| $x_i$ | $f_i$ | $x_i f_i$ |
| $x_1$ | $f_1$ | $x_1 f_1$ |
| $\vdots$ | $\vdots$ | $\vdots$ |
| $x_n$ | $f_n$ | $x_n f_n$ |
$ f_1 +f_2 +f_3 +\cdots+f_n =\displaystyle\sum_{i=1}^n x_i f_i =N$
$\begin{array}{rcl} \displaystyle \text{Mean }&:& m = \displaystyle \frac{\displaystyle\sum_{i=1}^n x_i f_i }{\displaystyle\sum_{i=1}^n f_i } =\displaystyle \frac{1}{N} \sum_{i=1}^n x_i f_i \\ && \\ \end{array}$
$\begin{array}{rcl} \displaystyle \displaystyle \text{Variance } : \sigma^2 &=&\displaystyle \frac{\displaystyle\sum_{i=1}^n(x_i-m)^2 f_i}{\displaystyle\sum_{i=1}^n f_i} =\displaystyle \frac{1}{N}\sum_{i=1}^n(x_i-m)^2 f_i =\displaystyle \frac{1}{N}\sum_{i=1}^n(x_i^2-2mx_i+m^2) f_i \\ &=&\displaystyle \frac{1}{N}\sum_{i=1}^n x_i^2 f_i - 2m \times \frac{1}{N}\sum_{i=1}^n x_i f_i +m^2 \times \frac{1}{N}\sum_{i=1}^n f_i \\ &=&\displaystyle \frac{1}{N}\sum_{i=1}^n x_i^2 f_i - 2m \times m +m^2 \times \frac{1}{N} \times N \\ &=&\displaystyle \frac{1}{N}\sum_{i=1}^n x_i^2 f_i - 2m^2 + m^2 =\displaystyle \frac{1}{N}\sum_{i=1}^n x_i^2 f_i -m^2\\ \end{array}$
$\begin{array}{rcl} \text{Standard Deviation } &:& \sigma =\sqrt{\sigma^2} \\ \end{array}$
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