Download Formulas for Central tendency Dispersion Index

MEASURES OF CENTRAL TENDENCY
DATA TYPES
Type 1
X
10
12
15
(Individual items)
Type 2
X f
20 4
25 6
30 2
(Discrete series)
Type 3 (Group data)
There are two different group data's
(1)Continuous group data
X
F
10------20 4
20------30 6
30------40 2
(2)Discontinuous group data
Group
F
10------19 4
20------29 6
30------39 2
Important Notes when you solve the questions
(1) When you solve Mode, Median, Quartiles, Deciles or Percentiles in Type 3, it must be in
continuous form. If it is discontinuous form you will convert into continuous form with the help of
Class Boundaries (C.B)
(2) When you calculate Median, Quartiles, Deciles or Percentiles in any data type, it must be in
arranged form
(3) When you calculate Median, Quartiles, Deciles or Percentiles in Type 2 or Type 3, you will make
the column of cumulative frequency (C.F) and put the value of "∑f " in the place of "n"
(4) When you calculate Mode, Median, Quartiles, Deciles or Percentiles in Type 2 and you have
continuous variable, then you will convert into Type 3 with the help of C.B and apply Type 3
formulas while Discrete Variables & Continuous Variables are
Discrete Variable
A variable which can assume only whole numbers is called discrete variable, e.g. No of students, etc.
Continuous Variable
A variable which can assume any value between the two specified intervals is called continuous
variable, e.g. Height, Weight, Wages etc.
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In Type 1
(I) Direct formula
ARITHMETIC MEAN (A.M or
In Type 2 & In Type 3
(I) Direct formula
A.M =
Where "n" is the total number of values
(II) Shortcut formula
Where "D= X – A", and "A" is any
arbitrary value
(II) Shortcut formula
A.M = A +
(III) Step deviation or coding formula
A.M = A +
(III) Step deviation or coding formula
Where "
value and
(i)
(ii)
(iii)
(iv)
", A is any arbitrary
is the class interval
WEIGHTED MEAN (
)
COMBINED MEAN (
)
Properties of A.M
Mean of a constant is constant itself
Sum of deviation from mean is always zero, i.e.
Sum of square of deviation from Mean is minimum, i.e.
If Y = aX + b, where "a" and "b" are constant, then
MEDIAN (
In Type 1 & In Type 2
In Type 3
Where
L = lower limit of the median group
h = class interval of the median group
f = frequency of the median group
c = C.f preceding the median class
MODE (
In Type 1 & In Type 2
Most repeated value
In Type 3
Where
L=lower value of the model class
h=class difference of the model class
fm=maximum frequency
f1=frequency above fm
f2=frequency blow fm
Relationship between A.M, G.M and H.M
A.M ≥ G.M ≥ H.M
Empirical relationship between Mean, Median and Mode
Mode = 3 Median – 2 Mean
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In Type 1
HARMONIC MEAN(H.M)
In Type 2 & In Type 3
GEOMETRIC MEAN(G.M)
In Type 1
In Type 2 & In Type 3
OR
QUARTILES(Q1,Q2 and Q3)
In Type 1 & In Type 2
In Type 3
Q1 is called lower quartile and Q3 is called upper quartile
DECILES(D1,D2,…..,D9)
In Type 1 & In Type 2
In Type 3
PERCENTILES(P1,P2,P3,……,P99)
In Type 1 & In Type 2
In Type 3
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MEASURES OF DISPERSION
DATA TYPES
Type 1
X
10
12
15
(Individual items)
Type 2
X f
20 4
25 6
30 2
(Discrete series)
Type 3 (Group data)
There are two different group data's
(1)Continuous group data
X
F
10------20 4
20------30 6
30------40 2
(2)Discontinuous group data
Group
F
10------19 4
20------29 6
30------39 2
Important Notes when you solve the questions
(1) When you solve Mode, Median, Quartiles, Deciles or Percentiles in Type 3, it must be in
continuous form. If it is discontinuous form you will convert into continuous form with the help of
Class Boundaries (C.B)
(2) When you calculate Median, Quartiles, Deciles or Percentiles in any data type, it must be in
arranged form
(3) When you calculate Median, Quartiles, Deciles or Percentiles in Type 2 or Type 3, you will make
the column of cumulative frequency (C.F) and put the value of "∑f " in the place of "n"
(4) When you calculate Mode, Median, Quartiles, Deciles or Percentiles in Type 2 and you have
continuous variable, then you will convert into Type 3 with the help of C.B and apply Type 3
formulas while Discrete Variables & Continuous Variables are
Discrete Variable
A variable which can assume only whole numbers is called discrete variable, e.g. No of students, etc.
Continuous Variable
A variable which can assume any value between the two specified interval is called continuous
variable, e.g. Height, Weight, Wages etc.
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RANGE & Coefficient of Range(In any type)
QUARTILE DEVIATION(Q.D) & Coefficient of Q.D(In any type)
MEAN DEVIATION(M.D) & Coefficient of M.D
In Type 1
In Type 2 & In Type 3
In Type 1
(i) Direct formula
(ii) Shortcut formula
VARIANCE(S2) & Coefficient of Variation(C.V)
In Type 2 & In Type 3
(i) Direct formula
(ii) Shortcut formula
(iii) Step deviation or coding formula (iii) Step deviation or coding formula
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STANDARD DEVIATION(S.D or S) & Coefficient of S.D
In Type 1
(i) Direct formula
(i) Direct formula
In Type 2 & In Type 3
(ii) Shortcut formula
(ii) Shortcut formula
(iii) Step deviation or coding formula (iii) Step deviation or coding formula
Moments about Mean (
In Type 1
First four moments about Mean
)
In Type 2 & In Type 3
First four moments about Mean
Raw Moments (
In Type 1
First four moments about Zero
First moment about zero is equal to A.M
In Type 2 & In Type 3
First four moments about Zero
First moment about zero is equal to A.M
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First four moments about any value
First four moments about any value
Using shortcut method
Using shortcut method
Using step deviation method
Using step deviation method
Relationship between moments about mean and raw moments
When you calculate raw moments with the help of step deviation or coding method
Moments Ratios
&
If b1 = 0, the distribution is symmetrical otherwise skewed
If b2 = 3, the distribution is normal or mesokurtic
If b2 < 3, the distribution is platykurtic
If b2 > 3, the distribution is leptokurtic
Sheppard's correction for moments
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(i)
(ii)
(iii)
Properties of Variance & S.D
The Variance and S.D is zero if all the observations have some constant value, i.e.
Var (a) = 0
&
S.D (a) = 0.
Where "a" is a constant
Variance and S.D do not change by change of origin, i.e.
Var(X + a) = Var(X)
or
Var(X – a) = Var(X)
S.D(X + a) = S.D(X)
or
S.D(X – a) = S.D(X)
Variance and S.D are affected by change of scale, i.e.
Var(aX) = a2Var(X)
(i)
Relationship between M.D, Q.D and S.D
,
,
(In any Type of data)
Pearson coefficient of Skewness
(ii)
Bowley's coefficient of Skewness
or
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INDEX NUMBERS
INDEX NUMBERS
UNWEIGHTED
WEIGHTED
FIXED
BASE
METHOD
ONE
COLUMN OF
PRICE
MORE THAN
ONE
COLUMNS
OF PRICE
CHAIN
BASE
METHOD
ONE
COLUMN OF
PRICE
MORE THAN
ONE
COLUMNS
OF PRICE
LASPEYRE'S METHOD
PAASCHE'S METHOD
PRICE
RELATIVES
ARE GIVEN
FISHER'S METHOD
MARSHAL'S METHOD
AGGRIGATIVE EXPENDITURE
METHOD
FAMILY BUDGET METHOD
Price relative
Link relative
Simple aggregative price index number
Simple average of price relatives
Laspeyre's price index number (also called base year weighted price index number)
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Paasche's price index number (also called current year weighted price index number)
Fisher's price index number
Marshall's price index number
Aggregative Expenditure method
Family Budget method
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PROBABILITY
Sample Space when a coin tossed
Sample Space when two coins are tossed
Sample Space when three coins are tossed
Sample Space when a die
Sample Space when two dice are rolled
CARDS
Red Cards
HEART ♥ DIAMOND ♦
♥2
♦2
♥3
♦3
♥4
♦4
♥5
♦5
♥6
♦6
♥7
♦7
♥8
♦8
♥9
♦9
♥10
♦10
♥K
♦K
♥Q
♦Q
♥J
♦J
♥Ace
♦Ace
Black Cards
SPADE ♠ CLUB ♣
♠2
♣2
♠3
♣3
♠4
♣4
♠5
♣5
♠6
♣6
♠7
♣7
♠8
♣8
♠9
♣9
♠10
♣10
♠K
♣K
♠Q
♣Q
♠J
♣J
♠Ace
♣Ace
Total Cards = 52
Black Cards = Red Cards = 26
Spade Cards = Club Cards = Diamond Cards = Heart Cards = 13
2's = 3's = 4's = 5's = 6's = 7's = 8's = 9's = 10's = J's = Q's = K's = A's = 4
Picture Cards = 12
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Factorial
Permutation
Combination
--------------------------------------------------------------------------OR
+
ᵁ
∩
AND
-----------------------------------------------------------------------------At least 10 mean
10, 9, 8, 7 …
At most 20 mean 20, 19, 18, 17…
------------------------------------------------------------------------------P (AUB) = P (A) + P (B)
when A and B are mutually exclusive events
P (AUB) = P (A) + P (B) – P (A∩B)
when A and B are not mutually exclusive events
P (A∩B) = P (A) P (B)
when A and B are independent events
P (A∩B) = P (A) P (B/A)
when A and B are dependent events
= P (B) P (A/B)
-----------------------------------------------------------------------------------Conditional Probability
---------------------------------------------------------------------------------------
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