What is Skewness and Four Measures of Skewness for a Distribution

 

There are majorly two measures of shape of the distribution. They are -

1. Skewness
2. Kurtosis

In this blog, we will cover the skewness measure of shape.

Skewness

Skewness is a measure of the symmetry of the shape of a distribution. If a distribution is symmetric, the skewness will be zero. If there is a long tail in the positive direction, skewness will be positive, while if there is a long tail in the negative direction, skewness will be negative.

The concept of skewness helps to understand the relationship between the mean, median, and mode. In a unimodal distribution (a distribution with a single peak or mode) that is skewed, the mode is the apex (high point) of the curve and the median is the middle value. The mean tends to be located toward the tail of the distribution because the mean is particularly affected by the extreme values. A bell-shaped or normal distribution with the mean, median, and mode all at the center of the distribution has no skewness.

Moment measure of skewness

A coefficient of skewness, independent of the units of measurement is given by –

In symmetrical distributions, β1 shall be zero. However, the coefficient β1, as a measure of skewness has limitations. β1 as a measure of skewness cannot tell about the direction of skewness that is whether the distribution is positively skewed or negatively skewed. This is because of the μ3 term being squared yields only positive value. While μ2 being the variance is always positive. Thus, β1 is always positive irrespective of μ2 and μ3. We can either use Karl Pearson’s γ1 as a measure to find out the sign of skewness or refer to the value of μ3. If μ3 is positive, we will have positive skewness and if μ3 is negative, we will have negative skewness.

Karl Pearson’s measure of skewness

If you have the mean, mode, and standard deviation then you can use Karl Pearson’s coefficient of skewness illustrated below –

Karl Pearson’s coefficient of skewness Sk is a relative measure, independent of the units of measurement. 

In situations where the mode is not defined for the distribution, we can make use of the empirical relation between mean, median, and mode which states that for a moderately symmetrical distribution, we have –

Hence, Karl Pearson’s coefficient of skewness Sk in terms of mean, median and standard deviation can be written as –

If the coefficient of skewness Sk is positive, the distribution is positively skewed. If the value of Sk is negative, the distribution is negatively skewed. The greater the magnitude of Sk, the more skewed is the distribution. For symmetrical distributions like the normal distribution, mean and median values are the same, therefore the coefficient Sk turns out to be 0.

Bowley’s measure of skewness

Bowley’s coefficient (SQ) is a relative measure of skewness based on quartiles. It is given by –

Here, Q3 and Q1 are the third and first quartiles respectively. Q2 is the second quartile or the median.

Kelly’s measure of skewness

Bowley’s coefficient of skewness makes use of quartiles and thus eliminates the first and last 25% of the observations. As an improvement over this, Kelly’s coefficient of skewness is based on percentiles. It eliminates the first ten and last ten percentiles of the total observations. It is denoted by SP and given by –

Here, P10 and P90 are the 10th and 90th percentiles respectively. P50 is the 50th percentile or the median.

Hope this blog gives you an idea about the skewness measures. In the next blog, we will cover the kurtosis measures which is another measure of the shape of the distribution.