### Theory:

Deviations from the mean:
Let $$x_{1}, x_{2}, x_{3}, … , x_{n}$$ be the given data for $$n$$ observations.

The mean of the given observations is given by $$\overline{x}$$.

Then the deviations from the mean is given by, $$\left(x_{1} - \overline{x}\right)$$, $$\left(x_{2} - \overline{x}\right)$$, $$\left(x_{3} - \overline{x}\right)$$, … , $$\left(x_{n} - \overline{x}\right)$$.
Squares of deviation from the mean:
Let $$x_{1}, x_{2}, x_{3}, … , x_{n}$$ be the given data for $$n$$ observations.

The mean of the given observations is given by $$\overline{x}$$.

Then the squares of deviation from the mean is given by, $$\left(x_{1} - \overline{x}\right)^2$$, $$\left(x_{2} - \overline{x}\right)^2$$, $$\left(x_{3} - \overline{x}\right)^2$$, … , $$\left(x_{n} - \overline{x}\right)^2$$ or $$\sum_{i = 1}^{n}\left(x_{i} - \overline{x}\right)^2$$
Variance:
The mean of squares of the deviation from the mean is called variance. It is denoted by the symbol $$\sigma^{2}$$.
Let $$x_{1}, x_{2}, x_{3}, … , x_{n}$$ be the given data for $$n$$ observations.

The mean of the given observations is given by $$\overline{x}$$.

Then, Variance $$\sigma^{2} =$$ Mean of the squares of deviation.

$$\sigma^{2} = \frac{\left(x_{1} - \overline{x}\right)^2 + \left(x_{2} - \overline{x}\right)^2 + \left(x_{3} - \overline{x}\right)^2 + … + \left(x_{n} - \overline{x}\right)^2}{n}$$

$$=$$ $$\frac{\sum_{i = 1}^{n}\left(x_{i} - \overline{x}\right)^2}{n}$$
Standard deviation:
The positive square root of variance is called standard deviation. It is denoted by the symbol $$\sigma$$.
Standard Deviation, $$\sigma$$ $$=$$ $$\sqrt{\text{Variance}}$$

$$\sigma$$ $$=$$ $$\sqrt{\frac{\sum_{i = 1}^{n}\left(x_{i} - \overline{x}\right)^2}{n}}$$