Question

How do you interpret mean and standard deviation?

Answer 1

Mean.

Mean of a set of observations is their sum divided by the number of observations. Mean give us an idea of the concentration of the observations about the central part of the distribution. For the set of observations $x_{i}i=1,2,3,.......n$,

$\mathit{M}\mathit{e}\mathit{a}\mathit{n}\mathbf{}\mathbf{\left(}\mathit{\mu }\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{\mathbf{n}}$

Standard deviation

Standard deviation, usually denoted by the Greek letter $\sigma$, is the positive square root of the arithmetic mean of the squares of the deviations of the given values from their arithmetic mean. A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. For the set of observations $x_i$, $i=1,2,3,......n$

$\mathit{S}\mathit{t}\mathit{a}\mathit{n}\mathit{d}\mathit{a}\mathit{r}\mathit{d}\mathbf{}\mathit{d}\mathit{e}\mathit{v}\mathit{i}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{}\mathbf{\left(}\mathit{\sigma }\mathbf{\right)}\mathbf{=}\sqrt{\frac{{\displaystyle \sum _{i=1}^n{(x_i-\mu )}^2}}{\mathbf{n}}}$

For example, For the set of observations $2,3,4,-1,6,7,$

$\begin{array}{rcl}\mu \mathbf{}& =& \frac{2+4+3+(-4)+6+7}{6}\\ & =& 3\end{array}$

$\begin{array}{rcl}\sigma & =& \sqrt{\frac{{(2-3)}^2+{(3-3)}^2+{(4-3)}^2+{(-1-3)}^2+{(6-3)}^2+{(7-3)}^2}{6}}\\ & =& \sqrt{\frac{1+1+16+9+16}{6}}\\ & =& \sqrt{\frac{43}{6}}\end{array}$

Hence, for the given set of observations, mean is $\mathbf{3}$ and standard deviation is $\sqrt{\frac{\mathbf{43}}{\mathbf{6}}}$