Skewness and Kurtosis in Statistics

The average and measure of dispersion can describe the distribution but they are not sufficient to describe the nature of the distribution. For this purpose we use other concepts known as Skewness and Kurtosis. The symmetrical and skewed distributions are shown by curves as
Skewness means lack of symmetry. A distribution is said to be symmetrical when the values are uniformly distributed around the mean. For example, the following distribution is symmetrical about its mean 3.
x                      :           1          2            3        4          5
frequency  (f ) :           5          9          12        9          5

In a symmetrical distribution the mean, median and mode coincide, that is, mean = median = mode.
Several measures are used to express the direction and extent of skewness of a dispersion. The important measures are that given by Pearson. The first one is the Coefficient of Skewness:

For a symmetric distribution Sk = 0. If the distribution is negatively skewed then Sk is negative and if it is positively skewed then Sk is positive. The range for Sk is from -3 to 3.

The other measure uses the b (read ‘beta’) coefficient which is given by,  where, m2 and m3 are the second and third central moments. The second central moment m2 is nothing but the variance. The sample estimate of this coefficient is  where m2 and m3 are the  sample central moments given by  

For a symmetrical distribution b1 = 0. Skewness is positive or negative depending upon whether m3 is positive or negative.

A measure of the peakness or convexity of a curve is known as Kurtosis.

It is clear from the above figure that all the three curves, (1), (2) and (3) are symmetrical about the mean. Still they are not of the same type. One has different peak as compared to that of others. Curve (1) is known as mesokurtic (normal curve); Curve (2) is  known as leptocurtic (leading curve) and Curve (3) is known as platykurtic (flat curve). Kurtosis is measured by Pearson’s coefficient, b2 (read ‘beta - two’).It is given by .
The sample estimate of this coefficient is  
 where, m4 is the fourth central moment given by m4

The distribution is called normal if b2 = 3. When b2 is more than 3 the distribution is said to be leptokurtic. If b2 is less than 3 the distribution is said to be platykurtic.