Measures of Asymmetry
3. Skewness – exercise

Instruction

As we already know from the previous exercise, skewness is given by the formula

Skewness=\dfrac{\frac{1}{n}((x_1-\overline{x})^3+(x_2 - \overline{x})^3 + \ldots + (x_n - \overline{x})^3)}{\sqrt{(\frac{1}{n-1}((x_1-\overline{x})^2+\ldots+(x_n-\overline{x})^2))^3}}

Consider the following set:

1, 2, 3, 3, 4, 4, 4, 5

The corresponding histogram is shown below. It looks like a left-tailed histogram, so we'd expect skewness to be negative.

We can calculate the skewness for this dataset using the above formula. First, we find the arithmetic mean \overline{x}=3.25. The rest of the calculations are given in the table below:

x_i x_i-\overline{x} (x_i-\overline{x})^2 (x_i-\overline{x})^3
1 -2.25 5.0625 -11.390625
2 -1.25 1.5625 -1.953125
3 -0.25 0.0625 -0.015625
3 -0.25 0.0625 -0.015625
4 0.75 0.5625 0.421875
4 0.75 0.5625 0.421875
4 0.75 0.5625 0.421875
5 1.75 3.0625 5.359375
SUM 0 11.5 -6.75
Skewness = \dfrac{\frac{1}{8}(-6.75)}{\sqrt{(\frac{1}{8-1}11.5})^3}=\frac{-0.84375}{\sqrt{1.642857^3}}=-0.4006951

A negative value implies an asymmetric histogram with a longer left tail. This histogram's asymmetry, however, is rather small.