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Measures Of Dispersion

Question
CBSEENST11024832
Wired Faculty App

Prove by an example that the variance is unaffected by the choice of the assumed mean.

Solution

We take the following example for proving that the variance is uneffected by the choice of the assumed mean.

Example : Calculate variance of 25, 50, 45, 30, 70, 42, 36, 48, 34 and 60 by actual mean assumed mean method.

(a) Calculation of Variance by Actual Mean Method

Values of X

(x)

x2

25

–19

361

50

+6

36

45

+1

1

30

–14

196

70

–26

676

42

–2

4

36

–8

64

48

+4

16

34

–10

100

60

+16

256

ΣX = 440

  

ΣX2 = 1710

(b) Calculation of Variance by Assumed Mean Method

Values X

(X–45) d

d2

25

–20

400

50

+5

25

45

0

0

30

–15

225

70

+25

625

42

–3

9

36

–9

81

48

+3

9

34

–11

121

60

+15

225

N=10

Σd = –10

Σd2 =1720

Some More Questions From Measures of Dispersion Chapter

In a town 25% of the persons earned more than Rs. 45000, whereas 75% earned more than Rs. 18000. Calculate the absolute and relative values of dispersion.

A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are:

X

Y

25

50

85

70

40

65

80

45

120

80

Which batsman should be selected if we want

(i) a higher run getter, or

(ii) a more reliable batsman in the team?

To check the quality of two brands of lightbulbs, their life in burning hours was estimated as under for 100 bulbs of each brand.

Life (in hrs)

No. of bulbs

Brand A

Brand B

0–50

15

2

50–100

20

8

100–150

18

60

150–200

25

25

200–250

22

5
 

100

100

(i) Which brand gives higher life?

(ii) Which brand is more dependable?

Average daily wage of 50 workers of a factory was Rs. 200 with a standard deviation of Rs. 40. Each worker is given a raise of Rs. 20. What is the new average daily wage and standard deviation? Have the wages become more or less uniform?

In the previous question, calculate the relative measures of variation and indicate the value, which in your opinion is more reliable.

If in the previous question, each worker is given a hike of 10% in wages, how are the mean and standard deviation values affected?

The sum of 10 values is 100 and the sum of their squares is 1090. Find the coefficient of variation.

Calculate the mean deviation about mean and standard deviation for the following distribution:

Classes

Frequencies

20–40

3

40–80

6

80–100

20

100–120

12

120–140

9
 

50

A measure of dispersion is a good .supplement to the central value in understanding a frequency distribution. Comment

Define dispersion.

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