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RANGE

It is the simplest measure for dispersion. For the observations x1, x2, ....... xnit is defined as

R = (largest value) - (smallest value)

For frequency distribution also, the same definition applies and frequencies do not taken into account.

Note. Of all the measures of dispersion, the S.D. is generally the one with the least sampling fluctuation. However; when the data contain a few extreme values widely different from the majority of the values, S.D. should not be used - Q.D. is the appropriate measure. 

Example 2. Find the mean deviation about the mean and variance for the following :

Solution. The calculations are shown in the following table :

Example 3. Calculate S.D., Q.D., and M.D. of the following data.

Solution.

Here N/4 12.5, then the cumulative frequency just greater than 12.5 is 19, so 9.5 - 14.5 is the  class containing Q1• L = 9.5, = 4, = 10, C = 9.

Again  3N/4 = 37.5, then the cumulative frequency just greater than 37.5 is 43, so 24.5 - 29.5 - the class containing Q3 . L = 24.5, = 4, = 9, C = 34

Example 4. Find out the S.D. of height (in em) of 10 persons given below : 

175, 167. 165. 171. 162, 165, 168, 170, 169 and 166 

Solution. Calculations using step deviation method given below :

Here = 10

 

Example S. For a set of 10 observations the A.M and S.D. were calculated and were found

to be 16 and 3.5 respectively. It was later found on serutiny that the last observation of the set should be 25 instead of 15. Calculate the correct A.M and S.D.

 

Example 6. Calculate mean and standard deviation from the following data :

 

Solution

Example 7. A factory produces two types of electric bulbs A and B. In an experiment relating  to their life, the following results were obtained.

Find which type of bulb is less variable in length of life. 

Solution.

Bulb A 

Bulb B

PROBLEMS

  1. Find the M.D. and variance of the following distribution.
        2. Find the M.D. and S.D. for the following data :
        3. With reference to the following data verify that the M.D. is the least when deviations are measured about the median instead of                               mean.
      4. Calculate the mean age and the variance of the ages of the members of a society from the following table :

5. Calculate the S.D. and M.D. about both mean and median for the first integers.

6. In a series of observations the values of mean and variance are 4.4 and 8.24. If three observations are I, 2 and 6, find the other two.

7. Two batsman A and B made the following scores in a series of cricket matches :

Who is ,more consistent player ?

8. From the data given below, find which series is more uniform.

9. Calculate (i) Q.D., and (ii) S.D. of wages from the following data

10. Measurements of the lengths in metres of 50 iron rods are distributed as follows;

Calculate (i) Q.D. and (ii) M.D. of the above data (iiz) Coefficient of Q.D. 

ANSWERS

1. M.D. = I .773, Variance = 4.432

2. M.D. = 2.03, S.D. = 2.4756

4. Mean = 48.15, Variance = 49.43

6. 4 and 9

7. C.V. (A) = 61. I% C.V. (B) = 58.47%, Batsman B is more consistent.

8. Series B. C.V. (A)= 47.05, C.V. (B) = 36.06

9. (i) Q.D. = 1.03, (ii) S.D. = 1.46

10. (i) Q.D. = 0.096, (ii) M.D. = 0.113, (iii) 0.035.