$\begin{array}{rcl}
\displaystyle \text{Mean : } m &=&\displaystyle\frac{x_1+x_2+x_3+\cdots+x_n}{n} \\
&=&\displaystyle \frac{1}{n}\sum_{i=1}^n x_i \\
\end{array}$
$\begin{array}{rcl}
\displaystyle \displaystyle \text{Variance : }
\sigma^2& =&\displaystyle\frac{(x_1-m)^2+(x_2-m)^2+\cdots+(x_n-m)^2}{n} \\
&=&\displaystyle \frac{1}{n}\sum_{i=1}^n(x_i-m)^2
=\displaystyle \frac{1}{n}\sum_{i=1}^n(x_i^2-2mx_i+m^2) \\
&=&\displaystyle \frac{1}{n}\sum_{i=1}^n x_i^2
- 2m \times \frac{1}{n}\sum_{i=1}^n x_i
+m^2 \times \frac{1}{n}\sum_{i=1}^n 1 \\
&=&\displaystyle \frac{1}{n}\sum_{i=1}^n x_i^2
- 2m \times m
+m^2 \times \frac{1}{n} \times n \\
&=&\displaystyle \frac{1}{n}\sum_{i=1}^n x_i^2 - 2m^2 + m^2
=\displaystyle \frac{1}{n}\sum_{i=1}^n x_i^2 -m^2\\
\end{array}$
$\text{Standard Deviation } : \sigma=\sqrt{\sigma^2}$
