Variance

159. Variance

Quantifies the spread or dispersion of a set of data points in a dataset

𝜎^{2} = \frac{1}{𝑁} \sum_{𝑖 = 1}^{𝑛} {(𝑥_{𝑖} - 𝜇)}^{2}

𝑠^{2} = \frac{1}{𝑛 - 1} \sum_{𝑖 = 1}^{𝑛} {(𝑥_{𝑖} - \bar{𝑥})}^{2}

Example

[10, 12, 13, 17, 20, 24]

Step 1: Find mean

\bar{𝑥} = \frac{10 + 12 + 13 + 17 + 20 + 24}{6} = \frac{96}{6} = 16

Step 2: Subtract the Mean and Square the result

\begin{matrix} {(10 - 16)}^{2} = {−6}^{2} = 36 \\ {(12 - 16)}^{2} = {−4}^{2} = 16 \\ {(13 - 16)}^{2} = {−3}^{2} = 9 \\ {(17 - 16)}^{2} = 1^{2} = 1 \\ {(20 - 16)}^{2} = 4^{2} = 16 \\ {(24 - 16)}^{2} = 8^{2} = 64 \end{matrix}

Step 3: Calculate variance

𝑠^{2} = \frac{36 + 16 + 9 + 1 + 16 + 64}{6 - 1} = \frac{142}{5} = 28.4