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Measures of Dispersion

Dr Anshul Singh Thapa

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INTRODUCTION

  • We have studied how to sum up the data into a single representative value. However, that value does not reveal the variability present in the data. Now we will study those measures, which seek to quantify variability of the data.

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Sr. No

Family I

Family II

Family III

1

12,000

7,000

0

2

14,000

10,000

7,000

3

16,000

14,000

8,000

4

18,000

17,000

10,000

5

-------------

20,000

50,000

6

--------------

22,000

--------------

Total income

60,000

90,000

75,000

Average income

15,000

15,000

15,000

It is quite obvious that averages try to tell only one aspect of a distribution i.e. a representative size of the values. To understand it better, you need to know the spread of values also.

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  • We can see that in Family I, differences in incomes are comparatively lower. In Family II, differences are higher and in Family III, the differences are the highest. Knowledge of only average is insufficient.
  • If we have another value which reflects the quantum of variation in values, our understanding of a distribution improves considerably.
  • Dispersion is the extent to which values in a distribution differ from the average of the distribution.

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  • To quantify the extent of the variation, there are certain measures namely:
    • Range
    • Quartile Deviation
    • Mean Deviation
    • Standard Deviation
  • Apart from these measures which give a numerical value, there is a graphic method for estimating dispersion.
  • Range and Quartile Deviation measure the dispersion by calculating the spread within which the values lie.
  • Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average.

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MEASURES BASED UPON SPREAD OF�VALUES

Range

  • Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus, R = L – S.
  • Higher value of Range implies higher dispersion and vice-versa.
  • Range is excessively affected by extreme values. It is not based on all the values. As long as the minimum and maximum values remain unaltered, any change in other values does not affect range. It cannot be calculated for open-ended frequency distribution.
  • Notwithstanding some limitations, Range is understood and used frequently because of its simplicity. For example, we see the maximum and minimum temperatures of different cities almost daily on our TV screens and form judgments about the temperature variations in them.

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Quartiles

  • Quartiles are the measures which divide the data into four equal parts, each portion contains equal number of observations. There are three quartiles. The first Quartile (denoted by Q1) or lower quartile has 25% of the items of the distribution below it and 75% of the items are greater than it. The second Quartile (denoted by Q2) or median has 50% of items below it and 50% of the observations above it.
  • The third Quartile (denoted by Q3) or upper Quartile has 75% of the items of the distribution below it and 25% of the items above it. Thus, Q1 and Q3 denote the two limits within which central 50% of the data lies.

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Quartile Deviation

  • The presence of even one extremely high or low value in a distribution can reduce the utility of range as a measure of dispersion. Thus, we may need a measure which is not affected by the outliers.
  • In such a situation, if the entire data is divided into four equal parts, each containing 25% of the values, we get the values of Quartiles and Median.
  • The upper and lower quartiles (Q3 and Q1, respectively) are used to calculate Inter Quartile Range which is Q3 – Q1.
  • Inter-Quartile Range is based upon middle 50% of the values in a distribution and is, therefore, not affected by extreme values. Half of the Inter-Quartile Range is called Quartile Deviation (Q.D.). Thus:
  • Q .D . = Q3 – Q1

2

  • Q.D. is therefore also called Semi-Inter Quartile Range

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Calculation of Range and Q.D. for ungrouped data

  • Calculate Range and Q.D. of the following observations:
    • 20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70
  • Range is clearly 70 – 20 = 50
  • For Q.D. = Q .D. = Q3 - Q1

2

  • Therefore for Q.D. we need to calculate values of Q3 and Q1.
  • Q1 is the size of n + 1 th value

4

  • n being 11, Q1 is the size of 3rd value.
  • As the values are already arranged in ascending order, it can be seen that Q1, the 3rd value is 29.
  • Similarly, Q3 is size of 3(n + 1) th value ; i.e. 9th value.

4

  • Hence Q3 = 51
  • Q .D. = Q3 - Q1= 51 – 29 = 11

2 2

​

  • Q.D. is the average difference of the Quartiles from the median.

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Example

  • Monthly wages:
    • 120, 150, 170, 180, 181, 187, 190, 192, 200, 210

​

  • Q1 = Size of (N + 1)th item = size of (10 + 1)th item

4 4

= 2.75th item

= 2nd item + ¾ (3rd item – 2nd item)

= 150 + ¾ (170 - 150)

Q1 = 165

  • Q3 = Size of 3(N + 1)th item = size of 3(10 + 1)th item

4 4

= 8.25th item

= 8th item + ¼ (9th item – 8th item)

= 192 + ¼ (200 - 192)

Q3 = 194

  • QD = Q3 – Q1 = 194 – 165 = 29 = 14.5

2 2 2

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Calculation of Range for a frequency distribution

Discrete Series

  • Calculate the Range of the following series:

Size:

10

11

12

13

14

15

16

18

Frequency

1

13

24

14

15

13

16

20

  • Here, H = 18 and L = 10

Range (R) = H – L = 18 – 10 = 8 (Ans)

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Calculation of Range for a frequency distribution

Continuous Series:

Find out Range of the following Series:

​

Size:

5 – 10

10 – 15

15 – 20

20 – 25

25 – 30

Frequencies:

4

9

15

30

40

Size

Mid- value

Frequency

5 – 10

7.5

4

10 – 15

12.5

9

15 – 20

17.5

15

20 – 25

22.5

30

25 – 30

27.5

40

Range = H – L = 27.5 – 7.5 = 20 (Ans)

​

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Calculation of Q.D. for grouped data

  • The following data shows daily wages of 199 workers of a factory. Find out the Q.D.:

​

Wages

10

20

30

40

50

60

70

80

90

100

No. of Workers

2

8

20

35

42

20

28

26

16

2

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  • Q1 = Size of N + 1 th item = 199 + 1 = 200 = 50th item

4 4 4

  • Q3 = Size of 3( N + 1 )th item = 3(199 + 1 )= 600 = 150th item

4 4 4

QD = Q3 – Q1 = 70 – 40 = 30 = 15

2 2 2

​

Wages

Frequency

Cumulative Frequency

10

2

2

20

8

10

30

20

30

40

35

65

50

42

107

60

20

127

70

28

155

80

26

181

90

16

197

100

2

199

​

N = 199

​

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Calculation of Q.D. for a frequency distribution

Continuous Series:

Find out Quartile Deviation of the following Series

Age (Yrs.)

0 – 20

20 – 40

40 – 60

60 – 80

80 - 100

Person

4

10

15

20

11

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Age (Years)

No. of Person

Cumulative Frequency

0 – 20

4

4

20 – 40

10

14

40 – 60

15

29

60 – 80

20

49

80 – 100

11

60

Q1 = Size of N/4 th item = 60/4 = 15th item

​

15th item lies in group 40 – 60 and falls within 29th cumulative frequency of the series.

Q1 = l1 + N/4 – c.f. x i

f

(l1 = lower limit of the class interval, N = sum total of the frequencies, c.f. = cumulative frequency of the class preceding the first quartile class, f = frequency of the quartile class, i = class interval)

​

Thus, Q1 = 40 + 15 – 14 x 20 = 41.33

15

​

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Age (Years)

No. of Person

Cumulative Frequency

33 – 35

14

​

35 – 37

62

​

38 – 40

99

​

41 – 43

18

​

43 – 45

7

​

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Q3 = Size of 3(N/4)th item = 3(60/4) = 45th item

​

45th item lies in group 60 – 80 and falls within 49th cumulative frequency of the series.

Q3 = l1 + 3(N/4) – c.f. x i

f

(l1 = lower limit of the class interval, N = sum total of the frequencies, c.f. = cumulative frequency of the class preceding the third quartile class, f = frequency of the quartile class, i = class interval)

​

Thus, Q3 = 60 + 45 – 29 x 20 = 76

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Having known the values of Q1 and Q3, Quartile Deviation QD is found as;

QD = Q3 – Q1 = 76 – 41.33 = 17.34

2 2

​