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KURTOSIS

This is a measure of peakness of a distribution and is defined in terms of second and fourth

central moments as follows (due to Karl Pearson) :

Example 1. Calculate the Karl Pearson :S coefficient of skewness of the following frequency distribution.

Solution.

To calculate mode \1/e require class boundaries. Since the maximum frequency is 17, this implies that 397.5 - 402.5 is the modal class.

which indicate the given distribution is negatively skewed.
Example 2. The first four moments of a distribution about the value 3 of a variable are 1, 5, 14 and 46.
Find the mean, variance and comment on the nature of the distribution.

Solution. Given A = 3,

Now, 

Since The distribution is positively skewed.

Since  The distribution is platykurtic.

Example 3. For a distribution, Bowley's coefficient of skewness is - 0.65, Q1=15.28 and median = 25.2. What is its coefficient of quartile deviation ?
Solution. Bowley's coefficient of skewness is

Then, coefficient of quartile deviation

                                                                                                                        PROBLEMS

1. The mean, median and the coefficient of variation of 100 observations are found to be 60, 56 and 30 respectively. Find the coefficient of skewness of the above system of 100 observations.

2. Compute the Karl Pearson's coefficient of skewness of the following distribution :

3. Compute mean, variance, β1 and β2 if the first four moments about a value 5 of a variable are given as 2, 20, 38 and 52.
4. The first three moments of a distribution about the value 3 are -1, 10, -28. Find the values of mean, standard deviation and the moment measure of skewness.
5. For a distribution the mean is 9, standard deviation is 5, √β1 = 1 and √β2 = 3. Obtain the first four moments about the origin i.e., zero.
6. Find the appropriate measure of skewness from the following data :

7. Find the skewness and kurtosis of the following distribution by central moments and comment on the type.

8. Find the C.V. of a frequency distribution given that its mean is 100, mode= 120 and Karl Pearson's coefficient of skewness = - 0.2.
9. Compare the skewness of two frequency distribution whose moments about the origin are as follows :

10. Calculate the coefficient of skewness and kurtosis of the following data:

                                                                                                             ANSWERS

  1. 0.67
  2. 0.176
  3. x¯= 7, μ2 =16, β1= 1.06, β2 = 0.70
  4. Mean= 2, S.D. = 3, β1 = 0
  5. µ´1= 9, µ´2 = 106, µ´3= 1529, µ´4 = 25086
  6. Bowley's coefficient of skewness is suitable and = 0.24
  7. β1 =0, β2 = 2.5, symmetrical and platykurtic
  8. 100
  9. A is symmetric, 8 is positively skew
  10. β1 = 0.038, β2 = 1.806.