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MEASURES OF CENTRAL TENDENCY DATA TYPES

Type 1 (Individual items)

X

10

12

15

Type 2 (Discrete series)

X f

20

25

30

4

6

2

Type 3 (Group data)

There are two different group data's

(1)Continuous group data

X F

10------20

20------30

30------40

4

6

2

(2)Discontinuous group data

Group F

10------19

20------29

30------39

4

6

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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ARITHMETIC MEAN (A.M or

In Type 1 In Type 2 & In Type 3

(I) Direct formula

Where "n" is the total number of values

(II) Shortcut formula

Where "D= X – A", and "A" is any

arbitrary value

(III) Step deviation or coding formula

Where "

", A is any arbitrary

value and is the class interval

(I) Direct formula

A.M =

(II) Shortcut formula

A.M = A +

(III) Step deviation or coding formula

A.M = A +

WEIGHTED MEAN ( )

COMBINED MEAN ( )

Properties of A.M

(i) Mean of a constant is constant itself

(ii) Sum of deviation from mean is always zero, i.e.

(iii) Sum of square of deviation from Mean is minimum, i.e.

(iv) 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 In Type 3

Most repeated value

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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HARMONIC MEAN(H.M)


GEOMETRIC MEAN(G.M)


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)


PERCENTILES(P1,P2,P3,……,P99)


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MEASURES OF DISPERSION DATA TYPES

Type 1 (Individual items)

X

10

12

15

Type 2 (Discrete series)

X f

20

25

30

4

6

2

Type 3 (Group data)

There are two different group data's

(1)Continuous group data

X F

10------20

20------30

30------40

4

6

2

(2)Discontinuous group data

Group F

10------19

20------29

30------39

4

6

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


VARIANCE(S2) & Coefficient of Variation(C.V)


(i) Direct formula

(ii) Shortcut formula

(iii) Step deviation or coding formula

(i) Direct formula



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STANDARD DEVIATION(S.D or S) & Coefficient of S.D


(i) Direct formula



(i) Direct formula



Moments about Mean ( )


First four moments about Mean

First four moments about Mean

Raw Moments (


First four moments about Zero

First moment about zero is equal to A.M

First four moments about Zero

First moment about zero is equal to A.M

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First four moments about any value

Using shortcut method

Using step deviation method

First four moments about any value

Using shortcut 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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Properties of Variance & S.D

(i) 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

(ii) 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)

(iii) Variance and S.D are affected by change of scale, i.e.

Var(aX) = a2Var(X)

Relationship between M.D, Q.D and S.D

,

,

(In any Type of data)

(i) Pearson coefficient of Skewness

or

(ii) Bowley's coefficient of Skewness

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INDEX NUMBERS

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)

INDEX NUMBERS

UNWEIGHTED

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

PRICE RELATIVES ARE GIVEN

WEIGHTED

LASPEYRE'S METHOD

PAASCHE'S METHOD

FISHER'S METHOD

MARSHAL'S METHOD

AGGRIGATIVE EXPENDITURE METHOD

FAMILY BUDGET METHOD

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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 Black Cards

HEART ♥ DIAMOND ♦ SPADE ♠ CLUB ♣

♥2 ♦2 ♠2 ♣2

♥3 ♦3 ♠3 ♣3

♥4 ♦4 ♠4 ♣4

♥5 ♦5 ♠5 ♣5

♥6 ♦6 ♠6 ♣6

♥7 ♦7 ♠7 ♣7

♥8 ♦8 ♠8 ♣8

♥9 ♦9 ♠9 ♣9

♥10 ♦10 ♠10 ♣10

♥K ♦K ♠K ♣K

♥Q ♦Q ♠Q ♣Q

♥J ♦J ♠J ♣J

♥Ace ♦Ace ♠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

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OR + ᵁ

AND ∩

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At least 10 mean 10, 9, 8, 7 …

At most 20 mean 20, 19, 18, 17…

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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)

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Conditional Probability

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