# What are the variance and standard deviation of {8, 29, 57, 3, 8, 95, 7, 37, 5, 8}?

Jan 21, 2016

$s = {\sigma}^{2} = 815.41 \to$ variance

$\sigma = 28.56 \to$ 1 standard deviation

#### Explanation:

The variance is a sort of mean measure of the variation of the data about the line of best fit.

It is derived from: ${\sigma}^{2} = \frac{\sum \left(x - \overline{x}\right)}{n}$

Where $\sum$ means add it all up

$\overline{x}$ is the mean value (sometimes they use $\mu$)

$n$ is the count of data used

${\sigma}^{2}$ is the variance (sometimes they use $s$)

$\sigma$ is one standard deviation

This equation, with a bit of manipulation end up as:

${\sigma}^{2} = \frac{\sum \left({x}^{2}\right)}{n} - {\overline{x}}^{2} \text{ }$ for variance

$\sigma = \sqrt{\frac{\sum \left({x}^{2}\right)}{n} - {\overline{x}}^{2}} \text{ }$ for 1 standard deviation
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Rather than building a table of values I used a calculator to do the work for me:

${\sigma}^{2} = \frac{\sum \left({x}^{2}\right)}{n} - {\overline{x}}^{2} \text{ }$

becomes:

${\sigma}^{2} = \frac{14759}{10} - {\left(25.7\right)}^{2}$

$s = {\sigma}^{2} = 815.41 \to$ variance

$\sigma = 28.56 \to$ 1 standard deviation