# Formula for Variance and Standard Deviation

Here, you will learn learn formula for variance and standard deviation and relationship between variance and standard deviation.

Let’s begin –

## Variance and Standard Deviation :

The variance of a distribution is, the mean of squares of deviation of variate from their mean. It is denoted by $$\sigma^2$$ or var(x).

The positive square root of the variance are called the standard deviation. It is denoted by $$\sigma$$ or S.D.

Hence standard deviation = + $$\sqrt{variance}$$

## Formula for Variance :

(i) for ungrouped distribution :

$${\sigma^2}_x$$ = $$\sum(x_i – \bar{x})^2\over n$$

$${\sigma^2}_x$$ = $$\sum{x_i}^2\over n$$ – $$\bar{x}^2$$

= $$\sum{x_i}^2\over n$$ – $$({\sum{x_i}\over n})^2$$

$${\sigma^2}_d$$ = $$\sum{d_i}^2\over n$$ – $$({\sum{d_i}\over n})^2$$, where $$d_i$$ = $$x_i$$ – a

(ii) for frequency distribution :

$${\sigma^2}_x$$ = $$\sum f_i(x_i – \bar{x})^2\over N$$

$${\sigma^2}_x$$ = $$\sum f_i{x_i}^2\over N$$ – $$\bar{x}^2$$

= $$\sum f_i{x_i}^2\over N$$ – $$({\sum f_i{x_i}\over N})^2$$

$${\sigma^2}_d$$ = $$\sum f_i{d_i}^2\over n$$ – $$({\sum f_i{d_i}\over n})^2$$, where $$d_i$$ = $$x_i$$ – a

$${\sigma^2}_d$$ = $$h^2$$[$$\sum f_i{u_i}^2\over n$$ – $$({\sum f_i{u_i}\over n})^2$$],  where $$u_i$$ = $$x_i\over h$$

(iii) Coefficient of Standard Deviation = $$\sigma\over \bar{x}$$

Coefficient of variation = $$\sigma\over \bar{x}$$ $$\times$$ 100   (in percentage)

Example : Find the variance and standard deviation of first n natural numbers.

Solution : We know that, $${\sigma^2}_x$$ = $$\sum{x_i}^2\over n$$ – $$({\sum{x_i}\over n})^2$$

= $$\sum{n}^2\over n$$ – $$({\sum{n}\over n})^2$$ = $$n(n + 1)(2n + 1)\over {6n}$$ – $$[{n(n + 1)\over {2n}}]^2$$ = $$n^2 – 1\over 12$$

Standard Deviation = $$\sqrt{variance}$$ = $$\sqrt{n^2 – 1\over 12}$$

Example : Find the Coefficient of variation in percentage of first n natural numbers.

Solution : We know that Mean $$\bar{x}$$ = $$n + 1\over 2$$, Variance = $${\sigma^2}_x$$ = $$n^2 – 1\over 12$$

Standard Deviation $$\sigma$$ = $$\sqrt{variance}$$ = $$\sqrt{n^2 – 1\over 12}$$

Coefficient of variation = $$\sigma\over \bar{x}$$ $$\times$$ 100

= $$\sqrt{n^2 – 1\over 12}$$ $$\times$$ ($$2\over n + 1$$) $$\times$$ 100

= $$\sqrt{(n – 1)\over 3(n + 1)}$$ $$\times$$ 100

Hope you learnt what is the formula for variance and standard deviation, learn more concepts of statistics and practice more questions to get ahead in the competition. Good luck!