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Unit 8: Skewness and Kurtosis: Karl Pearson, Bowley, Kelly's Methods

γ
For a normal distribution 2 = 0. If γ is positive, the curve is leptokurtic and if γ is negative, the Notes
2
2
curve is platykurtic.
Example 7: The first four central moments of a distribution are 0, 2.5, 0.7 and 18.75. Test the
skewness and kurtosis of the distribution.
Solution:
Testing Skewness
We are given μ 1 = 0, μ = 2.5, μ = 0.7 and μ = 18.75
2
3
4
Skewness is measured by the coefficient β 1
μ 2
β 1 = μ 3 2 3

Here μ 2 = 2.5, μ = 0.7
3
( )0.7 2
Substituting the values, β = = + 0.031
1
( )2.5 3
Since β = + 0.031, the distribution is slightly skewed.
1
Testing Kurtosis:
For testing kurtosis we compute the value of β . When a distribution is normal or
2
symmetrical, β = 3. When a distribution is more peaked than the normal, β is
2
2
more than 3 and when it is less peaked than the normal, β is less than 3.
2
μ
β 2 = μ 2 4 2

μ 4 = 18.75, μ = 2.5
2
18.75 18.75
∴ β 2 = ( )2.5 2 = 6.25 = 3

Since β is exactly three, the distribution is mesokurtic.
2
Self-Assessment

1. Fill in the blanks:
(i) If Q = 30, Q = 20, Med = 25, Coeff. of Sk. shall be ............
3 1
(ii) If X = 50, Mode = 48, σ = 20, the Coefficient of Skewness shall be ............
(iii) If Coeff. of Sk. = 0.8, Median = 35, σ = 12, the mean shall be ............
(iv) In a symmetrical distribution the coefficient of skewness is ............
(v) The limits for Bowley’s coefficient of skewness are ............

8.4 Summary

• The nature of distribution is further studied deeply by calculating ‘Moments’ which reveals
whether the symmetrical curve is normal, more flat than normal or more peaked than normal.

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