Mathematical measures of skewness can be calculated by :
(a) Bowley’s Method
(b) Karl-Pearson’s Method
(c) Kelly ‘s method
(a) Bowley’s Method
Bowley’s method of skewness is based on the values of median, lower and upper quartiles. This method suffers from the same limitations which are in the case of median and quartiles. Wherever positional measures are given, skewness should be measured by Bowley’s method. This method is also used in case of ‘open-end series’, where the importance of extreme values is ignored.
Absolute skewness = Q3 + Q1 – 2 Median
Coefficient of Skewness =
Coefficient of skewness lies within the limit ± 1. This method is quite convenient for determining skewness where one has already calculated quartiles.
(b) Karl-Pearson’s Method (Personian Coefficient of Skewness)
Karl Pearson has suggested two formulae;
(i) where the relationship of mean and mode is established;
(ii) where the relationship between mean and median is not established.
When the values of Mean When the values of Mean
and Mode are related and Median are related
Absolute skewness = Mean – Mode Absolute skewness = 3(Mean – Median)
Coefficient of skwenes = Coefficient of skweness =
Coefficient of skewness generally lies within + 1 Coefficient of skewness generally lies within + 3
Measures of Kurtosis
Measure of kurtosis is denoted by β2 and in a normal distribution β2= 3.
If β2 is greater than 3, the curve is more peaked and is named as leptokurtic. If β2 is less than 3, the
curve is flatter at the top than the normal, and is named as platykurtic. Thus kurtosis is measured by
β2 = where x = (X – )
R.A. Fisher had introduced another notation Greek letter gamma, symbolically.
γ2 = β2 – 3 = = 3.
In this case of a normal distribution, γ2 is zero. γ2 is more than zero (positive), then the curve is platykurtic and if γ2 is less than 0 (negative) then the curve is leptokurtic.
It may be noted that μ4 = is an absolute measure of kurtosis but β2 = is a relative measure of kurtosis. Larger the value of γ2 in a frequency distribution, the greater is its departure from normality.
β1 and β2 are measures of symmetry and normality respectively. If β2 = 0, the distribution is symmetrical and if β2 = 3, the distribution curve is mesokurtic.