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Formulas

Given a set of n numbers: $\left\{ x_{1},x_{2},x_{3},x_{4}......,x_{n}\right\}$

Mean

Mean$=\frac{\left( \underset{i=1}{\overset{n}{\sum }}x_{i}\right) }{n}$

Standard deviation

Standard deviation $=\sqrt{\frac{\underset{i=1}{\overset{n}{\sum }}\left( x_{i}-Mean\right) ^{2}}{n-1}}$

Example

Let $X$ be a set of numbers and
$X = \{1, -2, 3, 0, 2, 3, 1, -1, 3, 5\}$
The number of the elements is 10.

The sum of the elements is $\underset{i=1}{\overset{10}{\sum }}x_{i}=15$

Mean $=\frac{\underset{i=1}{\overset{10}{\sum }}x_{i}}{n}=\frac{15}{10}=\frac{3}{2}$

$\left( X-Mean\right) =\left\{ -\frac{1}{2}, -\frac{7}{2}, \frac{3}{2}, -\frac{3}{2}, \frac{1}{2}, \frac{3}{2}, -\frac{1}{2}, -\frac{5}{2}, \frac{3}{2}, \frac{7}{2}\right\}$

$\left( X-Mean\right)^{2}=\left\{\frac{1}{4}, \frac{49}{4}, \frac{9}{4}, \frac{9}{4}, \frac{1}{4}, \frac{9}{4}, \frac{1}{4}, \frac{25}{4}, \frac{9}{4}, \frac{49}{4}\right\}$

$\underset{i=1}{\overset{10}{\sum }}\left( x_{i}-Mean\right)^{2}=\frac{81}{2}$

Standard deviation $=\sqrt{\frac{81}{18}}=\allowbreak \frac{3}{2}\sqrt{2}$

Mean $=\frac{3}{2}$

Variance

Variance = standard deviation × standard deviation

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