What is Measure of Skewness?
A frequency distribution can be represented by drawing a curve or a graph. The measure of skewness is used to study the shape of a curve that can be drawn by plotting the data of a frequency distribution on a graph.
As you have learned in the preceding sections, through a measure of central tendency, you measure the concentration of values of a data series in the middle of a frequency distribution. Through a measure of dispersion, you measure the scattering of values near the middle value of the data series.
It may be possible that two data series, which are widely different in nature and composition, have the same mean and standard deviation. However, when you plot the data of such series on graphs, you obtain curves with different shapes. This shows that the measures of central tendency and dispersion are not sufficient to study the frequency distribution of a data series because they do not talk about the shape of the frequency distribution curves.
Therefore, you need skewness to gain understanding of the different shapes of various frequency distribution curves. The measure of skewness is used when the concentration of values of a data series is more on a single side that is either positive or negative. Skewness can be classified as positive skewness and negative skewness.
This is shown in Figure:
Positive skewness implies that the concentration of values is on the right side of the curve, whereas negative skewness implies that the concentration of values is on the left side of the curve. Skewness is calculated by taking the difference of mean and mode. In positive skewness, the values of these three measures of central tendency are in the following order:
Mean (X) > Median (M) > Mode (Z)
However, in the case of negative skewness, the values of these three measures of central tendency are in the following order:
Mean (X) < Median (M) < Mode (Z)
The formula to calculate skewness is as follows:
Skewness = X – Z
The coefficient of skewness is the relative measure of skewness that can be calculated by dividing skewness with standard deviation. The formula used to calculate the coefficient of skewness is as follows:
Coefficient of Skewness = k X – Z S =
For a moderately skewed, if there is more than one mode or if there is no mode, then you need to calculate skewness and the coefficient of skewness using the method of moments. Let us now calculate skewness and the coefficient of skewness with the help of an example.
Suppose you want to calculate the skewness and the coefficient of skewness of the data given in Table:
|People||Age (Years)||(Xi -X)||(Xi -X)2|
|Total||∑Xi = 89||∑(Xi –X)2 = 2.80|
The mean of age is calculated as follows:
Mean of Age, X = ∑Xi /n
X = 89/5
X = 17.8
The median of age is calculated as follows:
Median, M = Value of (n + 1/2)th observation
M = (5 + 1/2)th observation
M = 3rd observation = 18
Since the data contains two modes (17 and 18), you do not consider mode in this case.
The SD of age is calculated as follows:
σ = √∑ (Xi – X)2 /n
= √2.80/5 ≅ 0.75
Skewness is calculated as follows:
Skewness = 3(17.8 – 18) = 0.6
The coefficient of skewness is calculated as follows:
Coefficient of Skewness = 0.6/0.75 = 0.8
The skewness in the age of five friends is 0.6 and the relative measure of skewness is 0.8.