Elementary Statistics (1+1)

8.1.2 Measures of skewness

Various measures of skewness are

1) S_k = M – M_d 2) S_k = M – M_o

Where M is the mean, M_d, the median and M_o, the mode of the distribution.

3) S_k = (Q₃ – M_d) – (M_d – Q₁)

These are the absolute measures of skewness. As in dispersion, for comparing two series we do not calculate these absolute measures but we calculate the relative measures called the co-efficients of skewness which are pure numbers independent of units of measurement. The following are the co-efficients of skewness:

1) Prof. Karl Pearson’s Co-efficient of Skewness

where s is the standard deviation of the distribution

It has been shown that for any distribution, (M-M_d) / s lies between +1. Hence the limits for the co-efficient of skewness are +3. In practice, these limits are rarely attained.

Skewness is positive if M> M_o or M>M_d and negative if M<M_o or M<M_d.

2. Prof. Bowley’s Co-efficient of Skewness

Based on quartiles,

Thus Sk = + 1 if Md = Q1 and Sk = -1 if Q3 = Md. Bowley’s co-efficient of skewness lies between + 1

3. Based upon moments, co-efficient of skewness is

Where symbols have their usual meanings. Thus Sk = 0 if either b ₁ = 0 or b ₂ = -3. But since b ₂ = m ₄ / m ₂ ² , cannot be negative, Sk = 0 if and only if b ₁ = 0. In this respect b 1 is taken to be a measure of skewness. The co-efficient in (3) is to be regarded as without sign.

We observe in (1) and (2) that skewness can be positive as well as negative. The skewness is positive if the larger tail of the distribution lies towards the higher values of the variate (the right), i.e. if the curve drawn with the help of the given data is stretched more to the right than to the left and is negative in the contrary case.