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.