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Measures Of Dispersion

Question

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The distribution of the cost of production (in rupees) of a quintal of wheat in 50 farms in as follows:

Cost (in rupees)

40–50

50–60

60–70

70–80

80–90

90–100

Total

Number of farms

3

6

12

18

9

2

50

(a) Calculate the variance

(i) by direct method, (ii) by step deviation method and compare your results with the mean deviation about the arithmetic mean.

(b) Calculate the coefficient of variation by using:

(i) The standard devitation of cost and (ii) the mean deviation of costs about the arithmetic mean, and compare the two. What is your conclusion about variation of cost?

Solution

(a) (i) Calculation of Variance by Direct Method

Class	Frequency	Mid-points (X)	(xf)		f\|D\| df	fm	d²	fd²
40 – 50	3	45	135	–26	78	135	676	2028
50 – 60	6	55	330	–16	96	330	256	1536
60 – 70	12	65	780	–6	72	780	36	432
70 – 80	18	75	1350	4	72	1350	16	288
80 – 90	9	85	765	14	126	765	196	1764
90 – 100	2	95	190	24	48	190	576	1152
	50		Σnf = 3550	Σd = 90	Σ × P = 492			Σfd² = 7200

(ii) By Step Deviation Method :

Class	f	m	d(M–n)	d'²	fd'	fd'²
40 – 50	3	45	–3	9	–9	27
50 – 60	6	55	–2	4	–12	24
60 – 70	12	65	–1	1	–12	12
70 – 80	18	75	0	0	0	0
80 – 90	9	85	1	1	9	9
90 – 100	2	95	2	4	4	8
					Σfd'=-20	Σfd'²= 80

(b) Calculation of Coefficient of Variation

Some More Questions From Measures of Dispersion Chapter

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.

How many methods are there to calculate dipersion?

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