
9
A=B
2. Column matrix or Column Vector
That matrix consisting of one column. That is given Matrix A; it will be a column matrix if it
has only one column e.g.
1
A=
2
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3
3. Row Matrix as row vector
That which has one row/single row. Given Matrix A, it will be a row e.g.
A= 1 2
3
4. Square matrix
That which the number of rows and columns are equal. Given matrix A, then
A=
4
3
2
2
5
3
3
1
4
Since it has 3 rows and 3 columns.
Also
2
5
3
7
Is a square matrix
5. Diagonal Matrix
That which have zeros everywhere in the matrix except in the principle diagonal. At least one
element in the principal diagonal should be non-zero. E.g. Matrices A and B are diagonal
Matrices.
A=
3
0
0
0
1
0
0
0
7
B=
9
0
0
0
0
0
0
0
0
Matrices A and b above are 3 x3 diagonal matrices.
6. Identity Matrices/Unit Matrices
It is a diagonal matrix in which elements in the main/principal diagonal is a positive one. It is
represented by the symbol ‘I’ e.g. I3 and I2 are unit matrices. Whereby A= 3 x 3 and B = 2 x 2
I3
1
0
0
0
1
0
0
0
1
I2
3x3
NG’ANG’A S. I. 15TH DEC 2009
1
0
0
1
2x2
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7. Null or Zero Matrix
That which all elements are equal to zero e.g. 03 x 2 is a 3 x 2 null or zero matrix and
03 x 3 is a 3 x 3 zero matrix e.g.
03 x 2 =
0
0
0
0
0
0
03 x 3
0
0
0
0
0
0
0
0
0
8. Transpose Matrix
That matrix A denoted by M x N that has been transformed to n x m after inter-classifying the
rows and columns. It is denoted by AT e.g.
Find the transposes of the following matrices
(i)
A=
(ii)
1
5
7
2
1
4
0
9
3
2
4
1
3
6
7
B=
b1
b2
b3
b4
(iii)
C=
x1
D=
x2
x3
Solution
AT=
1
2
0
5
4
9
7
1
3
2
1
6
4
3
7
b1
BT=
b2
B3
B4
CT=
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DT= x1 x2 x3
9. Sub-matrices
It is another matrix obtained by deleting selected row or rows and column or columns of a
given matrix say A.
Example: Consider Matrices B, B1,B2 B3 and B4
B=
b11
b12
b13
b21
b22
b23
B31
b32
b33
b11
b12
B31
b32
Hence:
B1=
B2 =
b12
B22
B32
B3 =
B4 =
b11
b12
b13
B21
b22
b23
b11
b12
b13
As such,
A
a) B1 is a sub-matrix of B obtained by deleting row 2 and column 3 of B
b) B2 is a sub-matrix of B is obtained by deleting columns/and 3 of B.
c) B3 is a sub-matrix of B obtained by deleting row 3 of B.
d) B4 is a sub-matrix of B obtained by deleting rows 2 and 3 of B.
Question
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Given that matrix A as
7
9
8
2
3
6
1
5
0
How have the following matrices A1 and A2 have been obtained given that
A1
=
2
3
6
1
5
0
and A2 =
7
9
1
5
9. Principle sub-matrix
They are sub-matrices obtained from given square matrices whose diagonals are part of the
principle diagonal of the given square matrices e.g.
Matrices A1, A2 and A3 are three examples of the principles sub-matrices of A.
A=
a11
a12
a13
a14
a21
a22
a23
a24
a31
a32
a33
a34
a41
a42
a43
a44
Principal diagonal
Denoted by elements a11 a22 a33 and a11.
Hence:
A1 =
A2 =
A3 =
a11
a12
a13
a21
a22
a23
a31
a32
a33
a11
a12
a21
a22
a33
a34
a43
a44
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Exercise
1. Given that A=B and
A=
a
2
: B= 5
0
3
b
c
0
Find the values of a, d, c, and d
2. Given that
A=
a+ b
4
3
a–b
4b –a 4
B=
3
1
Find the values of a and b if A=B
3. The products of 3 motor vehicle companies are represented as follows by the following
Company 2
Company 3
Company 1
matrix.
3
10
0
saloons
7
2
5
Pick-ups
0
1
15
trucks
6
0
13
buses
Required:
a) State the company that has no buses?
b) How many pick-ups do the companies have in total?
c) How many saloons does company 3 have?
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4.3.3 Addition and Subtraction of Matrices
Two matrices can be added or subtracted only if they have the same order i.e. 2x2 or
3 x 3 e.t.c. to add or subtract two or more matrices, the corresponding elements are
added/subtracted.
e.g. if A =
2
3
8
0
and B=
1
4
5
6
Find A + B and B – A
Solution
A+B=
2 +1
3+4
1–2
4–3
5–8
6–0
7
13
6
-1
1
-3
6
2
6
and C= 4
2
1
4
1
5
0
2
=
8+5 0+6
B–A
3
=
Question
Given the matrices
A=
Find
3
0
5
1
2
4
: B=
i) A + B
iii) B + C
ii) A +C
iv) B + B + B
4.3.4 Multiplication of matrices by a real number
There are times when matrices or elements in matrices can be multiplied by a certain number
e.g. If
A=
3
0
2
and B = 6
4
2
1
4
1
5
2
0
+½
6
4
2
5
2
0
Find: (i) 3A + ½ B (ii) 2B – 3A
Solution:
i) 3A + ½ B = 3
3
0
2
1
4
1
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=9
0
7
3
12
3
= 12
2
7
5.5 13
(ii) 2A – 3B =
Given P=
+
3
2
1
2.5
1
0
3
0
2
3
26
4
2
5
2
0
1
4
2
= 12
8
4
9
0
6
10
4
0
-
3
12
3
=3
8
-2
7
-8
-3
4
2
6
4
1
3
2
0
(ii) 2P – ½ Q
(iii) 2(P + Q)
Find (i) 3P + 2Q
and Q =
-3
4.3.5 Multiplication of Matrices
Sometimes matrices can be multiplied. Suppose A is a matrix m x n and B is p x q matrix,
then the product n=p. if n = p, the order of AB will be m x q
e.g. Given that
A=
4
1
3
2
4
6
Then AB =
and B =
2
1
3
5
0
4
4
1
3
2
1
2
4
6
3
5
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0
4
(4 x2) + (1 x 3) + (3 x 0)
(4 x 1) + (1 x 5) + (3 x 4)
(2 x2) + (4 x 3) + (6 x0)
(2 x 1) + (4 x 5) + (6 x 4)
=
11
21
16
46
Given that A=
Find: (i) AB
2
3
1
1
(ii) CB
: B=
(iii) BC
3
1
4
and C= 2
1
5
0
2
4
0
1
3
(iv) (BC) A
4.3.6 Determinants
Determinants, in matrices are only found in square matrices. Containing matrix operations are
used to obtain determinant. Give a 2 x 2 matrix.
A=
a1
b1
a1
b2
Then the Determinant of A denoted as
or /A/ or Det A is given by a1 – a2b1
Example: Find the determinant of the following 2 x 2 matrices A, B and C whereby.
A=3
5
B= 2
3
2
4
3
4
and C=
6
8
3
4
A = (3 x 4) – (2 x 5) = 12 – 10 = 2
B = (2 x 4) – (3 x 3) = 8 – 9 = -1
C = (6 x 4) – (3 x 8) =24 – 24 = 0
Matrices such as C above which have determinants being equal to zero are called simple
matrices.
Determinants for 3 x 3 matrices
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Determinants for 3 x 3 matrices, say D are obtained by having the following operation.
A1 =
a1
b1
c1
a1
b1
c1
a1
b1
a2
b2
c2
a2
b2
c2
a2
b2
a3
b3
c3
a3
b3
c3
a3
b3
Add columns 1 and 2 to the end of the matrix D or any other.
Hence
=
(a1 x b2 x c3) + (b1 x c2 x a3) + (c1 x a2 x b3)
(a3 x b2 x c1) + (b3 x c3 x a1) + (c3 x a2 x b1)
Question
Find the determinants of the following matrices.
(i) A= 2
5
(ii) B= 2
3
5
(iii) C = 1
0
0
7
9
1
0
4
0
1
0
6
1
1
0
0
1
(iv) D = 3
0
0
0
0
0
0
0
2
2
5
7
9
2
3
5
2
3
1
0
4
1
0
6
1
1
6
1
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
3
0
0
3
0
Solution
(i)
(ii)
(iii)
A=
B=
C
(iv) D
= 18 – 35 = -17
NG’ANG’A S. I. 15TH DEC 2009
= (0+72+5) - (0 +8 +3) = 66
= (1 + 0 + 0) – (0 + 0 + 0) = 1
= (0 + 0 + 0) – (0 + 0 + 0) = 0
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0
0
0
0
0
0
0
2
0
0
4.3.7 MINORS
The minors of any square matrix A are the determinants of the square sub-matrices of A.
Suppose Matrix A is given as:-
A=
a11
a12
a13
a21
a22
a23
a31
a32
a33
Then the minors of A, normally defined with reference to the elements of A can be obtained
by deleting the rows and columns in which the elements appear e.g.
The minor of element a11 denoted
as M(a11) will be the determinant of the submatrix
obtained from A by deleting the first row and column in which element a11 appears, hence the
M (a11) = /A11/ or
M(a32 ) = A32 or
A11 = a22
a23
a32
a33
A32 = a11
a13
a21
a23
Example: Find the minors of the elements a32 and a21 of the matrix A below.
A=
3
4
2
1
6
3
1
5
0
Solution
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M (a32) =
M (a21) =
A32 =
3
2
1
3
A21 = 4
2
5
0
=9–2=7
=
0 – 10 = -10
Principle Minors
These are the determinants of principle sub-matrices of any square matrix.
Suppose:
A=
a11
a12
a13
a21
a22
a23
a31
a32
a33
The principle sub-matrices of A are
A11=
a22
a23
A22 = a11
a13
a32
a33
a31
a33
and A33 =
a11
a12
a21
a22
and the corresponding principal minors are:
M (a11 ) =
M (a22) =
M (a33) =
A11 =
a22
a23
a32
a33
A22 = a11
a13
a31
a33
a11
a11
A21
a22
A33 =
= (a22 a33) – (a32 a23)
= (a11 a33) – (a31 a13)
= (a11 a22) – (a21 a12)
4.3.8 Cofactor Matrix
This is the matrix of the cofactors corresponding to the elements of a given matrix. Given that
the matrix
D=
a1
b1
c1
a2
b2
c2
a3
b3
b3
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But A1 is given by
A2 =
A3 =
a1
a2 =
a3 =
+ b1
b3
b1
- b1
c1
b3
c3
+ b1
c1
b2
c2
B2 =
B3 =
C1 =
C 3=
C3 =
b1 =
a2
c2
a3
c3
b2 =
+a1
c1
a3
c3
b3 =
- a1
c1
a2
c2
c1 =
+a2
b2
a3
b3
c2 =
- a1
b1
a3
b3
c3 =
a1
b1
a2
b2
(b2 c3) – (b3 c2)
c3
B1 =
=
= (b1 c3) – (b3 c1)
= (b1 c2) – (b2 c1)
= (a3 c3) – (a3 c2)
= (a1 c3) – (a3 c1)
= (a1 c2) – (a2 c1)
= (a2 b3) – (a3 b2)
= (a1 b3) – (a3 b1)
= (a1 c2) – (a2 b1)
Example: Find the cofactor matrices corresponding to the following matrices.
(i) A =
1
2
4
2
3
1
4
1
5
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(ii) B =
2
4
3
5
Solution
(i) The factors of the elements of matrix A are
A1 = 14
B1 = -6
C1 = -10
A2 = -6
B2 = -11
C2 = 7
A3 = - 10
B3 = 7
C3 = -1
The cofactor Matrix A is:
Cof A =
(ii) B =
=
1
2
4
2
3
1
4
1
5
2
4
3
5
a1
b1
a2
b2
=
14
-6
-10
-6
-11
7
-10
7
-1
The respective cofactors of B are:
Cof B =
A1
B1
A2
B1
Whereby
A1
= a1 = M (a1) = +5, A2 =
B1 =
b1 = M(b1) = - 3: B2 =
Hence Cof B =
a2 = M(a2 ) = -4
b2 = M (b2) = + 2
A1
B1
A2
B2
=
5
-3
-4
2
Or
Given that B= 2
4
a1
b1
3
5
a2
b2
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(i) Get the Minors corresponding to elements in matrix B hence.
M (a1) = 5
M (a2) = 4
M (b1) = 3
M (b2) = 2
(ii) Get the cofactors or the signs corresponding to the elements in the matrix B i.e. for:
M (a1) = + ve hence +5
M (a2) = - ve hence – 4
M (b1) = - ve hence – 3
M (b2) = + ve hence + 2
Thus, Cof B = 5
-4
-3
2
Cofactor expansion of determinants: It is the process of getting determinants of a matrix by
summing up the products of cofactors and the elements of a given chosen row or column used
to get the determinant.
Steps:
1. Choose a row/column of a given matrix
2. Compute the cofactors corresponding to the elements in the row or column.
3. Multiply the elements of the row or column by their appropriate cofactors
4. Add
5. the sum is the determinant of the given matrix
Examples: Consider matrix A as follows.
A=
a1
b1
c1
a2
b2
c2
A3
b3
c3
and we choose the second row.
Then the expansion of A gives the following result.
A = - a2 /A2/ + b2 /B2/ - c2 /C3/
= - a2 b1
c1
b3
c3
+b2
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a1
c1
a3
c3
- c2
a1
b1
a3
b3
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Suppose we chose the first column, the following would be the result.
A
= a1 A1 – a2 A2 + a3 A3
A= - a1
b2
c2
b3
c3
- a2
b1
c1
b3
c3
+a3
b1
c1
b2
c2
Example: Using the cofactor expansion procedure, expand the determinant of Matrix A by
the 3rd column, where;
2
1
5
+
-
+
1
3
4
-
+
-
0
2
3
+
-
+
A=5 1
3
-4
2
1
+3
2
1
0
2
0
2
1
3
A=
= 5(2 -0) – 4(4 – 0) + 3 (6 – 1)
A
= 10 – 16 + 15 = 9
4.3.9 Adjoint Matrix
It is the transpose of the cofactor matrix. The adjoint of Matrix A is
Adj A = (Cof A)T
Example: Find the adjoint of Matrix A defined as:
A=
6
3
4
3
-5
2
4
3
-3
9
17
29
21
-34
-6
26
0
-39
Solution
Cof A =
Hence Adj A = (Cof. A) T
9
17
T
29
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=
=
21
-34
-6
26
0
-39
9
21
26
17
-34
0
29
-6
-39
-
Singular matrix – a square matrix with zero determinant.
-
Non-singular matrix – a square matrix with non-zero determinant
4.3.10 Inverse of a matrix
Inverse of a matrix, say A, and hence denoted by A-1 is given by adjoint of A divided by the
determinant of A. i.e.
A-1 =
AdjA
/ A/
Provided that A is a non-singular matrix.
Example: Find the inverse of Matrix A defined as
A=
6
-2
-3
-1
8
-7
4
-3
6
27
-22
-29
21
48
10
38
45
46
27
21
38
Solution:
/A/ = 293
Cof A =
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Adj A =
A-1 =
-22
48
45
-29
10
46
AdjA
/ A/
=
=
27
293
21
293
38
293
22
293
48
293
45
293
29
293
10
293
46
293
27
21
38
-22
48
45
-29
10
46
For a 2 x 2 matrix:
1. Interchange elements in main diagonal
2. Reverse the signs of element in the other diagonal
3. divide all elements by the determinant
Hence the inverse of A is
A=
1
A-1 = adbc
a
b
c
d
a
-d
-c a
NB: A-1 A =
1
0
0
1
Find the inverse of:
A=
Check
2
2
4
5
2
2
4
5
½
5
-2
-4
2
-2
1
2.5
-1
=
1
0
-2
2
0
1
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= 2.5
-1
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B=
2
-7
1
1.5
-3 8
8
7
3
2
=
-1.6
-1.4
-0.6
-0.4
4.3.11 Solutions of Linear Simultaneous Equation by Matrix Algebra
Consider the following system of two linear equations with two variables.
a1 x1 + b1 x2 = Q1
a2 x1 + b2 x2 = Q2
Cramer’s rule can be used to get the values of X1 and X1 then the following expressions are
used.
X1 =
=
X2 =
Q1 b2 – Q2 b1
a1
b1
x1
= Q1
a1 b2 - b1 a2
a2
b2
x2
Q2
Q1
b1
Q2
b2
a1
b1
a2
b2
a1 Q2 - a1 Q1
a1 b2 – b1 a2
=
a1
Q1
a2
Q2
a1
b1
a2
b2
Suppose
1 P-1 + 2 P- 2 = 1
Are two sentences equation
1 P1 + 2 P2 = 2
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By matrix algebra, it can be transformed to
1
2
1
2
-P
1
-
=
1
1
2
1
1
2
1
2
1
1
1
2
1
2
1
2
1
2
P2
Hence
-
P1 =
-
P2
=
12
- 2 2
1 2 -
=
12
12
- 1 1
1 2 -
12
Example: Solve the following systems of linear simultaneous equations by matrix and inverse
methods
(i) 2x1 + 3x2 = 7
(ii) x1 + 2x2 + 3x3 = 3
x1 + 5x2 = 14
2x1 + 4x2 +5x3 = 4
3x1 + 5x2 + 6x3 = 8
Solutions
(i)
x1
X2
2
3
x1
1
5
x2
7
3
14
5
2
3
1
5
2
7
1
14
=
=
=
7
14
=
-7
=
-1
=
3
7
=
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(ii)
X1
X2 =
2
3
7
1
5
1
2
3
x1
2
4
5
x2
3
5
6
x3
3
2
3
1
2
3
4
4
5
2
4
4
8
5
6
3
5
8 =2
1
2
3
1
2
3
2
4
5
2
4
5
3
8
6
3
5
6
1
3
3
2
4
5
3
5
6
1
2
3
2
4
5
3
5
6
=
B
q
=
3
=
4
8
7
X3
= -5
4.3.12 Solution of simultaneous equation by inverse method
Given 3 simultaneous equation i.e.
a1
A
b1
c1
X
x1
a2 x1 + b2 x2 + c2 x3 = r
a2
b2
c2
x2
r
a3 x1 + b3 x2 + c3 x2 = s
a3
b3
c3
x3
s
a1 x1 + b1 x2 = q
a1
b1
x1
a2
b2
x2
r
X
B
a1 x1 + b1 x2 + c1 x3 = q
then
then
a2 x1 + b2 x2 = r
A
Thus, A X = B
=
q
Whereby
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Hence solution of the equations by inverse method is given by
AX = B
Matrix rearranged – get the inverse of the given matrix by A-1 and multiply it on both sides,
hence:
A-1 A X
=
A-1B
X
=
A-1 B
Example
Find the solutions of the following equations by inverse method.
(i) 2x1 + 3x2 = 7
(ii) x1 + 2x2 + 3x3 = 3
x1 + 5x2 = 14
2x1 + 4x2 +5x3 = 4
3x1 + 5x2 + 6x3 = 8
Solution
(i)
2x1 + 3x2 = 7
x1 + 5x2 = 14
Step 1. Rewritten in the form of AX = B =
2
3
x1
1
5
x2
14
X
B
A
2. Get the inverse of the matrix -1
hence.
-1
-1
2
3
2
3
x1
1
5
1
5
x1
=
2
3
1
5
x2
5
7
3
7
1
7
2
7
=
=
x2
-1
=
2
3 -1
7
1
5
14
7
7
14
7
14
-1
3
(ii) x1 + 2x2 + 3x3 = 3
2x1 + 4x2 +5x3 = 4
3x1 + 5x2 + 6x3 = 8
Rewritten in the form of AX = B
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1
2
3
x1
2
4
5
x2
3
5
6
x3
8
X
B
A
3
=
4
(ii) Get the inverse of A and multiply on both sides hence
Hence x1 =7, x2 = -5, and x3 = 2
4.3.13Revision Questions
123
1. i) Given that A= 245 find A-1
356
10mks
ii) Hence or otherwise solve the following system of simultaneous equations
x 2 y 3z 5
2x 4 y 5z 4
6mks
3x 5 y 6 z 3
3
2 1
1 2 3
2. if A= 1 3 1
B 4 5 1
C 2 3 4
2 2 5
2 3 4
Find
i.) 3(A-C)
3mks
ii.) B1A
2mks
iii.) BC-B
3mks
iv.) AC-C
3mks
v.) AC1
3mks
-1
b.) i.) Determine A , showing all the necessary workings
5mks
ii.) Hence or otherwise determine the solution to the following systems of equations
2x-y+3z=2
-x-3y+z=-11
2x-2y+5z=3
5mks
4 5 6
3. Given matrix A 8 7 1
39 2
Compute
NG’ANG’A S. I. 15TH DEC 2009
9 2
B 8 4
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i.) BtA
ii.) AB
iii.) What is the rank of matrix B
2mks
2mks
1mks
4. a)
94 6
A 17 14 2
1 16 2
b) Determine A-1, showing all necessary workers
12mks
ii) Hence or otherwise solve the following systems of simultaneous equations
9 x 4 y 6 z 16
17 x 14 y 2 z 16
x
16 y 2 z 56
5.
123
123
(a) If A = 235
B = 2 34
345
34 5
Find
(i) AT
(ii) BT
(iii) A
(1mark)
(1mark)
(2marks)
(iv) B
(2marks)
(b) Solve by row operation or otherwise the simultaneous equations
x 2z 3y 3
2 x 3z 4 y 4
3x 4 z 5 y 8
(c). The relationship between Kenyan and Australian time is linear, such that if it is 7 am in
Kenya, it is 8 p.m in Australia. When it is 4 p.m in Kenya it is 5 am in Australia.
i.) Write an equation to express Australian time in terms of Kenyan time. 3mks
ii.) What will be the time in Kenya if it is 2 p.m in Australia?
3mks
3. Solve for x, y and z using any method:
3 x 5 y 6 z 255
4 x 7 y 8 z 310
9 x 8 y 3 z 287
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5.0 GRAPHS: (DATA PRESENTATION)
Introduction
Principles of Graph constructions
Types of Graphs and their uses
Construction of the Lorenz curve
Construction of z-curves
Construction of Semi-logarithm graphs
Revision questions
5.1 Introduction
Research data analysis, is followed, where necessary by a visual display of the data either in
the form of a chart, table, graph or a diagram to facilitate communication with readers. The
following section presents the various types of data presentation, visual display methods
commonly used in research. A researcher will then choose the method of presentation that
best presents the research data.
5.2Frequency distribution
Frequency distribution presents data by dividing them into classes and recording the number
of observation in each class. The number of classes in a frequency distribution is fixed
somewhat arbitrary but there should be between five and twenty classes.
A simple rule of the thumb is that (2c n) two raised to the number of classes (c) should be
slightly more or equal to the number of observations (n). The range of values formed within
each class called class interval (C.I) should be equal in all classes in a frequency distribution.
The class interval (C.I) can be established by dividing the range (R) (Largest value – the
smallest value) by the number of desired classes (C), So that:
C.I = R ?
C
The mid point (m) of each class is calculated by dividing the sum of the lower class boundary
and the upper class boundary by 2.
Example
A researcher has obtained the following data of the number of units of a product made per
month by each of the fifty employees sampled form a manufacturing firm in Eldoret.
Form a frequency distribution.
110
42
149
165
151
175
30
79
147
122
161
62
113
184
71
157
158
69
133
94
155
156
121
104
97
108
167
93
197
150
164
124
143
195
203
128
164
140
141
162
144
146
144
40
148
178
116
187
103
113
Solution
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Step 1 - Numbers of classes necessary.
2c 50 (number of respondents)
When c – 6, 2c = 64 hence 6 would be appropriate number of classes.
Step 2 - Class intervals (C.I)
C.I = R
C
R (Range) = 203 (highest number of units) – 30 (smallest number of units) = 173
C=6
C.I = 173 29
6
Step 3:- Forming the frequency distribution
Class
30 – 59
59 – 87
88 – 116
117 – 145
146 – 174
175 – 203
Frequency
3
4
10
10
7
50
5.3 Cumulative Frequency Distribution
The cumulative frequency distribution used to determine the number of observations that are
greater than or less than cumulative frequency distribution may be constructed as shown in
example 4.2.2
Example 4.2.2
Construct less than and more than cumulative frequency distribution from the frequency
distribution formed from data in example.
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Less than cumulative frequency distribution (CFD)
Class
Less than 30
59
88
117
146
175
204
Frequency
0
3
4
10
10
16
7
(CFD)
0
3
7
17
27
43
50
More than cumulative frequency distribution (CFD)
Class
Less than 30
58
87
116
145
174
203
Frequency
50
3
4
10
10
16
7
(CFD)
50
47
43
33
23
7
0
5.4 Ogive
An ogive is a cumulative frequency distribution displayed pictorially. It could be a less than
or more than ogive. To construct an ogive the limits of the class are plotted on the horizontal
axis (abscissa) while the cumulative frequencies are plotted on the vertical axis (ordinate) of a
Cartesian ordinate. Fig 1 shows a less than and fig. 2 a more than ogive constructed from data
in example.
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Example 4.2.3
From data in example 4.2.1 and 4.2.2 construct a less than or more than ogive.
Fig 4.2.1 Less than Ogive
Graphs of Frequency Distributions:
The graphs of a frequency distribution of continuous type are as under –
(a) Ogive curve
(b) Histogram
(c) Frequency polygon
(d) Frequency curve
These are explained as under:Ogive Curve:
An Ogive is the name given to the curve obtained when the cumulative frequencies of a
distribution are graphed. It is also called cumulative frequency curve. The following steps
are adopted to construct an ogive: (i) Compute the cumulative frequency of the distribution.
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(ii) Prepare a graph with the cumulative frequency on the vertical axis and class intervals
on the horizontal axis,
(i) Plot a starting point at zero on the vertical scale and the lower class limit of the first
class.
(ii) Plot the cumulative frequencies on the graph at the upper class limits of the classes to
which they refer,
(iii) Then join all these points by the help of a curve
An ogive curve is used to find out the values of deciles and percentiles graphically
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Example 9:
From the following information, draw an ogive curve:Class
Frequency
0 – 10
5
10- 20
10
20 – 30
15
30 – 40
8
40 – 50
7
Solution:
To draw an ogive curve, the frequency is to be converted into cumulative frequency as
follows- cumulative
Class
F
c.f
0 – 10
5
5
10- 20
10
15
20 – 30
15
30
30 – 40
8
38
40 – 50
7
45
Mark cumulative frequencies (c.f.) on the graph paper, c.f of each group is marked against
Cumulative frequency
upper limit of the respective group.
50
40
30
Cumulative Frequency Curve
20
10
0
0-10
10 - 20c
20-30
30-40
40-50
Class intervals
5.5 Relative frequency distribution
A relative frequency distribution expresses the frequency within a class as percentage of the
total number of observations in the sample as shown I example 4.2.4
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Example 4.2.4
Prepare a relative frequency distribution from the frequency distribution of the factory
workers in example 4.2.4
Class
30-58
Frequency
3
Relative frequency
59-87
4
4 x 100 = 8
50
88-116
10
117-145
10
146-174
16
175-203
7
3 x 100 = 6
50
10 x 100 = 20
50
10 x 100 = 20
50
16 x 100 = 32
50
7 x 100 = 14
50
A cumulative relative frequency distribution can be generated in a similar way.
5.6 Histograms and bar charts
Histograms place the classes of a frequency distribution on the horizontal axis and the
frequencies and the frequencies on the vertical axis. The area in each rectangular bar is
proportional to the frequency in that class. Fig 4.2.4 shows the histogram of the data in
example 4.2.1
Fig. 4.2.4 Histogram of units of a product produced by factory workers.
5.7 Frequency polygon
A frequency polygon expresses the distribution of data by means of a single line determined
by the midpoints of the classes. It starts with the mid point of a class lower and ends with
midpoint of a class higher than that data given as shown.
Fig. 4.2.5 Frequency polygon of units of a product produced by a factory worker
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5.8 Graphs
A graph is any pictorial representation of data where the Cartesian co-ordinates are used. The
independent variable is shown on the x-axis and the dependent variable along the y-axis. A
graph should have a clear and comprehensive title. It should be proportional with the
horizontal and vertical scales chosen carefully so as to give the best possible appearance. The
scales should accommodate the whole data and a false baseline may be used to avoid an
unnecessarily elongated axis. The table from which the data used to plot the graph should be
given alongside the graph and on index used to show the meaning of different curves used in
a graph. If the data plotted is not original, than the source of data or information must be
shown at the base of the graph.
Example 4.2.5
A researcher has obtained data on the total scales revenue and cost of production ABC Ltd
Company shown below; plot a graph for the data.
Year
Cost of production (000) Sh
Total Revenue (000) Sh
1
40
20
2
30
30
3
35
40
4
30
45
5
25
50
6
20
55
Fig 4.2.6 ABC Ltd Total Revenue – Cost graph for the lasts six years
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5.9Pie-Charts
A pie chart presents data in the form of a circle. The slices represent absolute or relative
proportions. A pie chart is formed by making of a portion of the pie corresponding to each
characteristic being displayed.
Example 4.2.6
A researcher studying the distribution of manufacturing costs in ABC Ltd found that 20% of
the firms unit cost is due to labour, 40% raw materials, 25% maintenance costs and 15% debt
servicing. Present this information in a pie chart.
Fig 4.2.7 A pie chart representing the distribution of ABC Ltd per unit manufacturing cost
during the year.
5.8Tables
The table is the most commonly used in presenting statistical data. Tables are classified into
general-purpose tables that are used for reference purposes. Examples of general-purpose
tables are mathematical tables such as the normal distribution (Z) tables, logarithm tables and
trigonometric tables. The other classification is the special purpose tables that provide
information for particular discussion.
All tables must contain the following parts;
(i)
Title
A title describes the content of a table and should indicate: What
Data
is
included
in
the
body.
Where - Area covered in data collection.
How - Data is classified.
When - Data will apply (period)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Captions – These are headings at the top of the columns
Stub – Describes the rows.
Body – content or statistical data a table is designed to present.
Head Notes – Written above the captions and below the heading are used to
explain certain points relating to the whole table.
Foot Notes – Placed below the stubs and are used to clarify some points included in
the table that is not explained in other parts.
Source – Usually written below the footnotes and indicates where the content of the
table is obtained from if not originally collected.
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There are two types of tables:
(i)
Simple or one-way table
This type of a table shows only one characteristic against which the frequency distribution
is given.
Example 4.2.6
Table 4.2.7 frequency distribution of number of units produced per worker in ABC Ltd in
2003.
Class No of Frequency
Units Produced
30-58
3
59-87
4
88-116
10
117-145
10
146-174
16
175-203
7
(ii)
Contingency Tables
Two or more characteristics are shown in one table and indicate the number of
observations for all variables that fall jointly in each category.
Example 4.2.7
Table 4.2.8
ABC workers level of salary and education and training in 2002
Level of
Earned
Salary High
Low
Total
Level of Education and Training
High
Low
10
5
10
25
20
30
A good table should be attractive and manageable. It should make it possible or easy to make
comparisons and should be prepared according to objectives. They should be prepared
scientifically so as to be clear and easy to understand.
Generally tables should be numbered, should not be over worded, should have figures
rounded to avoid unnecessary details and should not be too narrow. All parts should be
shown clearly with columns with figures to be compared close together. Units of
measurements should be shown and all contents should be visible at a glance.
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5.10Other Diagrams
There are other forms of visual presentation of statistical data that researcher may use to light
basic facts and relationships, such as:-Scatterplots
- Line diagrams
-Two Dimensional diagrams
- Three dimensional diagrams
- Pictograms and
- Cartogram
These are illustrated in fig 4.2.9
* * *
Y
*
*
*
*
*
*
X
Scatter plot/graph
Y
X
Line diagram
X
Two dimensional diagram
Cartogram
Key
wet lands
Dry land
……….. Semi arid lands
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100
80
60
Series1
Series2
40
Series3
20
0
1st qt 2nd qt 3rd qt
4th qt
Three dimension diagrams
It would be worthwhile to be conversant with all of them so as to add them to the variety of
choice when deciding on how to present research data.
It cannot be over- emphasized that visual display of research data breaks monotony, attracts
and captures readers’ attention and adds quality to the presentation of research data.
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5.11 SPECIAL TYPES OF GRAPHS
The following are the important types of graphs;1. Time series graphs or histogram (This is discussed on its own under topic 7).
2. Z – charts
3. Scattergraphs
4. Semi-logarithmic graphs or ration scale graphs
5. Lorenz curve
6. Graphs of frequency distribution
These graphs are explained as under
5.11.1 Z Charts
A Z chart is simply a time series chart incorporating three curves for
(i) Individual monthly figures.
(ii) Monthly cumulative figures for the year
(iii)A moving annual total.
Z chart takes its name from the fact that the three curves together tend to look like the letter
Z.
A Z chart is of great importance for presenting business data over a period of one year. The
information given in a Z chart can be explained under.
(i) Monthly totals – These simply show the monthly results at a glance together with any
rising or falling trends and seasonal variations.
(ii) Cumulative totals – These show the performance to date and can be easily compared
with planned or budgeted performance.
(iii)Annual moving totals – these show comparison of the current levels of performance
with those of the previous year If the line is rising then this year's monthly results
are better than the results of the corresponding month last year and vice versa.
Sometimes, separate vertical scales are used to plot the monthly data and the data for the
cumulative and the moving annual totals In some cases, the same vertical scale is used to plot
the monthly data and the data for the cumulative and the moving annual totals The decision to
take same vertical scale or separate vertical scales should be made in view of the nature of the
given data.
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Example 5:
The following are the sales of ABC Ltd for the years 1995 and 1996
1995
1996
January
400
420
February
480
450
March
420
600
April
580
640
May
600
580
June
800
700
July
750
800
August
600
750
September
550
600
October
500
480
November
600
550
December
900
950
(Source)
Construct a Z chart for the year 1996.
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Solution
Z Chart of sales 1996
Monthly
Moving
cumulative annual
1995
1996
for 1996
Total
January
400
420
420
7200
February
480
450
870
7170
March
420
600
1470
7350
April
580
640
2110
7410
May
600
580
2690
7390
June
800
700
3390
7290
July
750
800
4190
7340
August
600
750
4940
7490
September
550
600
5540
7540
October
500
480
6020
7520
November
600
550
6570
7470
December
900
950
7520
7520
7180
7520
Monthly cumulative totals are obtained as under;February = 420 + 450 = 870
March = 870 + 600 = 1470
April = 1470 + 640 = 2110 and so on
Moving annual totals are obtained as under:January = 7180 + 420 – 400 = 7200
February = 7200 + 450 – 480 = 7170
March = 7170 + 600 – 420 = 7350 and so on.
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It can be observed that the moving annual totals ran be easily obtained by adding the current
month's figure and subtracting the corresponding last year's figure to and from the preceding
month's annual total In this example, the total sales of 1995 are 7180. In order to obtajin the
moving annual total at the end of January 1996, add January 1996's sales into 7180 and
subtract from it. the sales of January 1995.
8000
7000
6000
5000
4000
3000
2000
1000
0
Monthly
Figures
Monthly
cumulative
figures
Moving
Annual
Totals
Ju
l
A
ug
Se
pt
O
ct
N
ov
D
ec
Ja
n
Fe
b
M
ar
A
pr
M
ay
Ju
n
1996 (Months)
The chart is constructed below -Y
Sales
5.11.2 Scatter Graphs
Scatter graphs are those graphs which are used to indicate the relationship between two
variables. The X-axis is used to represent the data of one variable and the Y-axis to represent
the data of other variable.
In order to construct a scatter graph or scatter diagram, we must have several pairs of two
variables. Each pair of these variables shows the value of one variable and the corresponding
value of the other variable. Each pair of data is plotted on a graph. The resulting graph will
show a number of plotted pairs of data scattered over the graph.
Scartergraphs are usually drawn to indicate the relationship between two variables. For this
purpose, a line of best fit is established from the scatter graph.
The line of best fit is that line from which the total deviation of the points plotted on a scatter
diagram is minimum. The line of best fit indicates the relation or association between two
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variables. It is one way of measuring correlation. In a scatter graph, the line of best fit is
drawn approximately. This line may have a rising or felling trend which shows positive and
negative relationship between two variables respectively.
Example 6: Sales and advertising expenditure of RST Ltd are given below for a period of
seven months.
Advertising 20
expenditure
(Sh 000’s)
Sales (Sh
650
000’s)
25
30
35
40
45
50
550
700
800
750
900
850
Draw a scatter graph
Solution
Sales (Sh 000's)
1000
800
600
Sales
400
200
0
0
10
20
30
40
50
60
Advertising Expenditure (Sh 000's)
in this example, the advertising expenditure is taken along – axis because it is independent
variable and sales are taken along Y – axis as these are dependent variable.
It can be observed from the graph that the plotted data, although scattered represent the rising
trend. It means the increase in advertising expenditure results in higher sales. This trend
shows there is a positive relationship between these two variables.
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5.11.3 Semi - logarithmic graphs:
A semi-logarithmic graphs is that graph on which the vertical scale is logarithmic. It is also
known as ratio scale graph. These graphs are useful to study the relative movements instead
of absolute movements.
Semi-logarithmic graphs are generally used when:1. Visual comparisons are to be made between series of greatly different magnitudes.
2. The series are quoted in non-comparable units.
3. The data are to be examined to see whether they are characterised by a constant rate
of change. A constant rate of change appears as straight line.
Ratio scale or semi-log graphs can be constructed in three ways:1. By using semi-log graphpaper
2. By using a slide rule
3. By plotting the logs of the variables.
Actual values can also be shown on the vertical scale. Zero has no log and Zero' should not
be inserted on the vertical scale of a semi-log graph.
In semi-log graphs, the horizontal scale is the same as on ordinary graph whereas the vertical
scale is the ratio scale or logarithmic values of the variable.
If the logarithmic curve is moving upward, it indicates that the rate of growth is increasing
and vice versa. If such a curve is a straight line, it means the rate of growth is constant.
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Example 7:
The following are the profits of Pombe Breweries Ltd over the calendar year 1996.
Month
Profits in '000' of Shillings
January
February
March
April
May
June
July
September
August
October
November
December
10
11
13
15
15
18
16
19
20
17
18
24
Using the ordinary graph paper, plot the time series for the profits using the
logarithmic values or ratio scale
Month
Profits (Sh. 000’s)
Logy
January
10
4.0
February
11
4.0
March
13
4.1
April
15
4.2
May
15
4.2
June
18
4.3
July
16
4.3
September
19
4.3
August
20
4.3
October
17
4.2
November
18
4.3
December
24
4.4
Note: Profit for January is Shs. 10,000 so the characteristics are 4 and so on.
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4.5
4.4
4.3
4.2
4.1
4
3.9
3.8
Sem- - log graph
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
1996
In a semi-logarithmic graph, one axis has a logarithmic scale and the other axis has a linear
scale.
Example 1: Variable Exponent
Plot the graph of y = 5x on normal and then semi-logarithmic paper.
Answer:
We first graph y = 5x using ordinary x- and y- linear scales (the space between each unit
remains fixed for both axes):
We see that the detail for anything less than x = 2 is lost.
Using a semi-logarithmic scale on the y axis gives:
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We can now see much more detail in the y values when x < 2.
Notice that the numbers along the x axis are evenly spaced, while along the y-axis, we have
powers of 10 evenly spaced.
Example 2: Variable Raised to a Fractional Exponent
Let's now graph y = x1/2 using all 3 axis types. This function is equivalent to y = √x.
Using rectangular axes, we can see that the graph of y = x1/2 is half of a parabola on its side
(i.e. its axis is vertical):
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We have seen this curve before, in The Parabola section.
Note 1: The detail near (0, 0) is not so good using a rectangular grid.
Note 2: The curve passes through (0, 0), (1, 1), (4, 2) and (9, 3). In each case, the y-value is
the square root of the x-value, which is to be expected.
Let's see the curve using a semi-logarithmic plot.
Now we have a lot better detail for small x. The lowest value of y that the graph indicates is y
= 0.1. We cannot show y = 0, since the logarithm of 0 is not defined.
We can see that the curve still passes through (1, 1), (4, 2) and (9, 3).
Application 1: Air pressure
1. By pumping, the air pressure in a tank is reduced by 18% each second. So the percentage
of air pressure remaining is given by p = 100(0.82)t.
Plot p against t for 0 < t < 30 s on
(a) A rectangular co-ordinate system
(b) A semi-logarithmic system.
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Try it on paper first, and then see what you get using the LiveMath example above.
The answer is given below.
Answer:
(a) Rectangular plot:
(b) Semi-logarithmic Plot:
5.11.4 LORENZ CURVE
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This is a graph to measure dispersion. It was devised by Dr. Lorenz to measure inequalities of
wealth distribution. So an important use of the Loren curve is in the measurement of the
extent to which income is unevenly distributed between the various income groups. The
disparity of proportions is a common economic phenomenon. This disparity can be
demonstrated by the help of Loren curve
A Lorenz curve is constructed as follows:1. Write down the values of the two variables being plotted
2. Express the variables as percentages of the total.
3. Compute the cumulative percentages of each variable.
4. Draw a horizontal and vertical axis and plot 0% to 100°o on .-.-u ii axis
5. Mark the cumulative percentages on the graph and join the points together by a free
hand curve. This is Lorenz curve.
6. Draw the line of equal distribution by joining 0% to the 1000 point by a straight line.
If the Lorenz curve is away from the line of equal distribution, there is greater disparity or
inequality and vice versa.
Example 8:
The following figures are taken from surrey on "Business Prospects" for 1996
Maize Flour Sales
Number of Establishment s
Net output(£'000')
23
104
26
450
24
860
19
1350
14
2190
6
3125
Draw a Lorenz Curve using the above data.
Solution:
Maize Flour Sales
Net output (£ ‘000’)
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Number of
%
Cumulative
%
Cumulative
23
20.5
20.5
104
1.3
1.3
26
23.2
43.7
450
5.6
6.9
24
21.4
65.1
860
10.6
17.5
19
17.0
82.1
1350
16.7
34.2
14
12.5
94.6
2190
27.1
61.3
6
5.4
100.0
2125
38.7
100.0
112
100.0
8079
100.0
establishments
Y
100
80
Line of Equal Distribution
60
Lorenz Curve
40
20
0
0
20
40
60
80
100
X
Number of establishment
This curve shows the greater disparity between the numbers of establishments and the net
output 20.5% establishments have only 1.3% net output and 5.4% establishments have 38.7%
share of net output.
The Lorenz curve is a graphical device used to demonstrate the equity of distribution of a
given variable such as income, asset ownership or wealth. For example, one might be
interested in the equity of cattle ownership since this is often taken as an indicator of the
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distribution of wealth, particularly for pastoral and agropastoral societies. The distribution of
cattle ownership is, however, often extremely difficult to determine so that the cattle holding
per household (or per holder) is often used as the proxy measure of wealth in the derivation
of a Lorenz curve. Holding implies the right and responsibility to manage on a day-to-day
basis but not necessarily to dispose of (e.g. by slaughter, sale or gift).
A graph for showing the concentration of ownership of economic quantities such as wealth
and income; it is formed by plotting the cumulative distribution of the amount of the variable
concerned against the cumulative frequency distribution of the individuals possessing the
amount.
A cumulative frequency curve showing the distribution of a variable such as population
against an independent variable such as income or area settled. If the distribution of the
dependent variable is equal, the plot will show as a straight, 45° line. Unequal distributions
will yield a curve. The gap between this curve and the 45° line is the inequality gap. Such a
gap exists everywhere, although the degree of inequality varies
In the following example, a Lorenz curve for cattle holdings to households holding cattle is
therefore constructed. The principles outlined in the derivation of the curve can be applied to
any data set in which the equity of distribution for a given variable is being calculated.
Derivation of the Lorenz curve
In the derivation of the example Lorenz curve, the following procedure has been adopted:
All individual units (households) are ranked from the lowest to the highest according
to the number of cattle held (Column 1, Table 2.A1) and the number of households in
each cattle-holding category is given (Column 2).
From this data, the percentage of households falling into each cattle-holder category is
derived (Column 3).
The cumulative percentage of households in each cattle-holder category is then
estimated (Column 4).
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By multiplying the number of cattle in each category by the number of households
holding those cattle (Column 1 x Column 2), we then obtain the total number of cattle
held within each category (Column 5).
From this latter figure, the percentage of total cattle held (Column 6) and the
cumulative per cent of cattle held in each category (Column 7) are obtained.
The cumulative percentage of cattle held in each category (vertical axis) is then
plotted against the cumulative percentage of households for each category (horizontal
axis) to derive the Lorenz curve (Figure 2A.1).
This plotted curve is then compared with the line of perfect equity (drawn at 45° from
the origin of the graph) to provide an indication of the equity of distribution of cattle
holdings within the area concerned.
5.12Revision questions
1. A hypothetical research on the average width of maize leaf in mm against the amount of
calcium potassium nitrate fertilizer applied in grammes yielded the following results.
Width of maize leaves (in mm)
Fertilizer in grams
20.0
19
20.5
16
21.0
18
21.5
17
22.0
18
22.5
19
23.0
14
23.5
12
24.0
11
24.5
11
Required
i.) using semi average method constructs a graph for the information above
6mks
ii.) using the graph in i.) Estimate the size of maize leaf when you apply 8 grams of the
fertilizer.
iii.) Assume the graph depicts a true situation, what will be your comment to the use of this
fertilizer with respect to the size of maize leaf?
2. The following represents the earnings period in shillings of 50 casual workers of a certain
company.
211
215
230
234
261
270
291
294
244
239
286
275
266
268
221
216
259
232
212
211
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290
265
218
246
268
276
246
229
250
254
272
273
280
261
219
238
225
240
257
231
263
241
274
271
270
267
447
248
254
257
a.) using the size of 8 shilling and beginning with 211-218 class
i.) form a frequency distribution
4mks
ii.) Construct a histogram and use it to estimate the modal earnings peer day 6mks
iii.) graphically and not otherwise determine the median earnings per day
8mks
3. The annual DAP fertilizer consumption in thousands of tonnes during 1995-2001 in
Lukuyani Division was recorded as given below.
Year
1995
1996
1997
1998
1999
2000
2001
Consumption 50
56
60
68
70
75
78
(‘000) tonnes
a.) i.) Use the semi average method to fit the trend line and use it to estimate the consumption
in 2005.
12mks
ii.) Indicate two major disadvantages of this method
4mks
b.) Construct semi logarithmic graph for the consumption of the Lukuyani and use it to
[comment on the rate of consumption.
8mks
4. The table shows the number of workers employed in two institutions REK and PEDI
respectively with regards to salaries paid to them in the year 2005.
REK
PEDI
Number of workers Salaries and
Number of workers Salaries and
allowances
allowances
60
2800
80
4200
70
4900
50
5100
55
6400
45
6600
50
7700
35
8800
40
8400
28
10600
20
6800
23
12300
10
5000
16
10000
5
4000
4
5000
i. Construct on the same graph Lorenz curve for the two institutions, round off your figure
to nearest whole number
10mks
ii. Using Lorenz curve estimate
iii. The production of salaries and allowances paid to the first 40% of workers in REK and
PEDI
4mks
iv. The proportion of salaries and allowances paid to the last 10% of the top cream workers
in the two institutions.
Join the origin ad the end point of Lorenz curves with a straight line, explain the importance
of the line with regards to income of workers with a view to pointing out the company with
better income distribution
5. a.) i.) Describe the advantages of using a graph as a means of data presentation 3mks
ii.) Distinguish between frequency distribution and frequency polygon
2mks
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iii.) Identify the differences between measures of central tendency and measure of variability
4mks
b.) The following information about the salaries of employees was obtained from a private
company in agricultural sector in Kitale.
Salary Per Month X (Ksh)
Frequency (f)
0-4000
9
4000-8000
36
8000-12000
91
12000-16000
147
16000-20000
87
20000-24000
22
24000-28000
8
Required
i. Draw on the same axes a “less than” and an “or more” ogives
6mks
ii. Using the ogives find the median salary
2mks
iii. Calculate the most frequently occurring salary
2mks
iv. Determine the mean and standard deviation of salary at the firm and describe the
distribution
5mks
6. In estimating the value of a plantation of cedar trees, the diameters of trees in a sample area
of 100 trees were measured in centimeters and recorded as follows;
14
5
7
8
5
18
4
15
8
14
9
8
11
14
9
14
18
15
13
16
19
11
19
12
11
9
14
17
7
15
13
17
14
18
16
12
5
11
15
19
10
7
16
6
16
8
18
9
17
10
14
8
6
19
13
16
16
15
10
11
7
6
19
16
9
9
8
17
13
9
10
12
14
4
14
7
14
18
5
10
7
11
18
9
11
10
15
13
18
17
12
13
17
19
16
6
4
15
18
13
Required
a.) Construct a histograph for the distribution and use it to estimate the modal size of the
diameter of the trees in the sample
6mks
b.) Using the class size width of 3cm, limit inclusive, form a frequency distribution table
for the diameter of cedar trees starting from 4cm.
6mks
c.) Form a cumulative frequency distribution of the diameter of the trees and construct a
more than ogive and use it to obtain the median
7mks
d.) Determine the quartile deviation of the distribution
5mks
7. The table below gives the production figures (in 000 of tonnes) of ceramic goods for 2006.
Month
Jan Feb Mar Apr May Jun July Aug Sep Oct Nov Dec
Production 335 325 310 354 360 338 333 270 375 395 415 373
i.) Plot the monthly production figures on a graph
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ii.) Which time series factor seems to influence the production of ceramic
goods?
3mks
iii.) Use the graph to estimate the production figures for the ceramic good in
February 2007.
4mks
8. In order to observe patterns and trends, data are often presented in the form of charts.
Discuss the type of chart that could be used in each case when it is relevant to use each type.
(20 marks)
9. State the points to be considered in the presentation of research data. (4 marks)
Explain with the aid of an example when and where each of the following may be used in the
presentation of research data.
Pie chart
(4 marks)
Ogive
(4 marks)
Bar chart
(4 marks)
Scatter diagrams
(4 marks)
10. The table below shows the frequency distribution of daily income earned in 1991 by a
sample of 50 workers of ABC Construction Company.
INCOMES
KSHS
50-54
55-59
60-64
65-69
70-74
75-79
80 - 84
85 – 89
90 – 94
NUMBER OF WORKERS
2
3
5
10
12
8
6
3
1
REQUIRED
Using the graph paper to:
i Construct a histogram and frequency polygon.
( 8 marks)
ii Construct a cumulative relative frequency polygon (8 marks)
Graphically determine;
iii The sample median
(2 marks)
iv The sample first quartile
(2 marks)
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6 FREQUENCY DISTRIBUTION: (12 HOURS)
Methods of Data collection,
Frequency Tables, Polygons and curves
Measures of Central Tendency
- Mode, mean and median (mention others too)
Measures of Dispersion
Range, Standard Deviation, Quartile Deviation, Variance.
Bivariate Data
6.1Sampling and sampling design
6.1.1 Sampling
Sampling is taking any portion or universe as represented of that population or universe.
Sample: just a part of the population selected according to some rule or plan.
Population: The totality of all possible values (measures, counts, or respondents) of a
particular characteristic for a specified group of objects.
Sampling means selecting a given number of subjects from a defined as representative of that
population or taking any portion of the universe as representative of that population or
universe. One type of population distinguished by researchers is called the target population
or universe- this means that all members of a real or hypothetical set of people, events or
objects to which results of a research are generalized.
Sample: a sample is a small proportion of a population selected for observation and analysis,
by observing the characteristics of a sample, one can make certain inferences about the
characteristics of the population from which it is drawn, and samples are chosen in a
systematic random way, so that chances or the operations of probabilities can be utilized.
Therefore a sample is a part of population selected according to some rule of plan.
The section of such a sample and collection of data from it would involve a tremendous
amount of work and expense. Instead a researcher must visually draw his sample from an
experimentally acceptable population such as all form three students in district schools. If the
researcher can demonstrate that the accessible population is closely comparable to the target
population of a few variables that appear most relevant he or she has done much to establish
population is reasonably representative of the target population.
Criteria of population validity
The criteria used to evaluate a sample of 460 articles in the field of marketing
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1. A Clear descriptions of the population to which the results are to be
generalized are given.
2. The sampling procedure should be specified in enough details so as another
investigator would be able to replicate the procedure. This should include at a
minimum (a) the type of sample (b) sample size and (c) geographical area. In
most educational studies other descriptive data, such as sex, age, grade level
and social economic status should also be included.
3. The sampling frame. That is, the list, indexes, or other population records from
which the sample was selected should be identified.
4. The completion rate – this is the proposition of the sample that participated as
intended in all the research procedures should be given.
Random does not mean haphazard section. What it does mean is that each member of the
population has some calculable chance of being selected- not always an equal chance. It also
means the converse that there is no identified population who could not be selected when the
sample is set up.
Why sample?
Reasons for sampling include;
a.
The population may be to large for complete enumeration.
b.
The enumeration or measurement process may be destructive
c.
Sample saves time and money
d.
Sampling allows more time to be spent on training, testing and checking.
N.B. the larger the sample the larger the potential level of confidence.
Sampling error
Errors due to inherent characteristics of the sampling procedure itself. Marked by the
difference between the sample estimates and the population parameters under study. Most
notable sampling errors are bias- the intuitional or systematic over or under – representation
of the qualities of interest.
Non- sampling error
Occur whether or not complete enumeration or sample remunerations is adopted. They arise
from failure to measure a certain phenomena, faulty questionnaire or ignorance.
N.B. sampling errors decrease with increase in sample size while error increase with increase
in sample size.
Reducing errors
To reduce errors, all of the following strategies must be adopted;
i.
Proper demarcations and identification of variables.
ii.
Suitable instruments.
iii.
Clear definition of concepts
iv.
Pre-testing of instruments
v.
Use of expert enumeration
vi.
Close supervision of enumerators
Methods of accessing and controlling non- sampling errors
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i.
ii.
iii.
iv.
v.
vi.
vii.
Check ups of the instruments
Inter- penetrating samples
Post census or post sample survey
Tracing techniques
Quality controls or instant checks
Study or recall lapse
Treatment of non- response cases.
Steps in sampling design
While developing a sampling design, the researcher must pay attention to the following
points:
a.
Types of universe: the first step in developing any sample is to clearly define
the set objective technically called the universe, to be studied. The universe can
be finite or infinite. In finite universe the number of items is certain, but in the
case of an infinite universe the number of items is infinite.
b.
Sampling units: a decision has to be taken concerning a sampling unit before
selecting samples. Sampling units may be a geographical one such as state,
district and village.
c.
Source list: it is also known as ‘sampling frame’ from which sample is to be
drawn. It contains the names of all items of a universe (in case of finite
universe only). If source list is not available, researcher has to prepare it. it is
extremely important for the source list to be as representative population as
possible.
d.
Size of sample: this refers to the number of items to be selected from the
universe to constitute a sample. This is a major problem before a researcher.
The size of sample should neither be excessively large, nor too small. It should
be optimal.
e.
Parameter of interest: in determining sample decisions, one must consider
the question of specific population parameter, which are of interest. for
instance we may be interested on estimating the proportion of person with
some characteristic in a population, or we may be interested in knowing some
average or the other measure concerning the population.
f.
Budgetary constraints: Cost considerations, from practical point of view,
have major impact upon decisions relating to not only the size of the sample
but also to the type of sample. This fact can even lead to the use of nonprobability sample.
g.
Sampling procedures: finally the researcher must decide about the technique
to be used in selecting the items for the sample.
Criteria for selecting a sampling procedure
in this context one must remember that two costs are involved in a sampling analysis viz.,
the cost of collecting the data and the cost of an incorrect inference resulting from the data
researcher must keep in view two causes of incorrect inferences viz., systematic bias and
sampling errors a systematic bias result from errors in sampling procedures, and it cannot be
reduced or eliminated by increasing the sample size. At best the causes responsible for these
errors can be detected and corrected. Usually a systematic bias is the result of one or more of
the following factors.
a.
Inappropriate sampling frame: if the sampling frame is inappropriate i.e.
bias representation of the universe, it will result in a systematic bias.
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b.
Defective measuring device: if the measuring device is constantly in error, it
will result in systematic bias. In survey work systemic bias can result if the
questionnaire or the interviewer is biased. Similarly, if the physical measuring
device is defective there will be systematic bias in the data collected through
such measuring device.
c.
Non respondent: if we are unable to sample all the individuals initially
included in the sample, there may raise a systematic bias.
d.
Indeterminacy principal: some times we find that individuals act differently
when kept under observation than what they do when they are kept in nonobserved situations.
Natural bias in the reporting of data: this is often the cause of a systematic
bias in many inquiries. There is usually a downward bias of data collected by
government taxation department. Whereas, we find an upward bias in the
income data collected by social organizations. People generally understate
there income if asked about it for tax purposes, but they overstate the same if
asked for social status or affluence.
Types of samples
There two types:
1.
probability sample
2.
non- probability (purposive) sample
Probability samples
In probability sampling, each element of the large population has a known probability of
being selected. There are several ways of drawing probability samples, as follows;(NB: each
element has an equal chance of being selected)
Simple random sample: the individual observation or individuals are chosen in such
a way that each has an equal chance of being selected and each choice is independent on any
other choice. If we wished to draw a sample of 50 individuals from a population of 600
names in a container and, blind folded draw one name at a time until the sample of 50 was
selected. This procedure is cumbersome and rarely used.
Random numbers: a more convenient way of selecting a random sample or
assigning individuals to experimental and control groups so that they are equated by use of a
table of random numbers as shown below
Typical Table of Random Numbers
2
3
4
Row 1
1
32388 52390 16815 69298
2
05300 22164 24369 54224
3
66523 44133 00697 35552
4
44167 64486 64758 75366
5
47914 05284 37680 20801
6
63445 17361 62825 39908
7
89917 15665 52872 73823
8
92648 45454 09552 88815
9
20979 04508 64535 31355
10
81959 65642 74240 56306
5
82732
35983
35970
76554
72152
05607
73144
16533
86064
00033
6
38480
19687
19124
31606
39339
91284
88662
51125
29472
67107
7
73817
11052
63318
12614
34806
68833
88970
79375
47689
77510
8
32523
91491
29686
33072
08930
25570
74492
97596
05974
70625
9
41961
60383
03387
60332
85001
38818
51805
16296
52468
28725
10
44437
19746
59846
92325
87820
46920
99378
66092
16834
34191
The use of random number tables, the researcher randomly selects a row or a column. If more
numbers are needed he proceeds to the next row or column until enough numbers have been
selected to make up the desired sample size. In effect the research may start at any random
pointing the table and select numbers from a column or row as she wishes.
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Systematic sampling: it is the way of selecting every nth item on the list. An element of
randomness is introduced into this kind of sampling by using random numbers to pick up the
nth item from which to start. For instance, if a four- percent sample is desired, the first item
would be selected randomly from twenty- five and thereafter every twenty fifth item would
be automatically be included in the sample. Thus in this sampling only the first unit is
selected randomly and the remaining units of the sample are selected at fixed intervals.
Stratified sampling: under this the population is divided into several sub-group populations
that are individually more homogeneous than the total population (the different subpopulations are called ‘strata’) and then we select items from each stratum to constitute a
sample. Since each stratum is more homogeneous than the total population, we are able to get
more precise estimates for each stratum and by estimating more accurately each of the
component part we get a better estimate of the whole in brief stratified sampling results in
more reliable and detailed information. The following question should be addressed in using
stratified sampling:
How should the strata be formed? (for example, it can be formed from the common
characteristics of the items to be put in each stratum)
How should items be selected from each stratum? (We can use simple random sampling or
systematic sampling can be used in certain situations)
How many items should be selected from each stratum or what is the sample size to each
stratum? (The method of proportional allocation under which the sizes of the sample from
different stratum are kept proportional to the size of the stratum is used)
Cluster sampling: if the total area of interest happens to be a big one, a convenient way in
which a sample can be taken is to divide the area into a number of smaller non overlapping
areas. Then, to randomly select a number of these randomly selected areas (usually called
clusters), with the ultimate sample consisting of all (or sample of) units in this smaller areas
or clusters. The respondents have heterogeneous characteristics in each cluster.
Area sampling: if clusters happen to be more geographical sub-divisions, cluster sampling is
better known as area sampling. In other words, cluster design where the primary sampling
unit represents a cluster of units based on geographical area sampling.
Multi- stage sampling: it is a further development of the principal of cluster sampling.
Supposed we want to find out the performance of the English subject in district school. The
first stage is to select large primary sampling unit – a district. Then we may select certain
divisions then interview all selected subjects in the division. This would represent a two stage
sampling design with the ultimate sampling unit being clusters of divisions.
Sampling with probability proportional to size: incase the cluster sampling unit do not
have the same number or approximately the same number of elements, it is considered
appropriate to use a random selection process. The probability of each item in the cluster
being included in the sample is proportional to the size of the cluster.
Sequential sampling: this sampling design is a somewhat complex sample design. The
ultimate size of the sample under this technique is not fixed in advance, but is determined
according to mathematical decision rules on the basis of information yielded as survey
progresses. This is usually adopted in the cases of acceptance sampling plan in context of
statistical quality control. When a particular lot is to be accepted or rejected on the basis of a
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single sample, it is known as single sampling. When the decision is to be taken on the basis of
two samples, it’s known as double sampling, and in the case where decision rests on the basis
of more than two samples but the number of sample is certain and decided in advance, the
sampling is known as multi-sampling. But when the number of sampling is more than two but
is neither certain nor decided in advance, this type of system is often referred to as sequential
sampling.
Non-probability (purposive sampling)
The common feature of getting a non-probability sample is not based on the probability with
which a unit can enter the sample, BUT, by other considerations such as common sense,
experience, intuition, and expertise. They have limitations of being biased, unconscious
errors of judgment, personal likes and dislikes, the attitude of the person sampling and so on.
There is no objective way of assessing the magnitude of these errors.
Non-probability or purposive sampling methods include;
i.
ii.
iii.
iv.
v.
Representative sample
Sample selected in general, represents a characteristic variable and may not represent
to other variables.
Judgment sample
The researcher after considering all the units of the population makes a judgment
selection of some units to form his sample
Accidental sample
Researcher selects any case he comes across. Method used to sample/survey quickly
public opinion.
Voluntary sample
Respondents volunteer to participate in a sample
Quota sampling
Kind of stratified judgment sampling. Samples of prefixed size are taken from each
stratum using judgment sampling techniques. Each enumerator fills his quota in each
stratum by taking advantage of any information that enables him it cover his quota
quickly and cheaply.
NB. It is not possible to know whether the sample is representative or not.
Sample size determination
Determined by:
i) resources
ii) Requirements of the proposed plan of analysis.
The sample size must be large enough to:i Allow for reliable analysis of cross- tabulations
ii Provide for desired level of accuracy in estimates of the larger population
iii.Test for significance of differences between estimators.
Minimum sample size
M = 50
when
M = minimum sample size
Ps
Ps = proportion of total cases expected in the smallest category of the
variable.
According to Krejcie (1990), reported by Michael et el (1971) the sample size is determined
by;
S=
x2 NP(1-P)
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D2(N-1) +x2 P (1-P)
Where
S
=desired sample size
N
=population
P
=population proportion (take 0.5)
D
=degree of accuracy reflected by the amount of error that can be tolerated in
fluctuation of a sample proportion (p) about the population. Take D = 0.05 equal to 1.96 6p at
95% confidence level.
6ᵨ = standard error of the proportion
.x = table chi square for one degree of freedom relative to the desired level of
confidence. (x = 3.841 for 95% confidence level.)
Substituting the constants in the relationship above.
.s
=
0.96025N
0.0025(N-1) + 0.96025
When
N = 318, the sample size is;
=
0.96025N
0.0025(371) + 0.96025
.s
=
305.3595
1.75275
=
174
NB. A sample size of 174 represents a proportion of 54.72% of the population, which is too
high and costly to survey.
According to Nassiuma (2000), the sample size can be determined by;
S
=
.N(cv2)
Cv2 + (N-1) e2
Where S = sample size
N = population
Cv = coefficient of variation (take 0.5)
.e = tolerance of desired level of confidence (take 0.05) at 95% confidence level)
Substituting the constants;
S
=
=
0.25N
0.25 + 0.7925
76
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A sample size of 76 would represent a proportion of 23.9% of the population.
Useful table for determining the sample size
Based on the above model by D. Morgan (1990), the following sample sizes are
recommended for corresponding populations.
Population
size
10
20
30
40
50
60
70
80
90
sample
10
19
28
35
44
52
59
66
73
Population
size
100
150
200
250
300
400
1500
2000
3000
Sample size
80
108
132
162
169
196
306
322
341
Population
size
4000
5000
10000
20000
50000
10000
Sample size
351
307
370
377
381
384
6.1.2 Sample Examination Questions -Sampling
1. a) A management Consulting firm based in Nakuru has been commissioned by BP Shell to
evaluate the company’s Human Resources (HR) capacity needs in relation to its performance.
The company has four major categories of staff as follows:
Finance and administration 300
Information technology
100
Marketing and production
450
Research and development 150
1) Giving reasons, suggest a suitable sampling techniques for the above study (4
marks)
2) Develop a suitable sampling design comprising of x 200 staff (8 Marks)
b) Explain why sampling is preferred to complete enumeration. (8 marks)
2
a. Explain four criteria for a representative sample that is suitable for an effective
survey research in Business management studies.
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b. A researcher engaged in a Business Management research study for an organization is
required to investigate consumer perceptions on product/services quality with a view
to establishing its impact on overall sales volumes in the company. The company has
a variety of customers on the basis of income groups, age-groups among other
characteristics. Soft Drink products and confectioneries constitute a major component
of the company’s sales volumes. However, the target population of customers is wide,
and no suitable sampling frame is available.
i)
ii)
Suggest with reasons, a suitable sampling method for the study. (6 Marks).
Using the selected sampling method (i) above, explain how the researcher
could obtain a sample of approximately 1,000 customers. (10 marks).
3
a) Differentiate between qualitative and quantitative research studies. (10 marks)
b) Write brief notes on the following:
i.
Stratified Sampling
(2 marks)
ii. Convenient Sampling
(2 marks)
iii. Sampling frame
(2 marks)
iv.
Systematic sampling
(2 marks)
a) By giving example, explain why a researcher may resort to samplings and not
conducting complete enumeration.
(8
marks)
Discuss at least three characteristics of a good sample
(6
marks)
Differentiate between probability sampling and non-probability sampling
(4 marks)
4
Distinguish between random and non-random sampling procedures. (8 marks)
State the difference between the terms “sample” and “population” as used in Business
Management Research
For the sample to be acceptable it must meet certain conditions. Which are these
conditions?
5
Abdul Onyango is a research worker with a reputable research consultancy firm that has
won the right to conduct a market research study for a client. He wishes to collect data for
the study from shopkeepers operating in the downtown shopping area of Nairobi.
What would be the most suitable technique?
(4
marks)
Justify the choice of ht sampling technique.
(10
marks)
Explain giving examples what is understood by:
(i) Stratified sampling
(ii) Judgment sampling
(4 marks)
(4 marks)
6
A researcher undertook to study how social background influences academic achievement.
He considered a population of 600 people. Whose composition was: 90 professionals. 115
managers. 150 skilled workers, 120 unskilled workers, and 125 businessmen.
What would be the most suitable sampling method for the study
NG’ANG’A S. I. 15TH DEC 2009
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Justify the use of such a sampling method. (4 marks)
Show how a sample of 240 would be drawn from the population. (10 marks)
Indicate the composition of each category of the population in the sample (2 marks)
7
As the first step to data collection. It is important to seek information from secondary sources
Explain in briefly the difference between primary and secondary data. (2 marks)
What are the merits and demerits of secondary data over primary data? (8 marks)
Discuss the factor that influences sample size in research.
8
Mary Mwangi is a K.I.M research student who wishes to collect primary data from a
population in meta estate in Nairobi. These is divided into 5 (five) blocks of 60, 75, 80, and
45 housing units.
What sampling procedure should Mary use? What?
Show how she could choose a sample of 75 units.
Can she use a sample random sampling? How?
10
Explain by giving examples what is meant by:
a.
Systematic random sampling.
b.
Convenience sampling
c.
Cluster random sampling
d.
Quota sampling
11
Research data is a reputable research consultancy firm that has worn the right to conduct a
study of the effectiveness of a newly developed teaching method at the Kenya institute of
management (KIM). If research data were to use DBMS classes (i.e. group a-d) as there
target population;
What would be the most appropriate sampling technique to use? (4 marks)
Justify the choice of (a) above (8 marks)
In what way can simple random sampling be used? (8 marks)
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6.2 Methods of Data collection
Instrumentation – research instruments
Instrumentation is the process of selecting and developing measuring devices and methods
appropriate to a given research problem. Research instruments are devices, which assist
researchers in collecting necessary information or data.
Requirement of the research instruments
Must be: i) valid – measure what it claims to measure. Relevant.( i.e. with respect to content
as expressed by objectives).
ii)
Reliable – stable, consistent, accurate, dependable and predictable.
Validity
Extent to which a research instrument measures what it is designed to measure. Three types
of validity; content validity, predictive validity and predictable
Content validity
Two varieties – face validity and sampling validity
Face validity is concerned with the extent to which the research instrument measures
what it appears to measure according to the researcher’s subject assessment.
Sampling validity refers to the extent to which the research instrument adequately
samples the content population of the property being measured.
Construct validity
Concerned with the extent to which a research instrument serves to predict some
meaning, traits or constructs in the candidate; data contained from a research
instrument should accurately reflect or represent a theoretical concept.
Predictive validity
Refers to the degree of correlation between test scores and some future outcome, such
as job success.
Concurrent validity predicts behavior of subjects in the present.
Validation of the research instrument
Process of collecting evidence to support the inference attached to the information obtained.
The presence or absence of systematic error in data largely determines validity.
Techniques of validating
i) Construct validity
a.
Variable being measured clearly defined
b.
Hypothesis based on a theory underlying the variable formed.
c.
The hypothesis tested – logically and empirically
Construct validity in a study can also be assessed if two or more different instruments
are used to measure the same concept.
Triangulation; methodological, source and or investigator
iii)
Content validity
Content validation is a matter of determining if the content that the
instrument contains is adequate. It also checks the format of the instrument
Use expert opinion to the content and format of an instrument to judge whether or not
it is appropriate.
iii) Criterion – related validity
Predictive and concurrent validity – not common in research
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Determining the reliability of the research instrument
Three methods:
i Test retest method
A research instrument administered to the same group of persons. The score on the two
sets of measures are then correlated to obtain an estimate computed –coefficient of
reliability.
ii Parallel – forms technique (equivalent form method)
The two sets of instruments administered to a group of persons. The score on the two sets
of measures are then correlated to obtain estimate reliability –coefficient of reliability.
iii Internal consistency method
Single instruments administered. There are three types.
a) Split - half method
The research instrument is separated into two sets of questions – even numbered and odd
numbered questions. The two sets of items scored separately and then correlated to obtain an
estimate of reliability. The reliability coefficient is calculated using the SPEARMAN –
BROEN prophecy formula:Reliability of scores on the total tests – 2(reliability for half test)
1+ reliability for ½ tests
Suppose that a test has a known reality. The spearman – brown formula
rn =
nr
1+ (n-1)r
Estimates the reliability of the score from a similar test n time as long with homogenous
content
Where r = the original reliability
Rn = reliability of the test n time as long
N = can be a fraction (shortened) or a whole number (lengthened) test
b) Kuder – Richardson approaches
Method of rational equivalence. The Kuder-richardson formulas 20 and 21 provide relatively
estimates of the coefficient of equivalence. Formula21, less accurate, but simple to compute.
r
RKR21 = k
k- 1
[
]
1- m(k-m)
Ks2
Where : items are scored 1 point if right and 0 point if wrong
M = mean
K = number of item
S = standard deviation
c) alpha coefficient (crowbach – {α} = KR20
α = KR20 =
K (S2 - ∑S2)
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S2(K- 1)
Where k = number of items used to measure
S2 = variance of all scores
S2 = variance of individual items
KR20 = reliability coefficient of internal consistence.
NB. High co efficiency implies that items correlate highly among themselves, i.e.
there is consistency implies that items correlate highly themselves, i.e. there is
consistence among the items in measuring concepts of interest.
Types of research instruments
Surveys are the most widely used technique in social science education and
the behavioral science for the collection of data.
They are as means of gathering information that describes the nature and
extent of a specified set of data ranging from physical counts and frequencies
to attributes and opinions.
Type survey includes: survey or records, mailed questionnaire, telephone
survey, group interviews, individual interviews.
Characteristics of survey techniques
I. Guiding principals underlying surveys are that they should be
II. Systematic – carefully planed and executed to injure appropriate content
coverage, sound and efficient data coverage.
III. Representative – closely reelecting the population of all possible cases or
occurrences, either by including everyone or everything, or by using scientific
sampling procedures.
IV. Objective – ensuring that the data are observable and explicit as possible.
V. Quantified – yielding data that can be expressed in numerical terms.
Limitations of survey techniques
Survey methods, with exception of record survey, run the risk of generating misleading
information due to:-
1. Survey only tabs respondents who are accessible and co-operative.
2. Surveys make respondents feel special or unnatural and this produce responses that
are artificial and/or slanted.
3. Surveys arouse ‘response sets’ such as acquiescence or a proneness to agree with
positive statement or questions.
4. Surveys are vulnerable to over-rate or under-rates bias the tendency for some
respondents to give consistently high or low ratings.
5. In case of interviews, biased reactions can be elicited because of characteristic of the
interviewer or respondents, or the combination that elicit an unduly favorable or
unfavorable pattern or responses.
General Guidelines for Designing Surveys
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1. Define the purpose and scope of the surveys in explicit terms
2. Avoid using an existing survey. If it was designed for a different purpose,
population circumstances.
3. In designing questionnaires or interviews, one often finds its helpful to sit down
with a group of potential respondents and explore what is meaningful or important
to them, and how best to phrase questions to reflect their attitudes or opinions.
4. Field test instruments, to spot ambiguous or redundant items and to arrive at a
format leading to ease of data tabulation and analysis.
5. Examine the merits of using machine-score answer sheets to facilitate tabulation
and analysis.
6. As often as possible, use structured questions as opposed to unstructured and
open-ended ones for uniformity or results and ease of analysis.
7. Do not ask questions out of idle curiosity.
8. Avoid loaded or biased questions be watchful or biased sampling.
9. Keep the final product as brief, simple, clear and straightforward as possible.
10. Brainstorm the analysis needs to insure the clarity and comprehensiveness of
instrument.
11. Consider the necessary and sufficient characteristics of the respondent that must
be collected at the time the survey is administered and on which the data analysis
will be based.
12. Imagine various outcomes that might result from the survey, including surprising
ones. This helps to anticipate gaps or shortcomings in the approach and may
indicate the need for more background information about the respondents or
additional questions.
The most common research instruments used social science survey technique include
i
Questionnaires
ii
Interview schedules
iii
Observational forms
iv
Standardized tests.
v
Records survey
The following is a summary of the types, characteristics advantages and limitations of the
research instruments.
i
Open-ended Questions
- Receives the Answer open to what a responded wishes to give.
Advantages: Free expressions, responses not biased and used to start a depth
interview, sets interview at ease.
Limitation: No specific answers: may digress; compiling, tabulating and
interpreting the responses could be difficult.
ii Dichotomous
Questions
- Receives only two types of responses – Yes/No; True/False; one or two
choices; can be varied to have a third or fourth opinion – not decided; Do not
know
e.t.c
- The responses can be scored by percentage.
Limitation: Opinion questions require a variation of approval or disapproval.
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iii
Multiple choice Questions
- A choice of responses offered. Respond by ticking/circling and/or fill in blank.
iv
Declarative Question
- Respondents give reactions to a given series of statements-Good,
Satisfactory, Fair or Poor.
Types of Questions to Avoid
1) Misleading questions
2) Leading questions
3) Double barreled questions
4) Embarrassing question
5) Ambiguous/argue questions.
6) Uninformative questions.
CHARACTERISTICS OF RESEARCH INSTRUMENTS
1.
2.
RESEARCH
INSTRUMENT
QUESTIONNAIRE
1) Open
ended
questions
2) Closed
and
pictorial
Multiple
choice
True/false
- Structured –
fill in blanks
INTERVIEW
SCHEDULES
1) Telephone
interviews
2) Group
interviews.
3) Individual
interviews.
CHARACTERISTICS
A set of carefully selected
an ordered questions used in
sampled studies.
- Indicate topic of study
- Should be attractive,
neatly, arranged, clearly
printed/typed
- Objective, simple and
clear question.
- - In a local or
understandable
language.
Should
be
accompanied by a letter
of transmittal.
-
May be
-
Unstruc
tured
Semi
structur
-
An interview is a
formal
meeting
or
communication
framework between two
parties whose primary
objective
is
the
procurement of factual
information
Plan what will be done
during the interview.
Kind of questionnaire
may be used to help
collect data required in
a standardized way.
NG’ANG’A S. I. 15TH DEC 2009
ADVANTAGES
-
-
Are inexpensive
Wide ranging
Can
be
well
designed,
simple
and clear.
Self – administering
Can
be
made
anonymous.
LIMITATIONS
-
-
-
-
-
-
-
-
Allow face to face
contact between the
researcher
and
respondent
Respondent
can
seek clarification of
a question not clear.
Researcher
can
evaluate sincerity
and insight of the
responded.
Allow
researcher
explain purpose of
research
-
-
-
Low response
rate can occur.
No
assurance
the
questions
were
understood.
Language may
not be to level
of respondent
Suspicious
respondents
deliberately
give
false
information.
May leave out
important
information
required
by
study.
Unstructured
interview often
yield data which
is difficult to
summarize or
evaluate.
Can be costly in
terms of money
and time.
Bias may creep
in personal class
Vulnerable to
personality
conflicts.
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-
ed.
Structur
ed
OBSERVATIONAL
FORMS
i. Systematic
ii.
Participant/ecol
ogical
iii. Archival
records
iv.
Simulations.
v.
Ethnography
vi.
Case studies
vii.
Content
analysis.
3
-
List desired questions in
a given order.
Record the interview –
note
taking,
tape
recording.
-
Systematic Observation
Recording and encoding a
set of natural behavior
usually in their natural
setting for the purpose of
uncovering
meaningful
relations
-
-
-
-
Steps
- Choose
natural
behavior to observe.
- Select
appropriate
observational setting.
- Decides on the mode of
recording observations.
- Determine
sampling
strategies.
- Train observers and
observe.
- Analyze data-structured
or unstructured.
-
Researcher
may
stimulate
respondent to a
greater extent
Appropriate
language
and
intellectual
difficulties exists.
Observation better
than self reports
obtained
from
questionnaires and
interviews.
Natural setting is
used.
Activities that could
not be investigated
easily like mob
justice,
natural
disasters
are
recorded and easily
access.
Yields
more
qualitative
information
-
Enable researcher to
obtain
detailed
information.
-
Ecological
observation can be
reflected
hence
verifiable.
-
Ecological
observation can be
reflected
hence
verifiable.
Economical
in
terms of time and
money.
Data real because it
is collected under
natural condition
Gathering
of
information
does
not require the
cooperation of the
individual/subject.
-
Requires
studied
and
trained
interviewers.
-
Hawthorne
effect – effect of
the observer on
the observed.
Observed may
question.
Halo effect on
the
observer
may lead to
confusion.
Sense
organs
may
be
inadequate
to
observe.
Costly in time
and money.
Some complex
behavior of the
subject may be
difficult
to
observe.
Information
obtained lacks
verification
(cannot
be
verified).
Lacks statistical
analysis.
Lacks
rules
which can be
understood in
order to collect,
analyze
qualitative
information.
Can lean to a lot
of bias
Can be risky.
-
-
-
Participant observation
- Used in field studies
- Non-experimental
- Researcher is there and
involved (naturalistic)
not very active by
passive.
Archival observation
- Statistical records that
allow the researcher to
observe the effect and
courses of real word
events
- National records –
statistical
records,
written
documents,
mass communication.
NG’ANG’A S. I. 15TH DEC 2009
-
-
-
-
-
-
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-
4
RECORD SURVEY
-
Differ from those in
other survey types
because they are nonreactive i.e they do not
involve a responsive
from people.
-
-
5
STANDARDIZED
TEST & SCALES
ii.
Intelligence &
Aptitude tests
Achievement
tests
iii. Attitude scales.
- value scales
- Lickert type or
summated rating
scales.
- Thurston type or
equal appearing
interval scale
- Gutiman type or
cumulative scale.
-
Tests are systematic
procedure in which
individuals
are
presented with a set of
constructed stimulus to
which they respond, the
responses enabling the
tester to assign the taste
a numeral or set of
numerals from which
inferences can be made
about
the
taste’s
possession of whatever
the scale is supposed to
measure
NG’ANG’A S. I. 15TH DEC 2009
-
-
Suitable for largescale
study
of
phenomena.
Records are nonreactive
They
are
inexpensive
Allow
historical
comparison
and
trend analysis.
Are accurate and up
to date they provide
an
excellent
baseline
for
comparison
-
May
involve
confidential
restrictions.
- Are
often
incomplete,
inaccurate, out
of
date
or
unavailable.
- Changing rules
for
keeping
records
often
makes year to
year comparison
invalid.
- Can
be
misleading
unless
knowledgeable
person
can
explain how the
records
were
complied.
- Purpose
of
records is often
unrelated
to
purpose
of
survey.
- Factual data (no
input on values
or attitudes) are
present.
Not common in Not
common
social
science standardized
research
tests/scales available
Mainly used in for social sciences
education
research.
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6.3 DATA ANALYISIS
6.3.1Introduction
One of the biggest challenges in research is designing a study with known and specified
variables whose measures are obtained in data collection, using an appropriate research
instrument. The data collected should be analyzed either qualitatively or quantitatively. This
section looks at how data is analyzed and presented. It will enable a researcher choose a data
analysis method to be employed in his/her study and thus design the study instrument to
collect the data as required.
6.3.2 Qualitative data analysis
Qualitative research investigates the quality of relationships, activities, situations and
materials where attributes and characteristics of interest are studied. Attributes are any
qualities ascribed to a person, subject or symbol and are essentially deemed to be a permanent
quality of a thing. Characteristics are distinguishing traits or features of the object under
study. In qualitative research, greater emphasis is placed on holistic description. This is
describing in details what goes on in a particular activity or situation rather on comparing the
effects of a particular treatment.
Qualitative research attempts to determine how people make sense of their lives in a natural
setting and the research is the key instrument. Data is collected in form of words or pictures
rather than numbers and concern is on both the process as well as the final results. Data is
analyzed through description and induction as outlined in the procedural steps below:
Step 1: Organization of data
Organize data to indicate how the data will be classified and tabulated according to research
questions and objectives and how the information will be analyzed, and synthesized and
presented in reports.
Step 2: Editing of data
Edit data to ensure accuracy and uniformity in report and to acquire maximum information
from the data. Check for inconsistencies; mistakes; lack of uniformity; illegibility and blank
or missing responses that should be disregarded. Check also for out layers that are likely to
distort the general picture portrayed by the sampled respondents and expected of the
population.
Step 3: Summarize data
Prepare summaries of data in questionnaires, interviews schedules and observation guides by:
a) Tabulating the number of responses received from the instrument for each item
b) Prepare a summary or a master questionnaire into which you put totals of responses onto a
blank instrument.
c) If endowed with or can access computer data analysis software such as the Statistical
Package for Social Scientists (SPSS), enter the data.
This makes analysis easier, presents concise summary statement of statistical findings,
facilitates comparisons, assists in interpretation of finding and provides a brief statement of
purpose, methods and data of a study.
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Step 4: Interpretation of Responses
Research questions are related to research problem, objectives and/or hypothesis and
responses may be interpreted with assistance of any of the following techniques.
More or less index:
This is computed to show the proportion of respondents whose choice is more or less
favorable. Thus:
Index = “more” – “less”
Total responses
The person answering the same or undecided is not included in the index. This index has a
shortcoming in that no measure is provided for the type of change required or expected.
Rating scale:
This scale rates the opinion of the respondents on a continuum such as the lickert scale. An
example is the 5 point scale where the frequency and percentage of respondents selecting a
particular response is computed indicating the general perception of the sampled respondents.
5
Strongly agree
4
agree
3
neutral
2
disagree
4
favorable
3
neutral
2
1
strongly disagree
OR
5
Very favorable
unfavorable
1
very unfavorable
It is also possible to obtain and test the overall position taken by all the respondents put
together by computing the mean score on each item and using a one sample t-test using a test
statistic of the highest rating plus the lowest rating divided by 2, to check the significance of
the difference between the mean and the test statistic. For example, in the 5 point lickert scale
above, the test statistic is 3. If the mean score in any item is higher than 3 numerically and
the test shows that the difference is significantly higher than 3, then all respondents put
together lean more towards agree and it is interpreted to mean that the respondents in general,
are in favor of the statement or construct in the question.
Interpretation weighting
As in the rating scale, respondents rate a statement/position on a variable. The ratings are
assigned to correspond to a score or weight. The weight and number of respondents,
frequency in favor of a certain position are then multiplied to give an interpretation
weighting. If on the five point lickert scale labeled 5, 4, 3, 2, and 1 are taken as weights, then
tabulating the results yields:
Scale
weight
(W)
Strongly agree
Agree
Undecided
Disagree
Strongly disagree
No of responses
(F)
5
4
3
2
1
∑W
NG’ANG’A S. I. 15TH DEC 2009
weight x no. of responses
30
25
20
15
10
∑F 100
150
100
60
30
10
350
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Average weight (mean) = ∑W *hF
∑F
= 350
100
= 3.5
The average weight 3.5 lies higher than 3, the midpoint of the lickert scale and it is
interpreted that the respondents agree with the proposition.
Indexes of fame or popularity
This is an index that can be used to rate the popularity or notoriety of an individual. To
develop this index, questions are asked that lead to a list of names of who ascribe to a given
concern. The list of names and the number of times the name is mentioned is then used to
calculate the index as shown table 5.1.
Table 5.1: Computation of an index of fame
Names No. of times the name mentioned Index of fame
F
=
F
∑F
A
60
0.30
B
40
0.20
C
50
0.25
D
20
0.10
E
30
0.15
TOTAL
200
This is a continuum ranging from 0 to 1. The closer a name is to 1, the more popular or
notorious the person is according to the respondents depending on the measure of interest.
Cross Tabulation
This is done to obtain and present more information that can be obtained in a single
classification. It contains a matrix of classes of values and may contain one or two variables,
original figures or percentages or both. It improves understanding of the data, cross effect of
the variables and forms a basis for comparison. An example can be seen in the composition
of residents in a residential area in Nairobi.
Table 5.2: Cross-tabulation of residents in a residential area in Nairobi in 1980 according to
race and gender
ETHICITY
Asians
European
Africans
TOTAL
POPULATION IN AN AREA/TOWN
MALE
FEMALE
3,000
5,000
600
800
15,000
16,000
18,600
21,800
TOTAL
8,000
14000
31,000
40,400
While cross tabulations are used to interpret data of various variables, they also form a basis
for comparisons; they can be used further for quantitative analysis in testing of relationships
between variables using chi-square. For example, if data leads to a cross tabulation of data on
the basis of two variables, visits by quality standards and assurance officers to secondary
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schools, and classification of schools according to performance in English, it can be used to
test whether there is an association between the number of visits and the performance in
English at the KCSE level as shown in table 5.3.
Table 5.3: Cross-tabulation of quality assurance and standards officers visits to schools and
the classification of schools in English according to performance at the KCSE level
QASO
number of
visits to
schools
Classification of schools according to performance in
English at the KCSE level
High
Average
Low
Total
Visited
Not visited
Total
Frequency distribution
Ratios and proportions in percentage of respondents in favor of a given response among
mutually exclusive responses are also used in the interpretation of data. The proportion of
respondents in favor of a certain response is calculated and tabulated as shown in table 5.4. If
a sample of registered voters were asked whether they would vote in favor of a certain
candidate the responses would be yes, no or no response.
Table 5.4: Frequency distribution of respondents in favor of a certain candidate in an election
Response
Yes
No
Missing
Total
frequency
80
70
0
150
Proportion
53.3
46.7
0
100
This would be interpreted to mean that the candidate is likely to win in the election although
s/he has a significant proportion (46.7%) of the respondents who are opposed to his/her
candidature.
If the researcher wishes to find out more about the electorate, the frequency distribution could
be broken down to show the distribution according to gender, age, level of education,
occupation of the respondents so that the candidate could be advised on the strategies to
adopt targeting specific stratum of respondents. The frequency distribution according to
gender is shown in table 5.5.
Table 5.5: Frequency distribution of respondents in favor of a candidate in an election
according to gender
RESPONSE MALE
F
Yes
30
No
40
Missing
0
Total
70
%
20
26.7
0
46.7
FEMALE
F
50
30
0
80
%
33.3
20
0
53.3
TOTAL
F
50
70
0
100
%
55.3
47.3
0
100
This may be interpreted to mean that more women (33.3%) as compared to men (20%) are in
support of the candidate.
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This data analysis can be done manually by use of tallies against individual responses and a
hand calculator to compute proportions. For example, if only 40 respondents were sampled in
the example above, then
Total respondents proportion (%)
Yes
28/40 =
70 %
No
12
= 30%
40
This can then be entered into the frequency distribution table.
A computer package (software) such as SPSS can also be used to analyze qualitative data
obtained from respondents in social science. Questions in the research instruments are coded,
variables defined and entered into the computer after responses are entered as either numeric
or string from which frequency distributions can be generated.
6.3.3 Quantitative data analysis
Quantitative research refers to the studies that make use of a numeric measure to evaluate an
aspect of a particular problem or situation. Such studies are done when a researcher wishes to
obtain a large body of data to perform statistical analysis and produce results that can be
generalized to the target population. Data is reduced to numeric scores and preference is
given for a random technique of obtaining meaningful samples.
In quantitative studies, data is statistically analyzed so that meaning is inferred. Quantitative
research is mainly concerned with the problem of estimation and testing statistically based
hypothesis. This is achieved through descriptive statistics from which population parameters
are estimated leading to generalizations. Quantitative research assumes that behavior of
people can be objectively measured and the cause and effect relationship between variables
determined through various techniques. The following sections explain how numeric data
obtained from quantitative research is analyzed.
The following is a general step by step procedure followed in quantitative research:
i)
Collect quantitative data
ii)
Obtain descriptive statistics
iii)
Estimate population parameter from the statistics
iv)
Test hypothesis
v)
Make inferences
6.3.4 Descriptive statistics
Descriptive statistics provide information on how data obtained in respect to variables of
interest relate to each other. There are four categories under which data can be analyzed to
provide descriptive statistics: measures of central tendency, measures of dispersion, measures
of skewedness and measures of peakedness.
6.4Measures of central tendency
These measures show how quantitative data obtained from respondents or from the study
tends to coalesce, or cluster towards a certain center. The most common measures of central
tendency used in research are the mean, the median and the mode.
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i)
The mean
The arithmetic mean is the measure of central tendency normally thought of as an average. It
is given by:
n
n
Sample mean x x1
for ungrouped data
i 1
n
x
fx
f
x
for grouped data
fm
f
for grouped continuous data
Where:
∑ = sum or summation of
f = frequency (number of time the same response is obtained)
x = observation
n = sample size
x = mean
m = is the midpoint of class obtained by adding the lower class limit and upper class limit
and dividing by two.
ii) The median
This is the middle observation after data have been put in an ascending ordered array. If the
number of observations (n) is odd, the median is the middle one e.g. in 32, 41, 59, 63, and 71;
59 is the median. If n is even, the median is the middle 2 divided by 2
59 63
61
e.g. in 41, 59, 63, and 71, the median is
2
For grouped data:
n
2 f
The median Lmd [
]c
f
md
When Lmd = lower class boundary of the median class.
f = cumulative frequency of the class preceding the median class.
fmd = the frequency of the median class.
c = class interval of the median class.
iii) The mode
The mode is the observation which occurs most often. In grouped data, the class with the
largest frequency is the modal class.
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Mode [ Lmd
Da
D D
b
]c
a
Where Lmd = the lower class boundary of the modal class.
Da = the difference between the frequency of the modal class and the class
preceding it.
Db = the difference between the frequency of the modal and the class after it.
c = the class interval of the modal class.
Examples 5.1
Kamau Otieno consultants conducted a study on the unemployment pattern in Nairobi which
produced the following results:
number unemployed (000’s)
Women
15
10
19
13
28
22
30
25
23
21
18
9
17
00
150
100
Age (years)
Men
15-19
20-24
25-29
30-34
35-39
40-44
45-49
Mean age of unemployed respondents
x
Class age
(years)
15-19
20-24
25-29
30-34
35-39
40-44
45-49
fm
f
MALE
frequency
f
15
19
28
30
23
18
17
∑f = 150
midpoint
m
17
22
27
32
37
42
47
FEMALE
fm
225
418
756
960
851
756
752
∑fm = 4748
f
10
13
22
25
21
9
0
∑f= 100
fm
170
286
594
800
777
378
0
∑fm =
3005
Mean age of unemployed men
x
fm 4748 31.65 years
f 150
Mean age of unemployed women x
NG’ANG’A S. I. 15TH DEC 2009
3005
30.05 years
100
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The mean age of all respondents together x
4748 3005
31.012 years
150 100
The median age of the unemployed
Class (age)
15-19
20-24
25-29
30-34
35-39
40-44
45-49
MALE
f
15
19
28
30
23
18
17
Cf
15
34
62
92
115
133
150
FEMALE
f
10
13
22
25
21
9
0
cf
10
23
45
70
91
100
100
The middle age of the unemployed men is the 75th which is in the class 30-34 as shown in the
cumulative frequency distribution.
n f
]c
Median age h Ma Lmd [ 2
f
md
Lmd = 30
f = 62 (cumulative frequency of the classes before the class containing the
median item).
fmd = 30
n = 150
c=5
150 _ 62
2
Median age of unemployed men 30 [
]5 32.17 years
30
100 45
2
Median age of unemployed women 30 [
]5 30.005 years
45
The mode of the age of the unemployed respondents
Lmd [
D
a
]c
Da Da
From the cumulative frequency distribution above, the modal class is 30-34
M
o
FOR MALE
NG’ANG’A S. I. 15TH DEC 2009
FOR FEMALE
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L
D
D
md
30
30
a
30 28
25 22
b
30 23
3 4
C 5
5
30 ( 2 )5 31.1 years
9
Mode age of unemployed men
Mode of unemployed women
h 30 ( 3 )5 31.5 years.
10
Deductions
On the whole: (i) The data suggests that there are more unemployed men than women in
Nairobi but the difference between the means of the two
independent samples Male and Female is small and should be
subjected to significance testing by use of t tests.
(ii) The unemployed are in their early thirties
6.5 Measures of Dispersion
These measures show how data tends to scatter, spread, disperse or vary. They show
variations or variability and as noted earlier there are three causes of variability in research
data. These are variations caused by systematic or natural causes also said to be attributable
causes, extraneous variables that require efforts to control or eliminate in a study or errors
either in measurements or in use of instruments. Efforts are made to control extraneous
variables by the research design and minimize variation due to errors by use of appropriate
methods; instruments and random sampling while systematic variations are the objects of the
study.
Dispersion or extent of spread is measured through computation of the range, quartile
deviation and percentiles, mean deviations, variance, standard deviation and coefficient of
variation.
i) The Range
The range is the difference between the highest observation and the lowest observation. In
frequency distribution, the range is taken to be the difference between the lower limit of the
class at the lower extreme of the distribution and the upper limit of the class at the upper
extreme. In example 5.1, the range is 49-15=34.
ii) Quartile Deviations
A quartile divides an array of data into four equal parts. Q1 gives the value of the item at the
1st quarter mark while Q3 gives the value of the item at the 3rd quarter mark. The semi-quartile
range or quartile range deviation (QD) is given by:
QD
Q Q
3
1
2
which means 50% of the distribution lie
with the interval defined plus or minus the
Quartile deviation.
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iii) Percentile
These are values of a variable, which divide a set of ordered observation into 100 equal parts.
The 25th percentile is also called lower quartile.
The 50th percentile is also called median
The 75th percentile is also called the upper quartile.
The coefficient of QD
Q Q
Q Q
3
1
3
1
To calculate Q1 and Q3 in grouped continuous data
Q L
1
1
And
Q L
3
3
[ N Lcf ]
1( 4
)C
fq
th
[3N Lcf ]
4
1(
)C
fq
th
Where L1 and L3 is the lower class limit of the class with the ¼ th item and the ¾ th item
respectively.
Lcf = lower cumulative frequency
fqth = frequency of the class that contains the ¼ or ¾ item.
The quartile deviation gives an indication about the uniformity or otherwise of the size of
items of a distribution. Q.D. is a distance on a scale and thus regarded as a measure of
partition.
iv) Mean deviation
This is the mean of the absolute values of the deviation from a measure of central tendency,
usually the mean.
xx
Mean absolute deviation (MAD)
OR MAD
n
f xx
f
NG’ANG’A S. I. 15TH DEC 2009
for grouped data
f xx
f
for grouped continuous data
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Mean Absolute Daviation
Mean
(If mean is the measure of central tendency used )
Coefficient of MD
The mean deviation is easier to understand and is affected by extreme values and is a better
measure of dispersion compared to the range and quartile deviations. However, it is not
suitable for further mathematical processing.
v. The variance
The variance is the mean of the squared deviation from their mean denoted by S2.
( x x)
2
s
Sample variance
OR
2
n 1
f ( x x)
S
f 1
2
2
for grouped data
2
OR
S
2
(m x )
f 1
for grouped continuous data
vi The standard deviation
The standard deviation denoted by (s) is the square root of the variance
Sample standard deviation
( x x)
(s)
2
n 1
f ( x x)
f 1
2
S
f (m x)
f 1
for grouped data
2
And
S
for grouped continuous data
The standard deviation is a good measure of dispersion since it takes into account all the data
and responds to the exact position of every score about the mean. It is also sensitive to
extreme score.
vi) Coefficient of variation
This is a measure of variability relative to the mean denoted by CV.
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S
*100
X
It is useful when comparing the spread of two distributions.
CV
Example 5.2
In the worked example 5.1 above, determine the group (male or female) which has greater
variability (related dispersion) in unemployment.
Solution
range ≥ upper class limit (upper extreme class) - lower limit (lower extreme class)
Male 49 16 33
Female 44 16 28
Suggest male have greater variability.
Q Q1
i)
Quartile deviation (Q.D) 3
2
Q
1
L1
[ N L4]C
4
F
150
[150 14]5
Q1 for male 25 428 25.63 years
100
[100 23]
4
for female 25
5 25.45 years
22
Q
3
[3N L fc]
C
Q3 L3 4F
112.5
150
[
92]
4
for male 35
5 40.69 years
18
[75 70]
5 36.19
3
21
Unemployed male respondents have a greater variability in age based on quartile
deviations.
Q
for female 35
QD for male
QD for female
Q Q
3
Q
3
1
2
Q
1
2
40.69 25.63
13.94 years
2
36.19 25.45
4.87 years
2
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ii) Coefficient of QD
For male
40.69 25.63 15.32
0.23 23%
40.60 25.63 66.32
For female
36.19 25.45 10.79
0.18 18%
36.19 25.45 61.64
iii) Mean absolute deviation (MAD)
f mx
MAD
f
From the worked example 5.1 x was found to be 31.6 years for male and 30.05 years for
male and 30.05 years for female.
class
15-19
20-24
25-29
30-34
35-39
40-44
45-49
MALE
f
15
19
28
30
23
18
17
∑f =
150
m
17
22
27
32
37
42
47
│m- x │
-14.65
-9.65
-4.65
=0.35
5.35
10.35
15.35
∑│m- x
│
=60.35
)2
(m- x
214.62
93.12
21.62
0.12
28.62
107.12
235.62
)2
F(m- x
3219.34
1769.33
608.43
3.68
658.32
1928.21
4005.58
12192.89
f
10
13
22
25
21
9
0
FEMALE
m- x
13.05
8.05
3.05
1.95
6.95
11.95
16.95
│m- x │=
41.95
(m- x )2
170.30
64080
9.30
3.80
48.30
142.80
287.30
F(m- x )2
1703
842.43
204.66
95.06
1014.35
1285.22
0
∑f(m- x )2=
5144.72
Mean deviation absolute deviation
60.35
For male
0.41
149
41.95
For female
0.42
99
From the calculation it can be noted that the interpretations have changed. The unemployed
female respondents are shown here to have greater variability. This also suggests the female
have higher deviation.
MAD
Coefficient of MD
X
0.41
For male
*100 1.30%
31.65
0.63
For female
*100 2.1%
30.05
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iv) The variance
( m x)
S f 1
2
2
12192.89
81.83
49
5144.72
for female
51.97
99
This measure suggests that male have greater variability.
For male
V) The standards deviation
f (m x)
f 1
2
S
for grouped data
= √variance
For male
S 81.83 9.05
For female
S 51.97 7.21
This suggests that the male have greater variability.
Coefficient of variation (cv)
s
cv * 100
x
9.05
For male cv
*100 28.6%
31.65
7.21
for female cv
*100 24.0%
30.05
The coefficient of variation (cv) also suggest that the male have greater variability.
Table measure of dispersion for unemployed in Nairobi
MALE
FEMALE
Range
34
29
Quartile deviation (QD)
13.94
4.87
Coefficient of QD
23%
18%
Mean absolute deviation (MAD)
0.41
0.63
Coefficient of (MD)
1.30
2.1%
Variance (S2)
81.83
51.97
Standard deviation (S)
9.05
7.21
Coefficient variation
28.6%
24.0%
The measure shows that the unemployed males have high relative variability in age compared
to female.
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6.6 Skewness and Peakedness
6.6.1 Skewness
A distribution is said to be skewed if it is not symmetric. Skew distributions that often arise
in practice are unimodal with one tail (upper or lower) longer compared with other tail. For
those distributions the mean tends to lie on the same side of the mode as in figure 3.5
Fig: 5.5 skew distribution
mode
median
mean
mean
median
mode
Fig 3.5 (a) is a positively skewed distribution with the lower (right) tail longer. The mean of
the date is larger than median.
Fig 3.5 (b) is a negatively skewed distribution with the lower (left) tail longer. The mean of
the date is smaller than the median.
In a normal distribution (one that is not skewed, all the three; the mean, mode and median
coincide at the same point as shown in fig 3.5(c))
Fig 3.5(c)
Mean,
Mode,
Median.
There are several ways of assessing skewness in a distribution: they include.i)
1
N
B
1
N
(x )
i 1
3
i
3
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where µ is the population mean
is the population size
is the population standard deviation.
For a sample, coefficient of skweness (b1)
1
b
n ( x x)
S
3
3
1
ii) Pearson’s coefficient of skewness (b1)
b
1
b
1
3( x median)
S
3(m median)
iii) Pearson’s absolute skewness = mean – mode
OR
( x median)
iv) Bowleys coefficient of skewness ( b1 )
b1
Q Q 2 median
Q Q
3
1
3
1
v) Bowleys absolute skewness Q Q 2 median
3
1
vi) Kelly’s coefficient of skewness (b1)
p90 p10
Where P50,P90 and P10 are the 50th, 90th and
p
b1 50
2
10th percentiles.
Note:- The coefficient of skewness are relative measures and have no dimension while
absolute measure have dimensions which are the unit for which x is measured.
b- The direction of skewness is given by the algebraic sign (+ or-) and the numeric value
gives the degree of skewness.
c- The relative coefficient of skewness usually lies between +1 and -1.
When b1 > 0, the distribution has a longer upper (right) tail and is very skewed.
When b1 < 0, the distribution has longer lower (left) tail and is negatively skewed.
When b1 = 0 the distribution is normal and is symmetric about the mean.
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6.6.2 Peakedness (kurtosis)
Frequency distribution also vary in regards to their Peakedness is the extent to which a
frequency distribution has a peak or is flat at the top.
Fig.4.6
Curve A is lepto kurtic
B is meso kurtic
C is platty kurtic
Kurtosis is measured by B2 where
1
B
2
n ( x x )
S
4
4
Note:
A – in a normal distribution, B2 will be equal to 3.
B – if B2 is greater than 3, the curve is more peaked (lepto kurtic).
C - if B2 is less than 3, the curve is flatter at the top than the normal curve and is said to be
(platy kurtic). A peak of a normal distribution is mesokurtic.
In research data analysis, coding and data entry is done in such a way that it allows
generating of descriptive statistics by use of SPSS or excel.
6.7 Bivariate Data
Measures of central tendency, variability, and spread summarize a single variable by
providing important information about its distribution. Often, more than one variable is
collected on each individual. For example, in large health studies of populations it is common
to obtain variables such as age, sex, height, weight, blood pressure, and total cholesterol on
each individual. Economic studies may be interested in, among other things, personal income
and years of education. As a third example, most university admissions committees ask for an
applicant's high school grade point average and standardized admission test scores (e.g.,
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SAT). In this chapter we consider bivariate data, which for now consists of two quantitative
variables for each individual. Our first interest is in summarizing such data in a way that is
analogous to summarizing univariate (single variable) data.
By way of illustration, let's consider something with which we are all familiar: age. It helps to
discuss something familiar since knowing the subject matter goes a long way in making
judgments about statistical results. Let's begin by asking if people tend to marry other people
of about the same age. Our experience tells us "yes," but how good is the correspondence?
One way to address the question is to look at pairs of ages for a sample of married couples.
Table 1 below shows the ages of 10 married couples. Going across the columns we see that,
yes, husbands and wives tend to be of about the same age, with men having a tendency to be
slightly older than their wives. This is no big surprise, but at least the data bear out our
experiences, which is not always the case.
Husband
36 72 37 36 51 50 47 50 37 41
Wife
35 67 33 35 50 46 47 42 36 41
Table 1: Sample of spousal ages of 10 White American
Couples.
The pairs of ages in Table 1 are from a dataset consisting of 282 pairs of spousal ages, too
many to make sense of from a table. What we need is a way to summarize the 282 pairs of
ages. We know that each variable can be summarized by a histogram (see Figure 1) and by a
mean and standard deviation (See Table 2).
Figure 1: Histograms of spousal ages.
Mean
Standard Deviation
Husband
49
11
Wife
47
11
Table 2: Means and standard deviations of spousal ages.
Each distribution is fairly skewed with a long right tail. From Table 1 we see that not all
husbands are older than their wives and it is important to see that this fact is lost when we
separate the variables. That is, even though we provide summary statistics on each variable,
the pairing within couple is lost by separating the variables. We cannot say, for example,
based on the means alone what percentage of couples have younger husbands than wives. We
have to count across pairs to find this out. Only by maintaining the pairing can meaningful
answers be found about couples per se. Another example of information not available from
the separate descriptions of husbands and wives' ages is the mean age of husbands with wives
of a certain age. For instance, what is the average age of husbands with 45-year-old wives?
Finally, we do not know the relationship between the husband's age and the wife's age.
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We can learn much more by displaying the bivariate data in a graphical form that maintains
the pairing. Figure 2 shows a scatter plot of the paired ages. The x-axis represents the age of
the husband and the y-axis the age of the wife.
Figure 2: Scatterplot showing wife age as a function of
husband age.
There are two important characteristics of the data revealed by Figure 2. First, it is clear that
there is a strong relationship between the husband's age and the wife's age: the older the
husband, the older the wife. When one variable (y) increases with the second variable (v), we
say that x and y have a positive association. Conversely, when y decreases as x increases, we
say that they have a negative association.
Second, the points cluster along a straight line. When this occurs, the relationship is called a
linear relationship.
Figure 3 shows a scatterplot of Arm Strength and Grip Strength from 149 individuals
working in physically demanding jobs including electricians, construction and maintenance
workers, and auto mechanics. Not surprisingly, the stronger someone's grip, the stronger their
arm tends to be. There is therefore a positive association between these variables. Although
the points cluster along a line, they are not clustered quite as closely as they are for the scatter
plot of spousal age.
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Figure 3: Scatter plot of Grip Strength and Arm Strength.
Not all scatter plots show linear relationships. Figure 4 shows the results of an experiment
conducted by Galileo on projectile motion. In the experiment, Galileo rolled balls down
incline and measured how far they traveled as a function of the release height. It is clear from
Figure 4 that the relationship between "Release Height" and "Distance Traveled" is not
described well by a straight line: If you drew a line connecting the lowest point and the
highest point, all of the remaining points would be above the line. The data are better fit by a
parabola.
Figure 4: Galileo's data showing a non-linear relationship.
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Scatter plots that show linear relationships between variables can differ in several ways
including the slope of the line about which they cluster and how tightly the points cluster
about the line. A statistical measure of the strength of the relationship between variables that
takes these factors into account is the subject of the next section.
Quantitative Variables:
Variables that have are measured on a numeric or quantitative scale. Ordinal, interval
and ratio scales are quantitative. A country's population, a person's shoe size, or a
car's speed are all quantitative variables. Variables that are not quantitative are known
as qualitative variables.
Histogram:
A histogram is a graphical representation of a distribution. It partitions the variable on
the x-axis into various contiguous class intervals of (usually) equal widths. The
heights of the bars represent the class frequencies.
Figure 5
See also: Sturgis's Rule
Sturgis's Rule:
One method of determining the number of classes for a histogram, Sturgis's Rule is to
take 1+log2N classes, rounded to the nearest integer.
Bivariate:
Bivariate data is data for which there are two variables for each observation. As an
example, the following bivariate data show the ages of husbands and wives of 10
married couples.
4
Husband 36 72 37 36 51 50 47 50 37
1
4
Wife
35 67 33 35 50 46 47 42 36
1
Table 3
Scatter Plot:
A scatter plot of two variables shows the values of one variable on the Y axis and the
values of the other variable on the X axis. Scatter plots are well suited for revealing
the relationship between two variables. The scatter plot shown in Figure 4 illustrates
data from one of Galileo's classic experiments in which he observed the distance
traveled balls traveled after being dropped on a incline as a function of their release
height.
Positive Association:
There is a positive association between variables X and Y if smaller values of X are
associated with smaller values of Y and larger values of X are assoicated with larger
values of Y.
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Negative Association:
There is a negative association between variables X and Y if smaller values of X are
associated with larger values of Y and larger values of X are assoicated with smaller
values of Y.
Linear Relationship:
If the relationship between two variables is a perfect linear relationship, then a
scatterplot of the points will fall on a straight line as shown in Figure 6.
Figure 6
With real data, there is almost never a perfect linear relationship between two
variables. The more the points tend to fall along a straight line the stronger the linear
relationship. Figure 2 shows two variables (husband's age and wife's age) that have a
strong but not a perfect linear relationship.
A dataset with two variables contains what is called bivariate data. This chapter discusses
ways to describe the relationship between two variables. For example, you may wish to
describe the relationship between the heights and weights of people to determine the extent to
which taller people weigh more.
The introductory section gives more examples of bivariate relationships and presents the
most common way of portraying these relationships graphically. The next five sections
discuss Pearson's correlation, the most common index of the relationship between two
variables. The final section, "Variance Sum Law II" makes use of Pearson's correlation to
generalize this law to bivariate data.
6.8 Revision Questions
1.
The following figures show the volume of commodity sales by three sales representatives (AC) at Manga-Craft Ltd. in a period of 5 days.
A
410
415
420
425
405
430
420
425
B
415
418
417
416
415
414
413
412
C
430
415
450
400
420
440
430
425
Calculate:
a) The mean deviation for each set of sales
b) The standard deviation for each set of sales
NG’ANG’A S. I. 15TH DEC 2009
(5 marks)
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c) What is the purpose of standard deviation?
(5 marks)
d) Comment on the standard deviation calculated in relation to the sales. (5 marks)
2.
The following sets of data refer to a sample of marks (out of 20) obtained in a class test by
two groups of the Diploma class at the Kenya Institute of Management
A
8
9
9
10
10
10
11
13
B
2
3
3
10
11
16
17
18
(a) For each set of data, calculate:
(i.) The mean (2marks)
(ii) The range
(2 marks)
(iii) The standard deviation
(8 marks)
(iv) The coefficient of deviation
(3 marks)
(b) Based on the values calculated in (a) above, comment on the data
(5 marks)
3.
The manager of a bank has ordered a study on the amount of time a customer waits before
being attended to by the bank personnel. The following data (minutes) was collected during a
typical day:
12
26
16
4
21
7
20
14
24
25
3
1
11
27
17
15
29
16
18
5
(e) Arrange the data in the array from the lowest to the highest
(3 marks)
(f) Comment the customers waiting time from the array
(3 marks)
(g) Construct a grouped frequency distribution using six classes
(9 marks)
(h) Based on the frequency distribution, what additional interpretation
can be given to the data?
(5 marks)
4.
Given the following sets (I & II) of data
I
5
6
II
7
6
8
8
12
5
12
9
For each set, calculate:
(i) The range
(1 mark)
(ii)The mean deviation
(iii)The standard deviation
marks)
(iv)The coefficient of deviation
(3 marks)
NG’ANG’A S. I. 15TH DEC 2009
(6 marks)
(6
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Based on the values calculated in (a) above, comment on the two sets of data. (4 marks)
5.
The following grouped frequency distribution shows the distance in kilometers covered by a
group of one hundred and twenty sales representatives in one week.
DISTANCES (Km)
400 - 420
420 - 440
440 - 460
460 - 480
480 - 500
500 - 520
No. of Sales Representatives
12
27
34
24
15
8
Required
Calculate:
(i)
The mean deviation
(8 marks)
(ii)
The standard deviation
Using the graph papers provided, construct;
(i)
A histogram
(5 marks)
(ii)
A frequency polygon
(2 marks)
(5 marks)
6. (a) Explain the difference between;
(i) Stratified sampling and clustered sampling
(2marks)
(ii) Qualitative data and Quantitative data
(2marks)
The ages of first year science education students in Moi University was found to be
Age
14-16
16-18
18-20
20-22
22-24
24-26
Frequency
5
16
13
7
5
4
Calculate
(i) The mean age of the Students
(ii) The coefficient of variation of the students age
(iii) The coefficient of skewness of the Students age
(v) Comment on the distribution of the Students age
(2marks)
(3marks)
(2marks)
(2marks)
7.
(a) For a given research data, “we can have two regression lines.” Explain this statement and
state clearly the suitability of using each line for estimation of the values
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(8 marks)
A research team, while studying the growth pattern of bacteria, recorded the following
observations:
Time since first infection (hours) x
15
20
25
30
Bacteria population y
40
70
5000
2000
i. Estimate an exponential curve Y ab x for the given data
(6 marks)
ii. Determine the bacteria population, 40 hours after the first infection
(2 marks)
iii. Using the exponential curve, estimate bacteria population; 25 hours after first
infection. Hence calculate the error of estimation
(4 marks)
8. i.)
State any two methods of data collection and indicate situations where
they can best be used.
4mks
ii)
Suppose measurement of an item with a metric micrometer A yield a mean
of 4.20mm and a standard deviation of 0.015mm and suppose
measurements of another item with an English micrometer B yield a mean
of 1.10 inches and a standard deviation of 0.005 inches. Which micrometer
is relatively “more” precise?
9. Explain the uses of statistics in research
3mks
a) What is a continuous variate?
3mks
b) State by giving examples situations where the median is more useful than the mean as
a measure of central tendency.
3mks
c) The lives of two models of refrigerators in a recent survey are given below:
Refrigeratory life (No of years)
Number of refrigerators
Model A
Model B
0-2
5
2
2-4
16
7
4-6
13
12
6-8
7
19
8-10
5
9
10-12
4
1
i.) Determine the average life of each model of these refrigerators?
4mks
ii.) Which model has less variation of life span?
6mks
iii.) Find the most common life span in years for each model
4mks
iv) Calculate the semi quartile range of the two models and interpret the results in relation to
your answer in (ii) above
6mks
v.) Based on your results in i.) to iv.) above and that the prices are the same for the models
which model would you recommend someone to purchase for use? 2mks
10.
a.)
A certain disease affects children in their early years and sometimes kills them. The
frequency table of the age at death in years of 96 children dying from this disease is shown
below.
Age of
1-2
2-3
3-4
4-5
5-8
8-10
deaths
0-1
(years)
Frequency 10
40
20
10
5
7
4
Using the data
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i.
ii.
iii.
iv.
Calculate the mean age of death
3mks
Determine the median age at death
2mks
Construct a histogram for the distribution
4mks
If the 96 children is a sample taken from a large population of children, what
general conclusion would you make about the impact of the disease 3mks
b.) Distinguish between Quantitative and Qualitative variables, giving examples in each
case
3mks
11.
a.) The weights (in kgs) and heights (in cm) of 50 students of a certain university were
measured and the table below shows the respective distribution.
Weights (kgs)
Frequency
Heights (cm)
Frequency
41-45
7
131-135
2
46-50
5
136-140
4
51-55
14
141-145
5
56-60
11
146-150
8
61-65
10
151-155
16
66-70
2
156-160
7
71-75
1
161-165
5
76-80
0
166-170
3
Total
50
Total
50
Determine
i.) The mean eight and height for the two distributions
6mks
ii.) The standard deviation for the weights and heights
8mks
iii.) The coefficient of variations and indicate which variables had a greater relative
dispersion?
6mks
iv.) Why is coefficient of variation as a relative measure of variation superior to
standard deviation?
4mks
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7.1
7. 0
TIME SERIES: (8 HOURS)
Definition of time series concepts
7.2
Examples of time series
7.3
Moving averages
7.4
Estimation of trend,
- Use of scatter diagrams.
7.1 Definition of Time series graphs
In a times series, values of a variable are given at a different periods of time. When a graph of
such a series is drawn it would give changes in the value of a variable with the passage of
time. The graphical presentation of such a series is called a histogram.
The aim of drawing such graphs is to have comparison to study the
(i) Changes in one variable over a period of time and
(ii) Changes if two or more variables over a period of time.
While constructing a histogram, time is taken along x – axis and the values along y – axix
then the data is plotted and points are joined by means of straight lines to get the histogram.
The main examples of time series are as under;a) Population of a country over a specific period of time.
b) Sales of a business enterprise over a period of one year.
c) Prices of some specific commodities over a period of time
d) Temperature over a period of time.
Example 3:
Monthly sales of AB stores for the year 19 – 8 were as follows:Month Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
Sales
40
60
70
50
80
100
90
110
80
70
120
50
(shs
000)
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Construct a graph from the above figures.
Sales
Graph of Time Series
140
120
100
80
60
40
20
0
Sales
(shs
000)
Jan
Feb Mar Apr May
Jun
Jul Aug Sept Oct Nov Dec
Year 19.8
Example 4
The following table gives the sales of a certain firm in 6 – years. Draw a graph of time series.
Years
1991
1992
193
1994
1995
1996
Sales Sh.
820
950
1000
950
900
1050
(000’s)
In this graph, false base line is required. When the fluctuations in a variable are relatively
small to its size then a definite break in the scale is shown between zero and the next number.
in this case, instead of showing the entire scale from zero to t
he highest value involved. Only as much is shown as is necessary for the purpose. The
portion which lies between zero and the lowest value of the variable is left out. This method
Sales (shs 000's)
is termed as False Line Approach Showing time series graphs.
1200
1000
800
600
400
200
0
East
1991
1992
1993
1994
1995
1996
Years
- Economists and businessmen have the task of making estimates about the future so that they
can be able to plan for various things such as:
- Sales
- Production
- Food supply
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- Jobs for the people
- Technology needs etc
- However, the step in making these estimates encompasses gathering information from the
past, which means that one deals with statistical data collected, observed, or recorded at
successive intervals of time. Such data are generally referred to as time series.
- Thus, when numerical data is observed at different points of time creates a set of
observations known as time series. Different points of time means over 5 years, 10 yrs, 20 yrs
etc.
- suppose production sales, exports, imports etc is observed at different points of time, say
over 5 or 10n yrs, the set of observations formed constitutes time series. Hence in the analysis
of time series time is very important because variable is related to time.
NB: time series refer to statistical data arranged chronologically, over successive increments,
in order of their occurrence etc.
Example: the data below shows sales of Radios by a firm in ‘000’ units:
Year
sale of Radios (000)
1999
40
2000
42
2001
47
2002
41
2003
43
2004
48
2005
65
2006
42
-
Observing the above series reveal that generally the sales have increased but for two
years a decline is also noticed.
7.2 Components of a time series
- The statistical analysis the effect of the various forces on data under 4 broad
categories.
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1. Secular movements: which refer to those changes that have occurred as a result of
general tendency of the data occurred or occurred. Secular trends do not include short
range movements but rather steady movements over along time. They are attributable
to factors such as population change, technological progress, large scale shift in
consumer tastes etc all which would lead to rising or falling trends in prices,
production, sales, incomes, employment, demand for food, clothing, shelter, discovery
and exhaustion of natural resources, mass production methods, improvement in
business organization etc. they cause, major growth or decline in time series.
-
sometimes a growth in one series involves a decline in another e.g. the displacement
of skin clothes by cotton clothes, better medical services have reduced death rates but
then contributed to rise in birth rates etc.
-
Also some series increase slowly and some increase fast. Others decrease at varying
rates; some remain relatively constant for long periods of time etc.
2. Seasonal variation: Which concerns changes that have taken place during a period of
12 months as a result of change in climate, weather conditions, festivals etc? They are
periodic movements in business activity occurring regularly every year as a result of
the nature of year itself. The variations repeat during a period of 1 year hence they can
be predicted fairly accurately factors known to cause seasonal variations include (a)
Climate and weather conditions e.g. which lead to climate or climate DD for woolen
clothes, hot drinks, cold drinks, planting season, harvest season etc. (b) customs,
traditions and habits e.g. Christmas leading to large for clothes, wheat flour, showers,
etc, money (withdrawals, etc.
3. cyclical variations:- which concern changes that have taken place as a result of booms
and deforestations. They are recurrent variations that cast longer than one year are
regular neither in amputable nor in length. Time series, mostly in economics and
businesses, fall under this category. They are known as business cycles which have
four phases:
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1. Prosperity
2. decline
3. deforestation
4. recovery
1
2
4
3
4
3
4. Irregular variations: also called erratic, accidental or random. This category concerns
changes that have taken place as a result of such forces that could be predicted like
floods, earthquakes, famines etc. these business variations do not recur in a definite
pattern. These variations include: all those others except those particularly scanter
trend, seasonal and cyclical variations have certain systematic movements e.g. sudden
fall in DD or rafoid technological movements can be included in this category.
NB: that the four variations explained above are also known as the components of time series.
Each of these components can be measured. However for this course we shall only measure
secular variations (trend).
Measuring trend
This is the determination and presentation of the direction which any long term series takes
i.e. is it growing or declining. Key reasons for measuring trend include:
1. To find trend characteristics about a given variable eg comparing the growth
of textile sector in Kenya with that of other countries, the growth of textile
sector in Kenya with that of the whole country.
2. To eliminator trend so as to study other components of time series such as
seasonal, cyclical and irregular variations.
Methods of measuring trend
1. Free hand smoothing or the graphic method
2. semi averages method
3. moving averages
4. Least squares method.
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1. Free hand smoothing/ Graphic method
It’s the simplest method of studying trend. The following procedure is followed
(a) Plot the time series on the graph.
(b) Examine the direction of the trend based on the plotted dots.
(c) Draw a straight line which will best fit to the data according to person’s judgment.
The line shows the direction of the trend.
Example: Fit a trend line to the following data by the free hand method
Year
production of steel (in millions)
1990
20
1991
22
1992
24
1993
21
1994
23
1995
25
1996
23
1997
26
1998
25
Fitting by free hand method
The trend line can be
extended to predict
future values. But
since the free hand
curve fitting is too
subjective, this
method should not be
used for predictions
Trench line
25
24
Production
23
22
21
Actual data
20
92
93
94 95
Years
NG’ANG’A S. I. 15TH DEC 2009
91
96
97
98
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The trend like can be extended to predict future values. But since the free hand curve fitting is
too subjective, this method should not be used for predictions
7.3 Method of semi averages
- When this method is used, the given data is divided into two parts, preferably with the same
No. of years.
- Then SAM of each part is taken/ calculated so that two points are obtained which are
plotted at the mid point of the class interval covered by the respective part and then the two
points are joined by a straight line. This straight line gives the needed trend line. The line can
be extended gives or gives to get intermediate value or predict future values.
Example: fit trend line to the following data by a method of semi – averages.
Year
sales of firm A (in thousand units)
1992
102
1993
105
1994
114
1995
110
1996
108
1997
116
1998
112
NB: since 7 years are given, the middle year shall be left out and an average of the first three
years be obtained. The average of first three years is;
102 + 195 +114 = 321 = 107
3
3
Average of last three years is
108 + 116 +112 = 336 = 112
3
3
Thus two points 107 and 112 will be gotten and plotted correspondingly to their respective
middle years i.e. 1993 and 1997. Plotting these points we get the needed trend line. The line
can be extended to predict or determine intermediate values.
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135
Trend line
130
Sales
125
120
115
110
Actual
data
105
100
1992
93
94
95
96
97
Years
98
9
8
99
00
NB: Where there even Number of years, two equal parts can easily be obtained. Hence, given
the years 1990, 91,92,93, 94, 95, 96, 97, the first part of years would be 90, 91, 92, 93 and the
second part would be 94, 96, 97, 98. The centering of the averages each part would be
between 91 and 92 for 1st part and 96 and 97 for second part.
Procedure:
1. Plot the actual data
2. divide data in two parts
3. get and plot the averages
4. connect the two points to get the trend line
3. Method of moving averages:
When trend is determined by this method, the average value for a number of years is secured.
It is therefore necessary to select the period of the moving average such as three yearly
moving averages; five yearly moving averages, 8 yearly moving average etc. the length of the
cycle determines the period of moving average. For instance, a 3 yearly moving average shall
be computes as follow;
a+ b + c; b + c + d ; c + d+ e; d + e + f ; etc
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3
3
3
3
Five yearly moving averages shall be;
a+ b + c +d + e; b + c + d +e +f ; c +d + e +f + g ; etc
3
3
3
Examples:
(a) Using a three year moving averages determine the trend and short term fluctuations.
Plot the original and trend values on the same graph paper
Year
production (‘000’ tones)
1989
21
1990
22
1991
23
1992
25
1993
24
1994
22
1995
25
1996
26
1997
27
1998
26
Graph of original trend values of products by moving averages methods
Production
30
28
Trend line
26
24
NG’ANG’A
S. I. 15TH DEC 2009
22
20
Actual data
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Sol.
Year
prod.
3 yr moving
(ooo tons)
3 yr moving
short term
(fluctuation y1 – yc)
totals
averages
1989 21
-
-
-
1990 22
66
22.00
0
1991 23
70
23.33
-0.33
1992 25
72
24.00
+1.00
1993 24
71
23.67
+ 0.33
1994 22
71
23.67
- 1.67
1995 25
73
24.33
+ 0.67
1996 26
78
26.00
0
1997 27
79
26.33
+ 0.67
1998 26
-
-
-
Graph of original and trend values of product by moving averages method
b) Calculate 5 yearly moving averages for the following data.
Year
product
1986
105
1987
107
1988
109
1989
112
1990
114
1991
116
1992
118
1993
121
1994
123
1995
124
1996
125
1997
127
1998
129
Sol.
Year
prod.
5Yrly totals
NG’ANG’A S. I. 15TH DEC 2009
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1986
105
-
-
1987
107
-
-
1988
109
547
109.4
1989
112
558
111.6
1990
114
560
113.8
1991
116
581
116.2
1992
118
592
118.4
1993
121
602
120.4
1994
123
611
122.2
1995
124
620
124.0
1996
125
628
125.0
1997
127
-
-
1998
129
-
-
Then, plot both the actual and moving averages data.
Even period of moving average:
If the moving average is an even period average say, 4 yearly or 6 yearly, the moving total
and moving averages are placed at the centre of the time span from which they are computed.
This placement is inconvenient since the moving averages so placed does not coincide with
the original time period. The two would then be synchronized i.e. moving averages and the
original data by process called centering which consists of taking a two period moving
average of moving averages.
Example:
Year
value
1985
12
1986
25
1987
39
1988
54
1989
70
1990
87
1991
105
1992
100
1993
82
1994
65
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1995
49
1996
34
1997
20
1998
7
Sol.
Yr
value 4y.M.T
1985
12
-
-
-
1986
25
-
-
-
130
32.5
1987
39
54.75
70.75
84.75
92.00
90.75
81.00
65.75
57.5
49
49.75
168
1996
74.0
65
230
1995
88.0
82
29.6
1994
93.5
100
352
1993
90.5
105
374
1992
79.0
87
362
1991
62.5
70
316
1990
47.0
54
250
1989
42.0
34
34.75
110
1997
20
1998
7
4y centered M.A
39.75
188
1988
4 M.A.
27.5
Then plot the actual data and the 4 yrly centered moving averages.
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Exercise
From the following data compute 3 yrly, 5 yrly and 7 yrly moving averages and plot them on
the graph paper with the actual data.
Year: 1984 85 86 87 88 89 90 91 92 93 94 95 96 97 97
+2
Yr
+1 0 -2 -1 +2 +1 0 -2 -1 +2 +1 0 -2 -1
fluctuations
3 M.A
5 M.A
7 M. A.
1984
+2
-
-
-
85
+1
+1.00
-
-
86
0
-0.33
0
-
87
-2
-1.00
0
+0.43
88
-1
-0.33
0
+0.14
89
+2
+0.67
0
-0.28
90
+1
+1.00
0
-0.43
91
0
-0.33
0
-0.14
92
-2
-1.00
0
-0.43
93
-1
-0.33
0
+0.14
94
+2
+0.67
0
-0.27
95
+1
+1.00
0
-0.43
96
0
-0.33
0
-
97
-2
-1.00
-
-
98
-1
-
-
-
7.4 Method of least squares:
This method is most widely used in practice. It’s a mathematical method and with its help a
trend line is fitted to the data in such a manner that the following two conditions are satisfied:
1. ∑ (Y – Yc) = 0 sum of deviations of the actual values of Y and the confronted values
of Y is zero.
2. ∑ (Y – Yc) 2 is least: i.e. the sum of the squares of the deviations of the actual and
computed value is least from this line and hence the name method of least squares.
The line of best fit.
This method of least squares is used to fit straight trend line or a paragraphed trend. The
straight line is represented by the equation Yc = a + bx
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Where Yc = The trend/ confronted values
a= Y Intercept
b= slope/ gradient of the trendline
x= the variable which represents time
In order to determine the values of the constants a and b the following two normal equations
are to be solved.
∑Y = Na + B∑ X……………….(i)
∑XY = a∑X +b∑x2…………….(ii)
Where N represents the number of years for which data are given. Two approaches;
Approach i
The variable X can be measured from any point of time in origin such as the first year. But
calculations are very simplified when the mid-point in time is taken as the origin because in
that case the –ve values in the half of the series balance out the +ve values in the 2 nd half so
that ∑ X = 0. The variable is measured as a deviation from its mean.
Since ∑X = 0
∑ Y = Na
the value of a and b can be determined easily.
∑ XY = b∑X2
Since ∑ Y = Na therefore a = ∑Y/ N
∑XY = b ∑x2 therefore b = ∑XY/
∑x2
Example:
Below are data of figures of production in tones from a factory.
Year:
Production:
1992
1993
1994
1995
1996
1997
1998
80
90
92
83
94
99
92
Required: (i) Fit a straight line trend to these figures
(ii) Plot these figures on a graph and show the trend line.
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Solution (i) NB:1995 is taken as origin
Year
production
X
XY
X2
Yc = a + bx
Trend values
1992
80
-3
-240
9
84
1993
90
-2
-180
4
86
1994
92
-1
-92
1
88
1995
83
0
0
0
90
1996
94
+1
+94
1
92
1997
99
+2
+198
4
94
1998
92
+3
+276
9
96
N= 7
∑Y = 630
∑X = 0
∑XY = 56
∑XY2 = 28
∑Yc = 630
The equation of a straight line trend is Yc = a + bx
Since ∑X = 0
a= ∑Y/N; b = ∑XY/∑X2
But: ∑Y = 630; N = 7; ∑XY = 56; ∑X2 = 8
a= 630/ 7 = 90; b = 56/28 = 2
Hence the equation of the straight line trend = Yc = 90 + 2x
Thus trend values (yc) for each year would
1992: Yc = 90 + 2 (-3)
= 90 + -6
= 90- 6 = 84
1993: Yc = 90 +2 (-2)
= 90 + - 4
= 86 etc
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Linear trend method by least squares
100
95
Trend
lines
90
85
Actual
data
80
75
1992 93
95
94
96
97
98
Suppose you took 1992 as the origin, the values of X will all be +ve after zero of 1992. ∑X =
218 not zero. Hence;
630 = 7a + 21b
1946 = 21 a + 91 b
The equation fitting trend line will change to Yc = 84 + 2x. the difference in origin. However
the trend values will be the same.
Example:
(a) Fit a straight line trend for the following series
(b) Estimate the value for 1999
(c) What is the monthly increase in production?
Year:
1992
1993
1994
1995
1996
1997
1998
Production:
125
128
133
135
140
141
143
Sol
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Year production
X
XY
X2
1992 125
-3
-375
9
1993 128
-2
-256
4
1994 133
-1
-133
1
1995 135
0
0
0
1996 140
+1
+140
1
1997 141
+2
+282
4
1998 143
+3
+429
9
∑X=0
∑XY=87
∑X2=28
N=7
∑Y=945
Yc= a+bx
The equation of the straight line trend is
Y = a +bx
Since ∑X = 0 a = ∑Y/N = 945/7 = 135
b= ∑XY/∑X = 87/28= 3.107
Hence: Yc = 135 + 3.107X
(b) For 1999 X will be + 4 thus Y 1999 = 135 + 3.107 (4)
= 147.428 tons
(c) Given the equation Yc = a+bx; b is the rate of change (in production) and in our trend
equation Yc = 135 + 3.107x the (annual) rate of change is 3.107 million tons. This monthly
increase would be given by 3.107/12 = 0.25 tons.
Example: using the method of least squares fit a trend line to the following data and find the
trend values and short term fluctuation.
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Year:
1990
1991
1992
1993
1994
1995
1996
1997
1998
Value:
232
226
220
180
190
168
162
152
144
X
XY
X2
Yc
short term -
Sol.
Year
value(Y)
Fluctuation (Y – Yc)
1990
232
-4
-928
16
232.8
-0.8
1991
226
-3
-678
9
221.1
+4.9
1992
220
-2
-440
4
209.4
+10.6
1993
180
-1
-180
1
197.7
-17.7
1994
190
0
0
0
186.0
+4.0
1995
168
+1
+168
1
174.3
+6.3
1996
162
+2
+324
4
162.6
-0.6
1997
152
+3
+436
9
150.9
+1.1
1998
144
+4
+576
16
139.2
4.8
N=9
∑Y=1674
∑x=0 ∑XY=702 ∑x2=60 ∑Yc =1674
Y=a+bx
a=
∑Y/N = 1674/9 =186
b= ∑xY/∑x2 = -702/60 = -11.7
Hence Y = 186
11.7x
Y1990 = 186 -11.7(-4) = 186 + 46.8 = 232.8 = 232.8
Approach ii
To obtain the regression equation y = a + bx values of a and b are obtained from
a=
y b x
n
b=
n xy x y
n x 2 ( x) 2
Which are derived from solving the simultaneous equations
an + bx = y…………………………………….(i)
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ax + bx2 = xy ………………………………...(ii)
The data obtained for x & y is tabulate to get the sums x, y, xy and x2 as follows.
x
y
x2
Xy
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
x
y
x
xy
Note:
(i)
The line obtained can be extended and used to predict (forecast).
(ii)
If the gradient or slope is negative (i.e. b is positive) the two variables x and y have
a positive relationship and y increases as x increases.
(iii)
If the gradient (slopes) in negative (b is negative) the two variables have a negative
relationship and y decreases as x increases.
Example
Example
Period
1
2
3
4
5
6
7
Actual demand
6
4
8
7
4
7
-
Forecasts
6
5
6
6.25
7.25
-
Forecast demand for period F7
From the example above, when the period stands for the independent variable x and actual
demand stands for the dependent variable y, the values a and b and the regression equation
are calculated as follows.
x
1
2
3
4
5
6
x=21
y
6
4
8
7
4
7
y=36
NG’ANG’A S. I. 15TH DEC 2009
x2
1
4
9
16
25
36
x2 =91
xy
6
8
24
28
20
42
xy =128
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IRD 101: QUANTITATIVE SKILLS I
From which,
b=
n xy x y
n x 2 ( x) 2
b
768 756
12
6(128) (21)(36)
=
=
= 0.114
546 441 105
6(91) (21)( 21)
a=
y b x
n
a
36 0.114(21)
36 2.394
33.606
=
=
= 5.601
6
6
6
From which,
This gives a regression Equation of
Y = a + bx
Y = 5.601 +0.114x.
The regression equation suggests a demand that increases slightly with increase in time and
may lead to a forecast for period 7 of;
Y = 5.601 + 0.114(7) = 6.40
7.4Revision questions
1.Explain four factors that might lead to a random variation in Kenya tea exports over a
period of one year.
4mks
2. Indicate the time series movement you will associate with the following events
i. Resistant of malaria parasite to quinine
1mk
ii. Presidential and parliamentary elections in Kenya after every five years
1mk
iii. Fire outbreak at petrol filling stations
1mk
3.
a) Explain why a supermarket may want to analyze the time series data it generates in its
sales
4mks
b.) The following time series data shows the annual production of sugar at a local sugar
processing firm in western Kenya or the years 1997-2005
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Year
1997
1998
1999
2000
2001
2002
2003
2004
2005
Production (000 tonnes )
20
22
24
21
23
25
23
26
25
Required
i.) Determine the trend lien using the method of centered moving average of order 4
6mks
ii.) on the same axes, plot the time series data and the trend line obtained in i.) above
4mks
iii.) Using the graph obtain estimates of sugar production for the firm for the years 2006 and
2007
4mks
4.With which characteristic movement of a time series would you mainly associate each of
the following independent cases or situations?
i.) An increase in sales for a supermarket during Christmas
1mk
ii.) The decline in the spread of HIV/AIDS in Kenya
1mk
iii.) The university lecturers union strike
1mk
iv.) A continually increasing demand for new information technology
1mk
v.) The heavy rains that caused floods in Kenya in November/December 2006
1mk
5. a) Explain the meaning of the following terms;
i.) Time series
ii.) Raw data
iii.) Median
iv.) Sample
12mks
b.) The table below gives the production figures (in 000 of tonnes) of ceramic goods for
2006.
Month
Jan Feb Mar Apr May Jun July Aug Sep Oct Nov Dec
Production 335 325 310 354 360 338 333 270 375 395 415 373
iv.) Plot the monthly production figures on a graph
5mks
v.) Which time series factor seems to influence the production of ceramic
goods?
3mks
vi.) Use the graph to estimate the production figures for the ceramic good in
February 2007.
4mks
6. (a) List and illustrate four components of a time series
NG’ANG’A S. I. 15TH DEC 2009
(4marks)
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IRD 101: QUANTITATIVE SKILLS I
(b) A firm has recorded the following sale data in (000)
Year
Time
Sales
1998
1
4
1999
2
2
2000
3
8
2001
4
12
2002
5
20
2003
6
18
2004
7
16
2005
8
30
2006
9
40
2007
10
36
2008
11
44
From the above data;
(i) Plot a scatter graph of time against sales
(2marks)
(ii) Find a trend line using the least square method
(3marks)
(iii) Plot the least square trend line on the same graph (i) above
(2marks)
(iv) Using the least square trend line, determine the sales forecast for year 2009 and
2010
(2marks)
7. The table below gives the production figures (in 000 of tones) of ceramic goods for 2006.
Month
Jan Feb Mar Apr May Jun July Aug Sep Oct Nov Dec
Production 335 325 310 354 360 338 333 270 375 395 415 373
i.) Plot the monthly production figures on a graph
5mks
ii.) Which time series factor seems to influence the production of ceramic
goods?
3mks
iii.) Use semi averages to plot a trend line on the graph and use it to estimate the
production figures for the ceramic good in February 2007.
4mks
iv.) Use the least squares method to plot a trend line on the same graph and
estimate production figures for Feb 2007.
v.) Which of the two methods do you think is more accurate and why?
The table below gives the production figures (in 000 of tones) of ceramic goods for 2006.
Month
Jan Feb Mar Apr May Jun July Aug Sep Oct Nov Dec
Production 335 325 310 354 360 338 333 270 375 395 415 373
i.) Plot the monthly production figures on a graph
5mks
ii.) Which time series factor seems to influence the production of ceramic
goods?
3mks
iii.) Use semi averages to plot a trend line on the graph and use it to estimate the
production figures for the ceramic good in February 2007.
4mks
iv.) Use the least squares method to plot a trend line on the same graph and
estimate production figures for Feb 2007.
v.) Which of the two methods do you think is more accurate and why?
8. a) Explain the following components in time series analysis
i.)
Seasonal variation
ii.)
Random variation
iii.)
Cyclic variation
NG’ANG’A S. I. 15TH DEC 2009
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IRD 101: QUANTITATIVE SKILLS I
b.) A firm has recorded the following levels of production in the last seven years.
Year
2002
2003
2004
2005
2006
2007
2008
Production 125
128
133
135
140
141
143
Required
i.)
Using the three years moving average calculate the projected level of production
for the year 2009.
ii.)
Plot a scatter graph for the data
iii.)
Fit a straight line trend for the data.
iv.)
Use the trendline to predict production levels for the year 2009.
v.)
Comment on the production level estimated by the moving average methods and
the trendline for the year 2009.
9. The annual DAP fertilizer consumption in thousands of tonnes during 1995-2001 in
Lukuyani Division was recorded as given below.
Year
1995
1996
1997
1998
1999
2000
2001
Consumption 50
56
60
68
70
75
78
(‘000) tonnes
a.) i.) Use the semi average method to fit the trend line and use it to estimate the consumption
in 2005.
12mks
ii.) Indicate two major disadvantages of this method
4mks
10. a) Explain
i.) The meaning of time series analysis
2mks
ii.) The importance of time series analysis
2mks
b)List and illustrate four components of time series
(4mks)
c)A firm has recorded the following sales data in (000)
Year
Time
Sales
1998
1
4
1999
2
2
2000
3
8
2001
4
12
2002
5
20
2003
6
18
2004
7
16
2005
8
30
2006
9
40
2007
10
36
2008
11
44
Form the data above
i.)
Plot a scatter graph of time against sales
3mks
NG’ANG’A S. I. 15TH DEC 2009
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IRD 101: QUANTITATIVE SKILLS I
ii.)
Use semi averages to plot a trend line on the scatter plot in i.) above
2mks
iii.)
Using the trend line forecast sales for the year 2009 and 2010
2mks
iv.)
Find the trend line using the least squares methods
3mks
v.)
Plot the least square trend line on the same graph i.) above
2mks
vi.)
Using the least squares trend lines determine the sales forecast for year
2009 and 2010
vii.)
2mks
Comment on the forecast by semi average trend line iii.) Above and those
by least squares trend line (vi.).
NG’ANG’A S. I. 15TH DEC 2009
2mks
Page 167
IRD 101: QUANTITATIVE SKILLS I
FORMULAE COMMOINLY USED IN IRD 101
1.
Arithmetic mean
2.
Median of grouped data
3.
Mode of grouped data
4.
Standard deviation
5.
Coefficient of variation
6.
Regression line of y on x
Y= a+bx
xf
f
N
f 1
2
c
L1
f median
1
L1
c
1 2
x f
f
xf
f
100 X S tan dard Deviation
Mean
n xy x y
b
n x 2 x 2
a
NG’ANG’A S. I. 15TH DEC 2009
2
2
y b x
n
n
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IRD 101: QUANTITATIVE SKILLS I
NG’ANG’A S. I. 15TH DEC 2009
Page 169
