Download QUANTITATIVE-APTITUDE

QUANTITATIVE APTITUDE
PERMUTATION AND COMBINATION
Permutation is used where arrangements of things is relevant for e.g. arrangement in a
queue, words, no’s, sitting arrangements etc. Permutation of n things taken 𝓇 at a time is 𝑛𝑃𝑟
𝑛
𝑛!
𝑃𝑟 = (𝑛−𝑟)!
𝑛
𝑃𝑟 has r factors.
For eg : 9𝑃3 has 3 factors i.e. 9𝑃3 = 9 × 8 × 7
10
𝑃2 has 2 factors i.e. 10𝑃2 = 10 × 9
Combination is used where selection is relevant for eg : Selection of a team, committee etc.
Combination of n things taken r at a time is 𝑛𝐶𝑟
𝑛
𝑛
𝑛!
𝐶𝑟 = 𝑟 !(𝑛−𝑟)! =
10
𝐶3 =
10
𝑃3
3!
=
𝑃𝑟
𝑟!
10×9×8
3×2×1
Note : Fundamental Principles of counting
a. Multiplication Rule – [AND] – When things are within same arrangement and all things
are required then choices are multiplied.
b. Addition Rule – [OR] – When there are different arrangements and either of the
arrangement can work then choices are added.
Factorial
0! =1
1!=1
2!=2
3!=6
4 ! = 24
5 ! = 120
6 ! = 720
7 ! = 5040
8 ! = 40320
Note : 1) Factorial cannot be calculated for a –ve no.
2) Factorial cannot be calculated for a fractional no. Example – 7
Note : 1) Sum of all 4 digit no. which can be formed using a,b,c,d is (a+b+c+d) X 6666
2) Sum of all 3 digit no. which can be formed using a,b,c is (a+b+c) X 222
Circular Permutation
1) n things can be arranged in a line in n ! ways.
2) n things can be arranged in a circle in (n – 1) ! ways.
1
3) n persons can be arranged in a circle so that no person has same 2 neighbors in 2 (𝑛 − 1)!
ways.
1
4) OR no, of necklaces that can be formed with n beads of different colors in2 (𝑛 − 1)! ways.
Permutation with Restriction : In case of restriction in an arrangement first of all position
with restriction should be considered.
Note : 1) No. of ways in which n things can be arranged so that 2 particular things are never
together = total ways – ways in which 2 particular things are always together.
2) Above formula cannot be applied for more than 2 things.
Properties of Combination :
1) 𝑛𝐶𝑜 = 𝑛𝐶𝑛 = 1
2) 𝑛𝐶𝑟 = 𝑛𝐶𝑛−𝑟
3) 𝑛𝐶𝑟 = 𝑛−1𝐶𝑟−1 + 𝑛−1𝐶𝑟
𝑛
𝐶𝑟 (𝑡𝑜𝑡𝑎𝑙 𝑤𝑎𝑦𝑠), 𝑛−1𝐶𝑟−1 (𝑜𝑛𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑡ℎ𝑖𝑛𝑔 𝑖𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑), 𝑛−1𝐶𝑟 (𝑜𝑛𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑡ℎ𝑖𝑛𝑔 𝑛𝑜𝑡 𝑠𝑒𝑙𝑒𝑐𝑡𝑒
Note : Permutation = Selection with arrangement
Combination = Selection without arrangement
Properties of Permutation
𝑛
𝑃𝑟 = 𝑛−1𝑃𝑟 + 𝑟. 𝑛−1𝑃𝑟−1
SEQUENCE AND SERIES
Sequence – 1, 3, 5, 7, 9, . . . . . . .
2, 4, 8, 16, 32, . . . . . .
Series – 1 + 3 + 5 + 7 + 9 .. . . . . . .
2 + 4 + 8 + 16 + 32 . . . . . . . .
ARITHMETIC PROGRESSION (A.P.)
a, (a+d), (a+2d), (a+3d), . . . . . . . .
nth term of A.P.
𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
𝑤ℎ𝑒𝑟𝑒 𝑎 = 𝐼 𝑡𝑒𝑟𝑚,
Sum of n terms
𝑛
𝑛
𝑆𝑛 = 2 [2𝑎 + (𝑛 − 1)𝑑] = 2 [𝑎 + 𝑙]
𝑑 = 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
Where 𝑙= 𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
Arithmetic Mean (A.M.)
A.M. between 2 no.s a & b is
𝑎+𝑏
2
and if n A.M. are inserted between 2 no.s a & b then a, A₁, A₂, A₃, A₄. . . . . . . . 𝐴𝑛 , b are in
A.P.
Note : 1) In case of A.P., T₂ - T₁ = T₃ - T₂ = d
2) T₁ + T₅ = 2T₃ or T₂ + T₁₀ = 2T₆ and so on.
3) If 3 terms are in A.P., then terms are (a-d), a, (a+d)
4) If 4 terms are in A.P., then terms are (a-3d), (a-d), (a+d), (a+3d)
5) If 4 terms are in A.P., then terms are (a-2d), (a-d), a, (a+d), (a+2d)
Note : 1) Sum of first n natural no.s
1+2+3+4.........n=
𝑛(𝑛+1)
2
2) Sum of squares of first n natural no.s
1²+2²+3²+4² . . . . . . . . . . . n² =
𝑛(𝑛+1)(2𝑛+1)
6
3) Sum of cubes of first n natural no.s
1³+2³+3³+4³+ . . . . . . . . . . . n³= [
𝑛(𝑛+1)
2
]²
Note : When d= -ve, we get two answers for n. In such case, first value of n better
option.
GEOMETRIC PROGRESSION (G.P.)
𝑎, 𝑎𝛾, 𝑎𝛾 2 , 𝑎𝛾 3 , … … … … …
nth term of G.P.
𝑇𝑛 = 𝑎𝑟 𝑛−1
Sum of n term of G.P.
𝑆𝑛 =
𝑆𝑛 =
𝑎(𝑟 𝑛 −1)
, 𝐼𝑓 𝑟 > 1
𝑟−1
𝑎(1−𝑟 𝑛 )
, 𝐼𝑓 𝑟 < 1
1−𝑟
Sum of Infinity
𝑎
𝑆∞ = 1−𝑟
, 𝐼𝑓 𝑟 < 1
Geometric Mean (G.M.)
G.M. of 2 no.s a & b is √𝑎𝑏
If n G.M. are inserted between a & b then a, G₁, G₂, G₃, . . . . . . . . . . 𝐺𝑛 , b are in G.P.
𝑇
𝑇
Note : 1) In G.P. 𝑇2 = 𝑇3 = 𝑟
1
2
2) T₁ + T₉ = (T₅)²
𝑎
3) 3 terms in a G.P. are 𝑟 , 𝑎, 𝑎𝑟
𝑎
𝑎
4) 4 terms in G.P. are 𝑟 3 , 𝑟 , 𝑎𝑟, 𝑎𝑟 3
𝑎
𝑎
5) 5 terms in a G.P. are 𝑟 2 , 𝑟 , 𝑎𝑟, 𝑎𝑟 2
Note : 1) For 2 equal no.s
A.M. = G.M. = H.M.
2) For 2 unequal no.s
A.M. > G.M. > H.M.
3) For 2 no.s
A.M. ≥ G.M. ≥ H.M
RATIO, PROPORTION, INDICES & LOGARITHMS
RATIO – 3 : 4 ≠ 4 : 3
Ratio has no unit
𝑎
𝑏
1) Inverse Ratio of 𝑏 𝑖𝑠 𝑎
2) Compound Ratio of
𝑎
𝑎
𝑏
𝑐
𝑐
𝑖𝑠 𝑑 & 𝑑 𝑖𝑠
𝑎 2
3) Duplicate ratio of 𝑏 𝑖𝑠 (𝑏)
𝑎
𝑏
𝑐
×𝑑
𝑎 3
𝑎
4) Triplicate ratio of 𝑏 𝑖𝑠 (𝑏)
1
𝑎
𝑎 2
5) Sub duplicate ratio of 𝑏 𝑖𝑠 (𝑏)
1
6) Sub triplicate ratio of
𝑎
𝑎 3
𝑖𝑠
(
)
𝑏
𝑏
7) Continued ratio is ratio of more than 2 things, 2 : 3 : 4
8) Ratio of 2 integers is said to be commensurable and ratio of 2 non-integers is
incommensurable.
2
:. 3 is commensurable and
√2
√3
is incommensurable.
Proportion – An equality of 2 ratios is called proportion.
𝑎
𝑐
E.g. : 𝑏 = 𝑑 then a, b, c, d are proportionate
i.e. a : b = c : d or a : b : : c : d
Product of Extreme = Product of mean
𝑎×𝑑 =𝑏×𝑐
Mean proportion of 2 no.s a, b is √𝑎 × 𝑏
Properties of Proportion :
𝑎
𝑐
1) 𝑏 = 𝑑 ⇒ 𝑎 × 𝑑 = 𝑏 × 𝑐
𝑎
𝑐
𝑏
𝑑
𝑐
𝑎
𝑐
𝑏
𝑐
𝑐
𝑑
𝑎+𝑏
𝑐+𝑑
𝑐
𝑏
𝑎−𝑏
𝑎
𝑐
𝑏
𝑎+𝑏
𝑎
𝑐
𝑒
2) 𝑏 = 𝑑 ⇒ 𝑎 =
𝑎
3) 𝑏 = 𝑑 ⇒
𝑎
4) 𝑏 = 𝑑 ⇒
𝑎
5) 𝑏 = 𝑑 ⇒
=
=
=
(𝐼𝑛𝑣𝑒𝑟𝑡𝑒𝑛𝑑𝑜)
(𝐴𝑙𝑡𝑒𝑟𝑛𝑒𝑛𝑑𝑜)
𝑑
𝑐−𝑑
𝑑
𝑐+𝑑
6) 𝑏 = 𝑑 ⇒ 𝑎−𝑏 = 𝑐−𝑑
(𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑑𝑜)
(𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑜)
(𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑑𝑜 & 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑜)
𝑎+𝑐+𝑒
(𝐴𝑑𝑑𝑒𝑛𝑑𝑜)
7) 𝑏 = 𝑑 ⇒ 𝑓 … … … … …. … … = 𝑏+𝑑+𝑓
𝑏
𝑥
Note : Third proportion of a X b is x such that 𝑎 = 𝑏
:. X =
𝑏2
𝑎
INDICES/INDEX
Laws :
1) 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
2) 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛
3) (𝑎𝑚 )𝑛 = 𝑎𝑚×𝑛
4) (𝑎 × 𝑏)𝑛 = 𝑎𝑛 × 𝑏 𝑛
5) 𝑎° = 1
6) 𝑎 𝑥 = 𝑏 𝑦 ⇒ 𝑥 = 𝑦
1
7) 𝑎 𝑥 = 𝑏 ⇒ 𝑎 = 𝑏 𝑥
LOGARITHM
1) If𝑎 𝑥 = 𝑛, then log 𝑎 𝑛 = 𝑥
log 𝑥
log 𝑒 𝑥
{ 10
}
𝑒 = 2.7183
2) log 𝑎 𝑚 × 𝑛 = log 𝑎 𝑚 + log 𝑎 𝑛
3) log 𝑎
𝑚
𝑛
= log 𝑎 𝑚 − log 𝑎 𝑛
4) log 𝑎 (𝑚𝑛 ) = 𝑛 log 𝑎 𝑚
5) log 𝑎 𝑎 = 1
6) log 𝑎 1 = 0
7) Base changing formula
log 𝑛 𝑚 =
log𝑎 𝑚
log𝑎 𝑛
common base means base = 10.
EQUATION
Quadratic Equation
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
𝑥=
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
= 𝑅𝑜𝑜𝑡𝑠 𝑜𝑓 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
8 Nature of Roots :
1) If 𝑏 2 − 4𝑎𝑐 < 0 𝑡ℎ𝑒𝑛 𝑟𝑜𝑜𝑡𝑠 𝑎𝑟𝑒 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑜𝑟 𝑢𝑛𝑟𝑒𝑎𝑙.
2) If 𝑏 2 − 4𝑎𝑐 = 0 𝑡ℎ𝑒𝑛 𝑟𝑜𝑜𝑡𝑠 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑎𝑛𝑑 𝑒𝑞𝑢𝑎𝑙.
3) If 𝑏 2 − 4𝑎𝑐 > 0 𝑎𝑛𝑑 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑡ℎ𝑒𝑛 𝑟𝑜𝑜𝑡𝑠 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙, 𝑢𝑛𝑒𝑞𝑢𝑎𝑙 𝑎𝑛𝑑 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙.
4) If 𝑏 2 − 4𝑎𝑐 > 0 𝑎𝑛𝑑 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑡ℎ𝑒𝑛 𝑟𝑜𝑜𝑡𝑠 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙, 𝑢𝑛𝑒𝑞𝑢𝑎𝑙 𝑎𝑛𝑑 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙.
Note : Irrational roots occur in pairs 𝑚√𝑛 , 𝑚 − √𝑛.
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Sum of 2 roots =
−𝑏
𝑎
𝑐
Products of 2 roots = 𝑎
Quadratic Equation – Formation
𝑥 2 − (𝑠𝑢𝑚 𝑜𝑓 𝑟𝑜𝑜𝑡𝑠)𝑥 + 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑟𝑜𝑜𝑡𝑠 = 0
Cubic Equation : In case of cubic equation, factors should be made if possible. If
factorization is not possible then only verification of answer should be preferred.
Co – ordinate Geometry
1. Distance between 2 points A (x, y) and B (𝑥2 , 𝑦2 )
𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
2. Equation of straight line
y = mx + c , m = slope of line
𝑦 −𝑦
𝑑𝑦
𝑚 = 𝑥2 −𝑥1 = 𝑑𝑥
2
1
Note : 1) Slope of a parallel lines is same.
2) Product of slope of 2 ⊥ lines = -1
𝑚1 × 𝑚2 = −1
3) Equation of line is (𝑦 − 𝑦1 ) = 𝑚(𝑥 − 𝑥1 )
4) Intercept form of Equation
𝑥
𝑦
+
=1
𝑎
𝑏
Cost slope = Variable Cost per unit =
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑐𝑜𝑠𝑡
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑢𝑛𝑖𝑡𝑠
Limits and Continuity
Some Important Limits
1) lim
𝑒 𝑥 −1
𝑥
𝑥→0
=1
L – Hospital Rule
0
For 0 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑟
∞
𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠
∞
𝐷𝑒𝑟𝑖𝑎𝑣𝑎𝑡𝑒 𝑜𝑓 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
Limit = 𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
2) lim
𝑥→0
3) lim
𝑥→0
𝑎𝑥 −1
= log 𝑎
𝑥
log(1+𝑥)
𝑥
=1
1
4) lim [1 + x] ˣ = e
5) lim
6) lim
𝑥→0
𝑥→∞
𝑥 𝑛 −𝑎𝑛
= 𝑛 𝑎𝑛−1
𝑥−𝑎
(1+𝑥)ⁿ−1
𝑥
=𝑛
Types of functions
1) Even function : If f(-x) = f(x)
2) Odd function : If f(-x) = -f(x)
3) Neither even nor odd : f(-x) ≠ f(x) ≠ -f(x)
Notes : A polynomial function is always continuous.
STATISTICAL DESCRIPTION OF DATA
Latin word “Status”
Italian word “Stastita”
German word “Stastistik”
French word “Statistique”
Definitition of Statistics can be in plural sense or singular sense. When satistic is used as a
plural noun then it may be defined as data qualitative and quantitative.
When used as singular noun then it is defined as scientific method of drawing conclusions
about some important characteristic It means “Science of counting” or “Science of averages”.
Application of Statistics
1) Economics
2) Business management
3) Commerce and Industry
4) Health care
Limitations of Statistics :
1) It deals with Aggregates (Total)
2) Statistics is concerned with quantitative data.
3) Future projections are valid under specific set of conditions.
4) Theory of Statistics is based on Random Sampling
COLLECTION OF DATA
DATA: is quantitative information about particular characteristic.
VARIABLE : A quantitative characteristic is called variable.
ATTRIBUTE : A qualitative characteristic is called Attribute.
For e.g. : gender, nationality, color etc.
VARIABLES are of 2 types :
A) DISCRETE VARIABLE : When a variable can assume finite no, of isolated valued then
it is discrete. E.g. : Petals in a flower no. of road accidents, marks in paper etc.
B) CONTINUOUS VARIABLE : When a variable can assume any value from a given
interval. E.g. : Weight, height, sale etc.
DATA can be of 2 types :
A) PRIMARY DATA – when data is collected by the person using the data.
B) SECONDARY DATA – When data is used by the person other than collecting the data
sources of secondary data are :
1) International sources – WHO (World Health Organisation)
ILO (International Labour Organisation)
IMF (International Monetary Fund), World
Bank, etc.
2) Govt. sources – CSO (Central Statistical Organisation), ministers,
etc.
3) Private and Quasi Govt. organisation – RBI, NCERT
4) Research Institute and Researcher
Collection of Primary Data : There are 4 methods of collection of primary data.
a) Interview Method
1) Personal Interview – Natural calamities
2) Indirect Interview – Accidents
3) Telephonic Interview – Maximum non response
b) Mailed Questionnaire – Maximum non response
– Widest coverage
c) Observation Method – Time consuming, expensive and applicable only for small
population.
d) Questionnaire filled by enumerator
NOTE: Scrutiny of data means checking data for any possible error. Two or more series of
figures which are related to each other can be compared for internal consistency.
PRESENTATION OF DATA
CLASSIFICATION OF DATA
a) Chronological or Temporal or Time series data
b) Geographical or spatial data.
c) Qualitative or ordinal
d) Quantitative or cardinal
MODE OF PRESENTATION
a) Textual presentation – Least preferred because it is monotonous.
b) Tabular presentation – (Page – 10.7)
1) Table has a serial no,
2) Table can be divided into 4 parts
→ Caption – is upper part of table describing columns and sub columns.
→ Boxhead – is entire upper part including column and sub column numbers, units of
measurement alongwith caption.
→ Stub – is left part of table
→ Body – is main part of table.
Example :
Status
Member of TU
M
F
T
Status of workers
Caption
Non members
M
F
T
Total
M
F
T
Body
Source
Footnote
C) Diagrammatic representation of Data
1) Line Diagram or Historiagram – Line diagrams are used when data vary over time.
Logarithmic or ratio chart is used where there is wide fluctuation. Multiple line chart is used
for 2 or more related time series with same units. Multiple axis line chart are used for 2 or
more series with different units.
2) Bar Diagram –
Horizontal Bar diagram – used for qualitative data or data varing over space
(Geographical)
→ Vertical Bar diagram – used for quantitative data or data varing over time.
→ Multiple or group Bar diagram – used to compare related series.
→ Component or subdivided Bar diagram (Percentage)
3) Pie Chart
FREQUENCY DISTRIBUTION
a) Discrete or ungrouped frequency distribution (Simple) – This is used for discrete variable
E.g. :
Marks
No. of students (f)
5
20
6
30
7
50
8
10
9
15
10
20
B) Grouped frequency Distribution – used for continuous variables.
E.g. :1)
Marks / Weight
f
0 – 10
10
10 – 20
5
Mutually exclusive classification – used
for
20 – 30
15
continuous variable
30 – 40
20
50
2)
Marks / Weight
0–9
f
10
10 – 19
5
Mutually inclusive classification – used
20 – 29
30 – 39
15
20
discrete variable
for
CLASS LIMIT – Lower Class limit (LCL), Upper Class Limit (UCL)
LCL
UCL
LCB
UCB
10 – 20
10
20
10
20
20 – 30
20
30
20
30
LCL =
LCB
30 – 40
30
40
30
40
UCL =
UCB
CLASS BOUNDARIES are real class limits
Lower class boundary (LCB), Upper class boundary (UCB)
LCL
UCL
LCB
UCB
10 – 19
10
19
9.5
19.5
20 – 29
20
29
19.5
29.5
LCL ≠
LCB
30 – 39
30
39
29.5
39.5
UCL ≠
UCB
Note : 29.5 – 39.5 includes 29.5 but excludes 39.5
CLASS MARK
10
10 − 20
15
10} Class width/ Interval
20 − 30
25} 𝐶𝑙𝑎𝑠𝑠 𝑚𝑎𝑟𝑘
10
30 − 40
35
Class Mark
Class width / Interval
10 – 19
14.5
9.5 – 19.5
10
20 – 29
24.5
19.5 – 29.5
10
30 – 39
34.5
29.5 – 39.5
10
CLASS WIDTH = CLASS SIZE = CLASS INTERVAL
Class
Cumulative
Frequency
frequency
interval
frequency
density
10 – 20
8
10
8
20 – 30
12
10
20
30 – 40
20
10
40
8
10
12
10
20
10
= .8
= 1.2
=2
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
* FREQUENCY DENSITY = 𝐶𝑙𝑎𝑠𝑠 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
* RELATIVE FREQUENCY = 𝑇𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
* PERCENTAGE FREQUENCY = 𝑇𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 × 100
Relative
frequency
8
Percentage
frequency
8
40
12
40
12
40
20
40
20
40
40
× 100
× 100
× 100
GRAPHICAL REPRESENTATION OF FREQUENCY DISTRIBUTION
1) Histogram
Mode – Most frequently occurring item.
2) Frequency polygon – is used for simple frequency distribution. It can be used for grouped
frequency distribution provided width of class interval is same.
Note : Frequency curve is limiting form of frequency polygon
3) Cumulative frequency curve or ogive (Two types)
Less than ogive can be used to calculate median, quartile deciles etc.
Frequency curve – It is limiting form of frequency polygon or histogram for which total area
is 1.
a) Bell shaped – Most common
b) U – shaped
MEASURE OF CENTRAL TENDENCY AND DISPERSION
CENTRAL TENDENCIES
ARITHMETIC MEAN (A.M.) =
∑𝑓𝑥
∑𝑓
𝑜𝑟 𝐴 +
∑𝑓𝑑𝑥
∑𝑓
Note : For calculation of A.M., mutually inclusive series need not be converted into
mutually exclusive series.
Properties of Arithmetic Mean :
1) A.M. is neither free of origin (affected by addition or subtraction of a constant) Nor free of
scale (multiplication and division of a constant).
2) Combined A.M. =
𝑛1 ̅̅̅
𝑋₁+𝑛2 ̅̅̅
𝑋₂
𝑛1 +𝑛2
GEOMETRY MEAN (G.M.)
G.M. = (𝑥1 × 𝑥2 … … … … … … … . 𝑥4 )1/4
log 𝐺. 𝑀. = log(𝑥1 × 𝑥2 … … … … … … … . 𝑥𝑛 )1/𝑛
1
log 𝐺. 𝑀. = 𝑛 ∑ log 𝑥𝑛
1
G.M. = A.L.[𝑛 ∑ log 𝑥𝑛 ]
Properties of G.M.
1) If all the observations are k then G.M. = k
2) If z = xy then G.M. of z = (𝐺. 𝑀. 𝑜𝑓 𝑥) × (𝐺. 𝑀. 𝑜𝑓 𝑦)
𝑥
𝐺.𝑀.𝑜𝑓 𝑥
3) If z = 𝑦 𝑡ℎ𝑒𝑛 𝐺. 𝑀. 𝑜𝑓 𝑧 = 𝐺.𝑀.𝑜𝑓 𝑦
HARMONIC MEAN (H.M.) =
∑𝑓
𝑓
𝑥
∑( )
Combined H.M. of two series =
𝑛₁+𝑛₂
𝑛₁ 𝑛₂
+
𝐻₁ 𝐻₂
E.g.: H.M. of 4, 6, 10
H.M. = 1
3
1 1
4 6 10
E.g. : x
f
+ +
= 5.77
2
3
H.M = 3
4
2
6
5
10
2 5
+ +
2 4 6
E.g. :
Distance (f)
Speed (x)
100 km
60km/h
Average of Speed =
200km
80km/h
100+200
100 200
+
60 80
Note : For average of speed, H.M. is used
E.g.: Home → College – 40 km/h
College → Home – 30 km/h
Average speed =
2
1
1
+
30 40
Relationship between A.M., G.M., H.M.- A.M. ≥ G.M. ≥ H.M.
a) For unequal no. A.M. > G.M. > H.M.
b) For equal no. A.M. = G.M. = H.M.
* For 2 numbers : (G.M.)² = A.M. x H.M.
* G.M. cannot be calculated for negative items.
* G.M. is used where compound growth is there.
* A.M. is most popular central tendency.
WEIGHTED MEAN =
∑𝑤𝑥
∑𝑤
𝑖𝑠 𝑢𝑠𝑒𝑑 𝑤ℎ𝑒𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑖𝑡𝑒𝑚𝑠 𝑎𝑟𝑒 𝑜𝑓 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑖𝑚𝑝𝑜𝑟𝑡𝑎𝑛𝑐𝑒.
E.g. :
food
Rent
Education
x
10
20
30
340
Weighted Average =
20
w
10
6
4
20
xw
100
120
120
340
= 17
MEDIAN – If all the items are arranged in increasing order then middle item is median.
𝑁+1 𝑡ℎ
Median ( in case of discrete items) = (
Median (for grouped items) = L₁ +(
) 𝑖𝑡𝑒𝑚
2
𝑁
−𝑁₁
2
𝑓
)×𝑐
Where L₁ = lower limit of median class
N = Total items
N₁ = Cumulative frequency of preceeding class
f = frequency of median class
c = class interval
E.g. :
f
c.f.
10 – 19
10
10
20 – 29
15
25
30 – 39
25
50 – Median class
40 – 49
20
70
70
2
( −25)×10
Median = 29.5 +
25
QUARTILE – Divide total items in 4 parts – Q₁ and Q₃
𝑁+1
𝑁+1
4
4
𝑄1 = (
) 𝑖𝑡𝑒𝑚, 𝑄3 = (
) × 3𝑡ℎ 𝑖𝑡𝑒𝑚
𝑁+1
DECILE (10 parts) – 𝐷𝑇 = (
10
) 𝑇𝑡ℎ 𝑖𝑡𝑒𝑚
PERCENTILE (100 parts)
MODE is most frequently occurring item.
For grouped data, Mode = L₁ + (2𝐹
𝐹𝑀 −𝐹𝑃
𝑀 −𝐹𝑃 −𝐹𝑆
)×𝑐
Where L₁ = lower limit of modal class
𝐹𝑀 = frequency of modal class
𝐹𝑃 = frequency of preceeding class
𝐹𝑆 = frequency of succeeding class
c = class interval
Note : For moderately skewed data
Mode = 3Median – 2Mean
Note : All central tendencies are neither free of origin nor free of scale
MEASURE OF DISPERSION
↙
↘
Absolute measure
Relative measure
1) Range
1) Coefficient of range
2) Mean Deviation
2) Coefficient of mean deviation
3) Standard Deviation
3) Coefficient of variation
4) Quartile deviation
4) Quartile deviation (Coefficient)
Note : All measures of dispersion are free of origin but not free of scale.
RANGE = L – S
L = Largest
S= Smallest
𝐿−𝑆
→ Coefficient of Range = [𝐿+𝑆] × 100
MEAN DEVIATION about mean =
→ Coefficient of mean deviation +
̅|
∑|X−X
0𝑟
̅|
∑ 𝑓|X−X
𝑛
𝑀𝑒𝑎𝑛 𝑑𝑒𝑣𝑎𝑡𝑖𝑜𝑛
𝑀𝑒𝑎𝑛
∑𝑓
× 100
∑(𝑥−𝑋̅ )2
STANDARD DEVIATION (𝜎)S.D. = √
𝑛
𝑆.𝐷.
→ Coefficient of variation =𝑀𝑒𝑎𝑛 × 100
𝑎−𝑏
S.D. of 2 numbers a and b = |
2
|
S.D. of first n natural numbers = √
∑ 𝑥2
= √
𝑛
∑𝑥
− ( 𝑛 )²
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = (𝑆. 𝐷. )²
∶. 𝑆. 𝐷. 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 − 𝑣𝑒
𝑛2 − 1
12
Combined S.D. of 2 groups
𝑛1 𝑠1 2 +𝑛2 𝑠2 2 +𝑛1 𝑑1 2 +𝑛2 𝑑2 2
√
𝑛1 +𝑛2
d₁ = 𝑋̅1 − 𝑋̅12
d₂ = 𝑋̅2 − 𝑋̅12
𝑋̅1= mean of first group
𝑋̅2= mean of second group
𝑋̅12= combined mean
𝑋̅12 =
𝑛1𝑋
̅ 1 + 𝑛2 𝑋
̅2
𝑛1 +𝑛2
QUARTILE DEVIATION =
𝑄3 −𝑄1
2
𝑄 −𝑄
→ Coefficient of quartile deviation = [𝑄3+𝑄1] × 100
3
1
Note : Range is not free of units (Rs., Kg)
Note : Measure of dispersion is free of origin but not free of scale.
1
Note : S.D. is 2of range of two numbers.
CORRELATION AND REGRESSION
Correlation and Regression
↓
↓
Extent of Relationship
Prediction
↓
-1 to 1
1) MARGINAL DISTRIBUTION – There are two marginal distribution
Marks in Maths (f)
Marks in Stats (f)
0 – 10
7
8
10 – 20
19
24
20 – 30
32
26
2) CONDITIONAL DISTRIBUTION – There are m + n conditional distribution where m
is no. of rows and n is no. of columns.
Marks in Maths [If marks in stats is 10 – 20]
f
0 – 10
2
10 – 20
7
20 – 30
15
Note : No. of cells = m X n
CORRELATION – Correlation analysis is establishing relation between two variable
(positive, negative or zero) and measuring the extent of relationship between two variables (1 to 1).
Correlation can be measured by four methods :
a) Scatter Diagram
b) Karl Pearson’s product moment correlation coefficient.
c) Spearman’s rank correlation coefficient.
d) Coefficient of concurrent deviations.
Scatter Diagram : can be used for linear as well as curvilinear variables. Exact measure of
correlation cannot be measured by this method.
KARL PEARSON’S PRODUCT MOMENT COEFFICIENT OF CORRELATION is
the best method for finding correlation between two variable having linear relationship.
𝛾𝑥𝑦 =
𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑥 𝑎𝑛𝑑 𝑦
𝑆.𝐷.𝑥 ×𝑆.𝐷.𝑦
Where Covariance =
∑ 𝑥𝑦
= 𝑛𝑆.𝐷.𝑥 ×𝑆.𝐷.𝑦
∑(𝑥−𝑥̅ )(𝑦−𝑦̅)
𝑛
=
∑ 𝑥𝑦
𝑛
→{
𝑥 = (𝑥 − 𝑥̅ )
}
𝑦 = (𝑦 − 𝑦̅)
Or
𝛾=
𝑁 ∑ 𝑋𝑌−∑ 𝑋 ∑ 𝑌
√𝑛 ∑ 𝑥 2 −(∑ 𝑥)2 √𝑛 ∑ 𝑦 2 −(∑ 𝑦)2
Note : Coefficient of correlation is free of origin as well as free of scale.
Note : r is free of unit.
SPEARMAN’S RANK CORRELATION : Rank correlation is used to study relationship
between Qualitative variables.
𝛾 =1−
1
∑(𝑡 3 −𝑡)]
12
𝑛3 −𝑛
6[∑ 𝐷 2 +
Where D = Difference of Ranks
t = no. of tied ranks
COEFFICIENT OF CONCURRENT DEVIATION
𝛾 = ±√±
(2𝑐−𝑚)
𝑚
where m = no. of pairs of deviations
c = No. of +ve signs = No. of concurrent deviations
REGRESSION ANALYSIS – There are two regression lines. Regression is concerned
with predicting dependent variable when independent variable is known. In simple,
Regression model, if y depends on x then line y on x is given by
y = a + bx
↓
↓
dependent
Independent
Regression line Y on X
𝑦 − 𝑦̅ = 𝑏𝑦𝑥 (𝑋 − 𝑋̅)
where 𝑏𝑦𝑥 Regression coefficient of y on x.
𝑆𝐷𝑦
𝑏𝑦𝑥 = 𝛾 × 𝑆𝐷𝑥 =
𝑁 ∑ 𝑋𝑌−∑ 𝑋 ∑ 𝑌
𝑛 ∑ 𝑥 2 −(∑ 𝑥 2 )
Regression line X on Y
𝑋 − 𝑋̅ = 𝑏𝑥𝑦 (𝑌 − 𝑌̅)
𝑆𝐷𝑥
𝑏𝑥𝑦 = 𝛾 × 𝑆𝐷𝑦 =
𝑁 ∑ 𝑋𝑌−∑ 𝑋 ∑ 𝑌
𝑛 ∑ 𝑦 2 −(∑ 𝑦 2 )
Properties of Regression
1) 𝑏𝑥𝑦 , 𝑏𝑦𝑥 𝑎𝑛𝑑 𝛾 ℎ𝑎𝑣𝑒 𝑠𝑎𝑚𝑒 𝑠𝑖𝑔𝑛.
2) 𝛾 = ±√𝑏𝑥𝑦 × 𝑏𝑦𝑥
3) 𝑏𝑥𝑦 𝑎𝑛𝑑 𝑏𝑦𝑥 𝑎𝑟𝑒 𝑓𝑟𝑒𝑒 𝑜𝑓 𝑜𝑟𝑖𝑔𝑖𝑛 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑓𝑟𝑒𝑒 𝑜𝑓 𝑠𝑐𝑎𝑙𝑒.
4) Arithmetic mean of x and y are solution of two regression equations.
5) Because √𝑏𝑥𝑦 × 𝑏𝑦𝑥 = 𝛾,
∶. 𝑏𝑥𝑦 × 𝑏𝑦𝑥 ≤ 1
PROBABILITY AND EXPECTED VALUE
Probability can be divided into two categories :
a) Subjective Probability
b) Objective Probability
Random Experiment – Experiment means performance of an Act. Random means that
Probability of all outcomes is equal.
Event – is result of and experiment
a) Simple or Elementary events
b) Composite or Compound events
An event is simple if it cannot be decomposed into further event.
Compound event are made of two or more simple event,
MUTUALLY EXCLUSIVE EVENTS – When happening of an event makes happening of
another event impossible then two events are mutually exclusive.
EXHAUSTIVE EVENTS : are set of all possible events i.e. events have to be from set of
Exhaustive events. Sum of probability of Exhaustive events is always equal to 1.
EQUALLY LIKELY or Equi Probable – Events having same probability.
CLASSICAL DEFINITION OF PROBABILITY
𝑁𝑜.𝑜𝑓 𝐹𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝐸𝑣𝑒𝑛𝑡𝑠
P (A) = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑜.𝑜𝑓 𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 𝑒𝑣𝑒𝑛𝑡𝑠
Note : 1) It is applicable when total no. of events is finite.
2) It can be used only when events are equally likely.
3) Above definition is also termed as “a priori” and is useful only in coin tossing,
dice throwing etc.
Note : E.g. : In one coin
H
T
1
1
2
2
In two coins
0H
1H
2H
1
1
1
4
2
4
In three coins
0H
1H
2H
1
3
3
1
8
8
8
8
Sum of no. on two dice
2
3
4
5
3H
6
7
8
9
10
11
1
2
3
4
5
6
5
4
3
2
12
1
36
36
36
36
36
36
36
36
36
36
36
STATISTICAL DEFINITION OF PROBABILITY
If a random experiment is repeated a very good no. of times say n under identical conditions
and an event occurs 𝐹𝐴 times then ratio of 𝐹𝐴 and n when n tends to infinity is defined as
statistical definition of probability.
P=
𝐹𝐴
𝑛
𝑤ℎ𝑒𝑟𝑒 𝑛 → ∞
1) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃 (𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
2) 𝑃(𝐴 ∩ 𝐵) = 𝑃 (𝐴 𝑎𝑛𝑑 𝐵)
For Independent events
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵⁄𝐴) = 𝑃(𝐵) × 𝑃(𝐴⁄𝐵 )
𝑃(𝐵⁄𝐴) = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐵 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑒𝑣𝑒𝑛𝑡 𝐴 ℎ𝑎𝑠 𝑡𝑎𝑘𝑒𝑛 𝑝𝑙𝑎𝑐𝑒
3) 𝑃(𝐴 ∪ 𝐵 ∪ 𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) − 𝑃(𝐴 ∩ 𝐵) − 𝑃(𝐴 ∩ 𝐶) − 𝑃(𝐵 ∩ 𝐶) +
𝑃(𝐴 ∩ 𝐵 ∩ 𝐶)
4) 𝑃(𝐴 ∩ 𝐵1 ) = 𝑃(𝐴 𝑎𝑛𝑑 𝑛𝑜𝑡 𝐵) = 𝑃(𝐴) − 𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴) = 𝑃(𝐴 ∩ 𝐵) + 𝑃(𝐴 ∩ 𝐵1 )
5) 𝑃(𝐴1 ∩ 𝐵1 ) = 1 − 𝑃(𝐴 ∪ 𝐵)
* Only A + only B + A or B + A¹ and B¹ = 1
6) P (A¹) = 1 – P(A)
Mutually exclusive events – 𝑃(𝐴 ∩ 𝐵) = 0
Exhaustive – 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) = 1
Equally/likely – 𝑃(𝐴) = 𝑃(𝐵) = 𝑃(𝐶)
𝑃(𝐵⁄𝐴) =
𝑃(𝐴⁄𝐵 ) =
𝑃(𝐴∩𝐵)
𝑃(𝐴)
}
𝑃(𝐴∩𝐵)
Conditional Probability
𝑃(𝐵)
EXPECTED VALUE is sum of product of different values and its probability.
𝜀(𝑥) = 𝜀𝑥 × 𝑝(𝑥) = 𝑚𝑒𝑎𝑛
𝜀(𝑥 2 ) = 𝜀𝑥 2 × 𝑝(𝑥)
Variance = (S.D.)² = 𝜀(𝑥 2 ) − [𝜀(𝑥)]2
= 𝜖(𝑋 − 𝑋̅)2
Properties of Expected value :
1) 𝜖(𝑥 + 𝑦) = 𝜖(𝑥) + 𝜖(𝑦)
2) 𝜖(𝑘. 𝑥) = 𝑘𝜖(𝑥) 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
3) 𝜖(𝑥. 𝑦) = 𝜖(𝑥). 𝜖(𝑦)𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟 𝑥 𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡
Mean = 𝜇 = 𝑥̅ = 𝜀(𝑥) = ∑ 𝑥 𝑃(𝑥)
∑(𝑥−𝑥̅ )2
S.D. = √
𝑛
∑ 𝑥2
or √
𝑛
∑𝑥
2
−( 𝑛 )
∑ 𝑃(𝑥) = 1
V = 𝜖(𝑥 − 𝑥̅ )2 = ∑(𝑥 − 𝑥̅ )2 × 𝑃(𝑥)
V = 𝜖(𝑥 2 ) − [𝜖(𝑥)]2 = ∑ 𝑥 2 𝑃(𝑥) − [∑ 𝑥 𝑃(𝑥)]2
THEORETICAL DISTRIBUTION
Theoretical Distribution exists only in theory. It is theoretical probability distribution of
experiments.
BINOMIAL DISTRIBUTION : Features are
1) There are 2 possible outcomes.
2) Trials are independent.
3) n is small
4) It is Biparametric Distribution (n, p)
5) Probability of 𝛾 successes out of n trials is
𝑃(𝛾) = 𝑛𝐶𝛾 . 𝑃𝛾 . 𝑞 𝑛−𝛾
6) Mean = np, variance = npq
7) It can be unimodal or bimodal
Mode = Largest integer in (n+1) P if (n+1) p is a non integer.
= (n+1) P and (n+1)p – 1 if (n+1)p is integer.
𝑛
8) Variance is maximum when p = q = .5 Maximum variance = 4
Additive Property
If x and y are two independent variables such that
𝑋~𝐵(𝑛1 , 𝑝)
𝑌~𝐵(𝑛2 , 𝑝)
X + Y ~𝐵(𝑛1 + 𝑛2 , 𝑝)
* {p = success ,
q = failure}
POISION DISTRIBUTION : Properties
1) Probability of success in very small time interval (t, t+ dt) is kt where k is constant.
2) Probability of success is independent.
3) Probability of success in this time interval is very small.
4) This distribution is called distribution of rare events.
5) n is large and p is very small such that n X p is finite.
6) Mean = Variance = n X p = m
7) It is uniparametric distribution (m).
8) There are two possible outcomes.
9) Probability of r success out of n
𝑃(𝑟) =
𝑒 −𝑚 .𝑚𝑟
𝑟!
10) Mode is largest integer in m if m is non integer
Mode is m and m – 1 if m is an integer.
Additive Property :
If x and y are two independent poisson variables and
z=x+y
x ~𝑃(𝑚1 )
y ~𝑃(𝑚2 )
z = x + y~𝑃(𝑚1 + 𝑚2 )
NORMAL OR GAUSSIAN DISTRIBUTION
P(x) = f(x) = Probability density function
f(x) =𝜎
1
√2𝜋
×𝑒
−(𝑥−𝑚)²
2𝜎²
m = mean
𝜎 = S.D.
Properties of Normal Distribution
1) Mean = median = mode
2) Mean deviation = .8 S.D.
3) First Quartile = Q₁ = Mean – .675 S.D.
Third Quartile = Q₃ = Mean + .675 S.D.
4) Point of inflexion are
a) Mean – S.D.
b) Mean + S.D.
5) In standard normal variate, Mean = 0 , S.D. = 1
6) Area under normal curve
𝑋−𝑋̅
z =|
𝑆.𝐷.
|
CHI-SQUARE DISTRIBUTION (X²-DISTRIBUTION)
Properties :
1) It is continuous, positively skewed probability distribution.
2) Mean = n
3) S.D. = √2𝑛
4) When n is large it follows normal distribution.
STUDENTS T – DISTRIBUTION
1) It is a continuous symmetrical distribution.
2) Mean = 0
𝑛
3) S.D. = √𝑛−2
𝑖𝑓 𝑛 > 2
4) For large n (n>30) t distribution is identical to z distribution.
F Distribution
1) F Distribution is positively skewed.
2) It is continuous.
Note :
z
Area under the normal curve
1
.6826
1.64
.90 (90%)
1.96
.95 (95%)
2.33
.98 (98%)
2.58
.99 (99%)
3
.9973 (99.73%)
SAMPLING
BASIC PRINCIPLES OF SAMPLE SURVEY :
1) Law of Statistical regularity : States a large sample drawn at random from population
possesses characteristics of population at an average.
2) Principle of inertia : States that results drawn from samples are likely to be more reliable,
accurate and precise as sample size increases, other things remaining same.
3) Principle of Optimization : means maximum efficiency at given cost or minimum cost for
optimum level of efficiency.
4) Principle of Validity : States that sampling designs is valid only if it is possible to obtain
valid results.
Probabilistic sampling ensures this validity.
COMPARISON BETWEEN SAMPLE AND COMPLETE ENUMERATION (CENSUS)
1) Speed 2) Cost 3) Reliability 4) Accuracy 5) Necessity
ERRORS IN SAMPLING :
1) Sampling errors
a) Errors due to defective sampling design.
b) Errors arising out due to substitution.
c) Errors due to faulty demarcation.
d) Errors due to wrong choice of statistic.
e) Variability in the population.
2) Non-Sampling Errors : Memory lapse, preference for certain digits, ignorance, psychological
factors like vanity, non-response etc.
SAMPLING DISTRIBUTION AND STANDARD ERROR OF A STATISTIC
Standard Error (SE) is standard deviation of sample statistic.
𝜎
1) SE of means = SE (𝑥̅ ) = 𝑛(with replacements)
√
where 𝜎= population S.D.
2) SE (𝑥̅ )(𝑊𝑂𝑅) =
3) SE (𝑥̅ ) =
4) SE (𝑥̅ ) =
𝑠
√𝑛−1
√
𝑁−𝑛
𝑁−1
where N = population size
s = Sample S.D.
√𝑛−1
𝑠
𝜎
√𝑛
n = Sample size
𝑁−𝑛
√
𝑁−1
𝑃×(1−𝑃)
5) Standard error of proportion = √
𝑃×(1−𝑃)
6) SE (P) (WOR) = √
𝑛
𝑛
𝑁−𝑛
√
𝑁−1
INTERVAL ESTIMATION OF MEAN AND PROPORTION
1) Interval estimation of mean =𝑋̅ ± 𝑍 × 𝑆𝐸(𝑥̅ )
2) Interval estimation of mean if
a) Sample size is less than 30.
b) Population S.D. is not known.
then interval estimation of mean 𝑋̅ ± 𝑡 × 𝑆𝐸(𝑋̅)
3) Interval estimation of proportion = 𝑃 ± 𝑍 × 𝑆𝐸(𝑃)
Sample size
𝜎×𝑧
1) Sample size for population mean = |
2) Sample size for proportion =
𝜖
|²
𝑝×𝑞×𝑧 2
𝜖2
TYPES OF SAMPLING
I) PROBABILITY SAMPLING
II) NON-PROBABILITY SAMPLING
III) MIXED SAMPLING
PROBABILITY SAMPLING : When each member of population has equal chance of
selection.
a) Simple Random Sampling (SRS) – should be used when
1) Population is small. 2) Sample size is not large. 3) Population is homogeneous.
b) Stratified Sampling (strata = layer) : is used when
1) Population is large.
2) Population is heterogeneous
c) Multi stage Sampling : If population is very large then it is cost effective and flexible
system of sampling.E.g. : Estimation of foodgrain production in India.
PURPOSIVE OR JUDGEMENT OR NON PROBABILISTIC SAMPLING : Probability of
selection of each unit is not equal.
MIXED SAMPLING : (Systematic Sampling) is partly probabilistic and partly non
probabilistic.
E.g. : Sampling done by Auditors.
Systematic Sampling has a drawback – If there is unknown or undetected periodicity in
sampling frame and sampling interval is multiple of that period then we get most biased
samples.
CRITERIA FOR IDEAL ESTIMATION
1) Unbiasedness and minimum variance (MVUE) Minimum variance unbiased estimation
2) Consistency and Efficiency
3) Sufficiency
INDEX NUMBERS
Index Numbers is an average of ratios expressed as percentage. Two or more time periods are
involved one of which is base time period. The value at the time of base period serves as
standard of comparison.
INDEX NUMBER ARE OF FOLLOWING TYPES :
1) PRICE INDEX
2) QUANTITY INDEX
3) VALUE INDEX
4) COST OF LIVING INDEX OR CONSUMER PRICE INDEX
METHODS OF CONSTRUCTION OF INDEX NUMBER (PRICE INDEX)
1) SIMPLE AGGREGATIVE METHOD (0 is base year, 1 is current year)
𝑃° 1 =
∑ 𝑃1
∑ 𝑃°
× 100
𝑃
2) SIMPLE AVERAGE OF PRICE RELATIVE(𝑃1 )
°
𝑃° 1 =
𝑃
∑( 1 )
𝑃°
𝑛
× 100
3) WEIGHTED METHODS
a) Laspeyre’s Price Index (L)
𝑃° 1 =
∑(𝑃1 ×𝑄° )
∑(𝑃° ×𝑄° )
× 100
b) Passche’s Price Index (P)
𝑃° 1 =
∑(𝑃1 ×𝑄1 )
∑(𝑃° ×𝑄1 )
× 100
c) Marshall –Edge worth
𝑃° 1 =
∑ 𝑃1 (𝑄° +𝑄1 )
∑ 𝑃° (𝑄° +𝑄1 )
× 100
d) Fisher’s ideal Index no. = √𝐿 × 𝑃
e) Bowley’s Index no. =
𝐿+𝑃
2
4) WEIGHTED AVERAGE OF PRICE RELATIVE METHOD
𝑃° 1 =
∑ 𝑃1 ×𝑄°
∑ 𝑃° ×𝑄°
× 100
[Similar to Laspeyre’s Index no.]
5) Chain Index =
𝐿𝑖𝑛𝑘 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑒𝑎𝑟×𝑐ℎ𝑎𝑖𝑛 𝑖𝑛𝑑𝑒𝑥 𝑜𝑓 𝑙𝑎𝑠𝑡 𝑦𝑒𝑎𝑟
100
𝑃𝑛
Link Relative =𝑃
𝑛−1
× 100
QUANTITY INDEX – REPLACE P BY Q AND Q BY P IN PRICE INDEX
FORMULA.
VALUE INDEX
𝑉°1 =
∑ 𝑃1 ×𝑄°
∑ 𝑃° ×𝑄°
× 100
COST OF LIVING INDEX (CLT) OR CONSUMER PRICE INDEX (CPI) =
weighted average of indices]
or
∑ 𝑃1 ×𝑄°
∑ 𝑃° ×𝑄°
∑ 𝑊𝐼
∑𝑊
[is
× 100 [same as Laspeyre’s]
DEFLATING
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 ×100
Deflated Value = 𝑃𝑟𝑖𝑐𝑒 𝑖𝑛𝑑𝑒𝑥 𝑜𝑓 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑒𝑎𝑟
or
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 ×𝐵𝑎𝑠𝑒 𝑦𝑒𝑎𝑟 𝑝𝑟𝑖𝑐𝑒4
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑒𝑎𝑟 𝑝𝑟𝑖𝑐𝑒𝑠
BASE SHIFTING
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑃𝑟𝑖𝑐𝑒 𝐼𝑛𝑑𝑒𝑥 ×100
Shifting Price Index = 𝑃𝑟𝑖𝑐𝑒 𝐼𝑛𝑑𝑒𝑥 𝑜𝑓 𝑦𝑒𝑎𝑟 𝑜𝑛 𝑤ℎ𝑖𝑐ℎ 𝑖𝑡 𝑖𝑠 𝑠ℎ𝑖𝑓𝑡𝑒𝑑
Note : Index no. of base year is always 100.
325
Real wages of Base year = 110 × 100 = 295.45
500
Real wages in Current year = 200 × 100 = 250
Decrease in Real wages = 295.45 – 250
= 45.45
TEST OF ADEQUACY
1) UNIT TEST – This test requires index no. to be free of unit. All index no. expect simple
aggregative method satisfy this test.
2) TIME REVERSAL TEST is cleared if 𝑃° 1 × 𝑃1 ° = 1
a) Fisher Index no.
b) Marshall Edgeworth sastisfy this text.
3) FACTOR REVERSAL TEST is satisfied if 𝑃° 1 × 𝑄° 1 = 𝑉° 1
a) Fisher Index no.
b) Simple Aggregative method satisfy this text.
4) CIRCULAR TEST – is satisfied if
𝑃° 1 × 𝑃12 × 𝑃23 = 𝑃° 3 𝑜𝑟 𝑃° 1 × 𝑃12 × 𝑃20 = 1
It is extension of time reversal test
a) simple Geometric mean of Price relative
b) Weighted Aggregative with fixed weight
c) simple aggregative meet this test.
DIFFERENTIAL CALCULUS
Derivative of y w.ɣ.t. x is change in y w.r.t. x. when change in x is very small
𝑑𝑦
𝑑𝑥
∆𝑦
= lim
∆𝑥→0 ∆𝑥
𝑓(𝑥+ℎ)− 𝑓(𝑥)
= lim
ℎ
ℎ→0
Standard Results
𝑑
1) 𝑑𝑥 (𝑥 𝑛 ) = 𝑛 𝑥 𝑛−1
𝑑
2) 𝑑𝑥 (𝑒 𝑥 ) = 𝑒 𝑥
𝑑
3) 𝑑𝑥 (𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡) = 0
𝑑
4) 𝑑𝑥 (𝑎 𝑥 ) = 𝑎 𝑥 log 𝑎
𝑑
5) 𝑑𝑥 (log 𝑥) =
1
𝑥
Product Rule
𝑑
𝑑𝑥
(𝐼 × 𝐼𝐼) = 𝐼 ×
𝑑
𝑑𝑥
𝑑
× (𝐼𝐼) + 𝐼𝐼 × 𝑑𝑥 (𝐼)
Quotient Rule
𝑑
𝐼
( )=
𝐼𝐼×
𝑑𝑥 𝐼𝐼
𝑑
𝑑
(𝐼)−𝐼× (𝐼𝐼)
𝑑𝑥
𝑑𝑥
(𝐼𝐼)2
𝑑𝑦
Gradient = Derivative = 𝑑𝑥
* Ordinate = value of x
Abscissa = value of y
INTEGRATION
𝑑
1
𝑥−𝑎
𝑑
1
𝑎+𝑥
1) ∫ 𝑥 2 −𝑎2 = 2𝑎 log (𝑥+𝑎) + 𝐶
2) ∫ 𝑎2 −𝑥2 = 2𝑎 log (𝑎−𝑥) + 𝐶
𝑑
3) ∫ √𝑥 2
+𝑎2
𝑑
4) ∫ √𝑥 2
−𝑎2
= log(𝑥 + √𝑥 2 + 𝑎2 ) + 𝐶
= log(𝑥 + √𝑥 2 − 𝑎2 ) + 𝐶
5) ∫ 𝑒 𝑥 (𝑓(𝑥) + 𝑓 1 (𝑥)) = 𝑒 𝑥 𝑓(𝑥)
𝑥
6) ∫ √𝑥 2 + 𝑎2 𝑑𝑥 = 2 √𝑥 2 + 𝑎2 +
𝑎2
2
log[𝑥 + √𝑥 2 + 𝑎2 ] + 𝐶
SETS, FUNCTIONS AND RELATIONS
Sets : A set is defined as collection of well defined distinct objects. For e.g. :
A = {a,e,i,o,u} = {x : x is a vowel in the alphabets}
B = {2, 4, 6, 8, 10} ={ x : x = 2m and m is an integer o<m<6}
Note : Each set has 2𝑛 subsets where n are no. of items in a set.
Each set has2𝑛 − 1 proper subsets.
For e.g. : A={1, 2, 3},
Subsets are {1},{3},{2},{1,2},{1,3},{2,3},{1,2,3} { } =
8 = 2³
If 1,2,3 is excluded then remaining subsets are proper subsets. Null set = = ∅
𝑛(𝐴 ∪ 𝐵) = 𝑛 (𝐴) + 𝑁 (𝐵) − 𝑁 (𝐴 ∩ 𝐵)
1) Singleton set – contains only one element a
2) Equal set – A = {1, 3, 4}
B = {1, 3, 4}
3) Equivalent set – A = {1, 3, 4}
B = {2, 3, 5}
n (A) = n (B)
:. A & B are Equivalent set.
4) Power set is collection of all possible subsets of a set.
Cartesian product of Sets – If A and B are 2 non empty sets then A x B is set of all ordered
pairs (a, b) such that a belongs to A and b belongs to B.
For E.g. : A ={1, 3, 6} ,
B ={3, 5}
A x B = {(1, 3), (1, 5), (3, 3), (3, 5), (6, 3), (6, 5)
n (A x B) = n (A) x n (B)
Domain of a function y = f (x) is all possible values of x.
Range of a function y = f (x) is all possible values of y.
Relations
1) Reflexive relation is a relation if a is related to a. [a=a]
2) Symmetric relation : If a = b and b = a then a and b has symmetric relation.
3) Transitive relation : If a = b and b = c then If a = c then relation is transitive.
4) Equivalence relation : If a relation is reflexive, symmetric and transitive then it is
equivalence relation.