MEASURES OF CENTRAL TENDENCY

(Mean , Median , Mode)

(Business Statistics)

RAM SINGH

ROLL NO.– 85

M.B.A.- 2.1

MEANING OF AVERAGE An average is a single value which represents the whole set of figure and individual item concentrate around it.

DEFINATION “An average is a single figure that represents the whole group.’’ clark

CHARACTERSTICS OF AVERAGE

It should be easy to understand.It should be simple to compute.It should be uniquely define.It should be based on all observation.It should be capable of future algebraic

treatment.

Arithmetic meanArithmatic mean may be defined as a value

which is obtained by adding all the item of a series and dividing this total by the number of item.

Simple Arithmatic Mean-

Direct method-

n

XX

Example- Direct methodThe pocket allowance is given. Calculate arithmatic mean

STUDENT POCKET ALLOWANCE (X)

1 15

2 20

3 30

4 22

5 25

6 18

7 40

8 50

9 55

10 65

N=10 ∑X=340

n

XX

= 340/10=34

(Short cut method) mean=A+∑d/N

Mean= 40+(-60)/10=34

students X d=X-A(A=40)

1 15 15-40=-25

2 20 20-40=-20

3 30 30-20=-10

4 22 22-40=-18

5 25 25-40=-15

6 18 18-40=-22

7 40 40-40=0

8 50 50-40=10

9 55 55-40=15

10 65 65-40=25

N=10 ∑d=-60

Discrete series-1. direct method n

XfX

Wages (Rs.)

10 20 30 40 50

No. Of workers

4 5 3 2 5

Wages (x) No of worker(f) fx

10 4 40

20 5 100

30 3 90

40 2 80

50 5 250

N=19 ∑fx=560

Mean=560/19=29.47

2. Short cut method-Mean=A+∑fd/N, (here, d=X-A) n=∑f

Wages (x) 10 20 30 40 50

F 4 5 3 2 5

Wages (x)

f fx d=X-A(A=30)

fd

10 4 40 10-30=-20

-80

20 5 100 -10 -50

30 3 90 0 0

40 2 80 10 20

50 5 250 20 100

∑f=19 ∑fd=-10

Mean=30+(-10)/19=29.47

Continuous series-1. Direct methodMean=∑fm/N, (here, m= mid-value, N=∑f)

Mean=3620/140= 25.85

marks 0-10 10-20 20-30 30-40 40-50

No. Of student

20 24 40 36 20

marks f Mid value

fm

0-10 20 5 100

10-20 24 15 360

20-30 40 25 1000

30-40 36 35 1260

40-50 20 45 900

∑f=140 ∑fm= 3620

solution

Short-cut method-Mean=∑fm/NExample

Mean= 25+120/140=25.85

Marks No. of students

Mid-value

A=25d=M-A

fd

0-10 20 5 5-25=-20 -400

10-20 24 15 -10 -240

20-30 40 25=A 0 0

30-40 36 35 10 360

40-50 20 45 20 400

N=140 ∑fd=120

MEDIAN-Median is defined as the middle value of the series when they arranged in ascending and descending order.

MedianMedian Location = (N +1)/2 th item

Here, M= Median, N= No. of items

Example- Calculate median from the following data- 22,16,18,13,15,19,17,20,23

Median Location = (N +1)/2 th item

= (9+1)/2 th item

5th item=18

Hence, M=16.

Z Item (X)

1 13

2 15

3 16

4 17

5 18

6 19

7 20

8 22

9 23

N=9

Discrete series-Median Location = (N +1)/2 th item

Here, N=total of frequency

Example- from the following data calculate median.

X 10 12 14 16 18 20 22

Y 2 5 12 20 10 7 3

Solution-X f c.f.

10 2 2

12 5 7

14 12 19

16 20 39 =M

18 10 49

20 7 56

22 3 59

N=59

Median Location = (N +1)/2 th item59+1/2=30th item=16Hence, M=16.

Continuous Series-

Median Location = (N +1)/2 th item

)(2 if

CFn

LMedian

Calculate median-

Movies showing

Frequency Cumulative Frequency

1 up to 3 1 1

3 up to 5 2 3

5 up to 7 3 6

7 up to 9 1 7

9 up to 11 3 10

Movies showing

1 Up to 3 3 Up to 5 5 Up to 7 7 Up to 9 9 Up to 11

frequency 1 2 3 1 3

From the table, L=5, n=10, f=3, i=2, CF=3

33.6)2(3

32

10

5)(2

if

CFn

LMedian

MODE Mode is defined as the value which occurs most frequency in a series.

In other words, it is a value which has the highest frequency in a distribution. Mode is denoted “Z”.

Individual Series1. Inspection method-Example-

Find the Mode from the following data-

8,10,5,8,12,7,8,9,11,7.

Solution- An inspection of the series shows that the value 8 occurs most frequency in the Series

Hence, mode (Z) =8.

2. By changing the individual Series in Discrete Series-Example- Find the mode from the following data-

11.1,10.9,10.7,11.1,10.6,11.3,1o.6,10.7,10.6,10.9,10.6,10.5.10.4,10.6.

Solution. Firstly we convert the given series into a discrete series in ascending order as follows;

Size; 10.4 10.5 10.6 10.7 10.9 11.1 11.3

Frequency;

1 1 5 2 2 2 3

The modal value is 10.6. Since it appears maximum number of times in the series.

Continuous Series:- In this series we calculate the mode with the help of

this formula:

Here, L1 = Lowest valuef1=frequence , h or i = class-interval difference

Example :-calculate the mode from the following data:-

Wages:

0-5 5-10 10-15 15-20 20-25 25-30 30-35(f)

No. Of workers:

3 7 15 30 20 10 5

wages: f

0-5 3

05-10 7

10-15 15f0

L1=15-20 30 f1

20-25 20f2

25-30 10

30-35 5

Solution:

Here, L1=15 , f1=30, f0=15, f2 =20, i =5 Z=15+{30-15/2(30)-15-20}×5

=18

Thus, mode= 18.

..

Thank you