Measures of Central

tendanCiescalculations of

Mean, median and Mode

statistiCs topiC- 1

1.Measures of Central tendenCy

average

Mathematical

averagePositional

average

Arithmetic

mean

Geometric

mean

Harmonic

mean Median Mode

average

An average is a single figure that represent whole

group.

Mathematical

average

Arithmetic

mean

Geometric

mean

Harmonic

mean

Mathematical average

A.Arithmetic mean-

Generally if we talk about average ,it signifies

arithmetic mean.

It is based on all observations.

It is also known as mean

It is calculated value and not based on the position of

the series.

Features-

Simple Arithmetic mean-

1.Direct method 2.Short cut method

x̄̄̄̄= ∑X

N

x ̄̄= Arithmetic mean.

∑X-sum of the values of

the item of a series.

N= number of observations.

X= A+ ∑d

N

x ̄̄= Arithmetic mean.

A= Assumed mean.

∑d= sum of deviation.

N= number of observations.

Properties of Arithmetic mean-

1.The sum of deviation of item from AM is always zero.

∑(X-X)=0

2.The sum of squared deviation of AM is minimum.

B-Geometric Mean

GM define as nth root of the product of all the n values

of the variable.

If there are two items we take square root .

Two items there and their values are 4 and 9.

GM= n X1.X2.X3………Xn

GM will be- GM = 4*9 = 36 = 6

if there are three items , we take cube root.

Three items and their values 2,4,8

GM = 2*4*8 = 64 = 4

It is based on all the items of the series.

It gives less weight to large items.

It is best measure of ratio change.

C. Harmonic mean

HM is based on reciprocal of the items.

It is the reciprocal of the AM.

HM=N

1/X1 + 1/X2 + 1/X3 ………..+ 1/Xn

It is also called as – sub contrary mean.

2.Median

The median is that value of the variable which divides

the group into two equal parts.

Median is determined by first arranging the series in an

ascending or descending manner.

Median is denoted as M

M= N+1

2

1.If odd number series-

9,5,,3,6,10,12,7.

Arrange items in either ascending or descending order.

Ser.no

1

2

3

4

5

6

7

Items

3

5

6

7

9

10

12

M= N+1

2

= 7+1

2= 4

M= 7

4 7

1.If even number series-

12,16,14,18,24,20.

Arrange items in either ascending or descending order.

Ser.no

1

2

3

4

5

6

Items

24

20

18

16

13

12

M= N+1

2

= 6+1

2= 3.5

M= 17

= 2

= 3rd +4th 18+16

2

34

2=

3.Mode

Mode is define as value which occur most frequently.

8,6,14,12,8,5,10,8,14,3,8,4.

Mode is denoted as Z.

Z= 8

Relationship b/w mean, median ,and mode.

1.Perfectly symmetrical distribution.

In this case x ̄̄,M, and Z. are equal.

+VE-VE

x ̄̄=M=Z

In this case normal

distribution is bell shaped.

In normal distribution mean

is zero and variance is 1.

Relationship b/w mean, median ,and mode.

2. When distribution is positively skewed.

In this case x ̄̄,>M> Z. .

+VE-VE x ̄̄

In this case normal

distribution is +vly skewed..

MZ

Relationship b/w mean, median ,and mode.

3. When distribution is negatively skewed.

In this case x ̄̄,<M< Z. .

+VE-VE x ̄̄

In this case normal

distribution is -vly skewed..

M Z

Empirical Relationship b/w mean, median ,and

mode.

In asymmetrical distribution the difference b/w x ̄̄ and z

is 3 times the difference bw x̄̄ and M.

x ̄̄-z = 3 (x̄̄ - M)

x ̄̄=1/2(3M-Z)

Mean Median Mode

M=1/3(2x ̄̄-z) Z= 3M-2x ̄̄

prev. years ques.

Relationship b/w mean, median ,and mode.

2. When distribution is positively skewed.

In this case x ̄̄,>M> Z. .

+VE-VE x ̄̄

In this case normal

distribution is +vly skewed..

MZ

Empirical Relationship b/w mean, median ,and

mode.

In asymmetrical distribution the difference b/w x ̄̄ and z

is 3 times the difference bw x̄̄ and M.

x ̄̄-z = 3 (x̄̄ - M)

x ̄̄=1/2(3M-Z)

Mean Median Mode

M=1/3(2x ̄̄-z) Z= 3M-2x ̄̄

Measures of dispersion

&skewness

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