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