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!

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