What is Variance, standard deviation? — Mathematics

Definition 17.6 High School Mathematics · Chapter 17 — Descriptive Statistics

The variance of the dataset is the mean of the squared deviations from the mean:

V = \frac{1}{n}\sum_{i=1}^{n} (x_i - \bar x)^2,

and the standard deviation is $\sigma = \sqrt{V}$ , expressed in the same unit as the data.

Examples

Example 17.8

For the running dataset, the deviations from $\bar x = 12$ are $-4$ , $-3$ , $-2$ , $-2$ , $-1$ , $0$ , $0$ , $0$ , $1$ , $2$ , $4$ , $5$ ; their squares sum to

16 + 9 + 4 + 4 + 1 + 0 + 0 + 0 + 1 + 4 + 16 + 25 = 80 .

Hence

V = \frac{80}{12} = \frac{20}{3} \approx 6.67, \qquad \sigma = \sqrt{20/3} \approx 2.6 \text{ marks}.

Roughly speaking, a typical mark sits about $2.6$ marks away from the mean.

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Examples

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