Skewness and Kurtosis - Measures of Skewness

Statistics >> Skewness

The absence of symmetry in a distribution is known as skewness. When a distribution is not symmetrical, it is called skewed or asymmetrical. The symmetry of a distribution shows' that at equal distances on either side of the centre, the frequencies are evenly distributed. We study skewness to have an idea about the shape of the curve which we can draw with the help of given frequency distribution.  

In a symmetrical distribution

(i)     Mean = Median = Mode

(ii)    Q₃ - Median = Median – Q₁  

(iii)    The curve of the given data is symmetrical.

A distribution is said to be positively skewed if the longer tail lie towards the right side. Here mean> median> mode.

Again a distribution is said to be negatively skewed if the longer tail lie towards the left side. Here mean < median < mode.

 

Measures of skewness

Absolute measure

The larger the distance between the mean and the mode, the greater is the skewness.

Skewness (Sk) = Mean - Mode

This measure is not suitable for comparison 

Relative Measure

1.     Karl Pearson's co-efficient of skewness

It is given by

        Sk = (Mean - Mode) / SD

If mode is not well defined we use the formula

    Sk = .3(Mean - Mode) / SD

Here - 3 £ Sk £ 3.

In practice it is rarely obtained. 

2.     Bowley's co-efficient of skewness

This is based on quartiles and is given by

                Sk = (Q3+Q1+2Q2) / Q3+Q1

        Here -1 <= Sk <= 1

The co-efficient of skewness is a pure number and is zero for a symmetrical distribution.

 

3.     Kelly's co-efficient of Skewness

                Sk = (P₉₀ + P₁₀ + 2P₅₀) / P₉₀- P₁₀

      = (D9 + D1 - 2D5) / D9 - D1

        Where P_i and D_i denote ith percentile and decile respectively. 

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