Notes On Standard Deviation - CBSE Class 11 Maths
Standard deviation is the positive square root of the variance. It is denoted by s.   The standard deviation of the observations x1, x2, x3,..., xn =   σ = $\sqrt{\frac{\text{1}}{\text{n}}\text{}\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}$   Variance =  σ2 = $\frac{\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$                       = $\frac{{\text{(}{\text{x}}_{\text{1}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}\text{+}{\text{(}{\text{x}}_{\text{2}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}\text{+ .... +}{\text{(}{\text{x}}_{\text{n}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$                       = $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{) - 2}\stackrel{_}{\text{x}}\text{(}{\text{x}}_{\text{1}}\text{+}{\text{x}}_{\text{2}}\text{+ .... +}{\text{x}}_{\text{n}}\text{) + n}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$                       = $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}\frac{\text{2}\stackrel{_}{\text{x}}\text{(}{\text{x}}_{\text{1}}\text{+}{\text{x}}_{\text{2}}\text{+ .... +}{\text{x}}_{\text{n}}\text{)}}{\text{n}}\text{+}\frac{\text{n}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$   Sum of the observations divided by the number of observations is the mean of the observations.   =  $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}$ $\text{2}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}\text{+}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}$                                            {∵  $\stackrel{_}{\text{x}}$ = $\frac{\text{(}{\text{x}}_{1}\text{+}{\text{x}}_{2}\text{+ ... +}\text{}{\text{x}}_{\text{n}}\text{)}}{\text{n}}$} =  $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}$ $\text{}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}$          σ2 = $\frac{\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{{\text{x}}_{\text{i}}}^{2}}{\text{n}}\mathrm{\text{-}}\text{}{\left(\frac{\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\text{x}}_{\text{i}}}{\text{n}}\right)}^{2}$ σ = $\sqrt{\frac{\sum _{\text{i = 1}}^{\text{n}}{{\text{x}}_{\text{i}}}^{\text{2}}}{\text{n}}\text{-}{\left(\frac{\sum _{\text{i = 1}}^{\text{n}}{\text{x}}_{\text{i}}}{\text{n}}\right)}^{\text{2}}}$   Therefore, standard deviation of an ungrouped data can also be calculated by using this formula.   Variance of a discrete frequency distribution: σ2 = $\frac{\text{1}}{\text{N}}$ $\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}$     xi’s are the n observations.  fi’s are the frequencies of the observations. N is the total frequency. $\stackrel{_}{\text{x}}$ is the mean of the n observations. Standard deviation of a discrete frequency distribution: σ = $\sqrt{\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}$ Alternate formula: σ = $\frac{\text{1}}{\text{N}}$ $\sqrt{\text{N}\sum {{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}}^{\text{2}}\text{-}{\text{(}\sum {{\text{f}}_{\text{i}}\text{x}}_{\text{i}}\text{)}}^{\text{2}}}$   Variance of a continuous frequency distribution: σ2 = ${\frac{1}{\text{N}}\mathrm{\text{}}\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}\mathrm{\text{}}{\mathrm{\text{}}\mathrm{\text{}}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\mathrm{\text{}}\mathrm{\text{-}}\mathrm{\text{}}\stackrel{_}{\text{x}}\text{)}}^{2}$ xi’s are the mid-values of the class intervals. fi’s are the frequencies of the class intervals. $\stackrel{_}{\text{x}}$ is the mean of the frequency distribution observation. N is the total frequency. Standard deviation of a continuous frequency distribution: σ = $\sqrt{\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}$ Alternative formula for the standard deviation of a continuous frequency distribution:   Variance of a continuous distribution = σ2 = ${\frac{1}{\text{N}}\mathrm{\text{}}\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}\mathrm{\text{}}{\mathrm{\text{}}\mathrm{\text{}}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\mathrm{\text{}}\mathrm{\text{-}}\mathrm{\text{}}\stackrel{_}{\text{x}}\text{)}}^{2}$ = $\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{f}}_{\text{i}}\text{(}{{\text{x}}_{\text{i}}}^{\text{2}}\text{- 2}{\text{x}}_{\text{i}}\stackrel{_}{\text{x}}\text{+}{\stackrel{_}{\text{x}}}^{\text{2}}\text{)}$ = $\frac{\text{1}}{\text{N}}\text{[}\sum _{\text{i = 1}}^{\text{n}}\text{}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^{\text{2}}\text{-}\sum _{\text{i = 1}}^{\text{n}}\text{2}\stackrel{_}{\text{x}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}\text{+}\underset{\text{i = 1}}{\overset{\text{n}}{\sum {\text{f}}_{\text{i}}}}{\stackrel{_}{\text{x}}}^{\text{2}}\text{]}$ =$\frac{1}{\text{N}}\mathrm{\text{[}}\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\text{f}}_{\text{i}}\text{}{{\text{x}}_{\text{i}}}^{2}\mathrm{\text{-}}\text{2}\stackrel{_}{\text{x}}\text{}\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\text{f}}_{\text{i}}\text{}{\text{x}}_{\text{i}}\mathrm{\text{+}}{\sum \text{}{\text{f}}_{\text{i}}}_{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\stackrel{_}{\text{x}}\text{}}^{2}\text{]}$ = (Since, ${\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}\mathrm{\text{}}\mathrm{\text{}}\mathrm{\text{}}{\text{f}}_{\text{i}}\mathrm{\text{}}{\text{x}}_{\text{i}}\mathrm{\text{}}$  = N $\stackrel{_}{\text{x}}$ ) =  $\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{n}}\sum }}\text{(}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^{\text{2}}\text{-}{\stackrel{_}{\text{x}}}^{\text{2}}\text{N)}\text{}$ = σ2 = $\frac{1}{{\text{N}}^{2}}\left[\text{N}\sum _{\mathrm{\text{i= 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}^{2}\mathrm{\text{-}}{\left(\sum _{\text{i = 1}}^{\text{n}}{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}\right)}^{\text{2}}\right]$ Standard derivation (σ) = $\frac{1}{\text{N}}\sqrt{\left[\text{N}\sum _{\mathrm{\text{i= 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}^{2}\mathrm{\text{-}}{\left(\sum _{\mathrm{\text{i = 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}\right)}^{2}\right]}$ This is the alternative formula for the standard deviation. Note that there is no need to find the mean and deviations of the frequency distribution.   Note: The standard deviation of an arithmetic series a, a+d, a+2d, a+3d,.....,a+(n-1)d is $\sqrt{\left(\frac{{\text{n}}^{2}\mathrm{\text{-}}1}{12}\right)}$ where c.d. is the common difference of the series and n is the number of terms in the series. If the standard deviation of the observations, x1, x2, x3,....,xn is σ and k is some constant, then: The standard deviation of the observations x1 ± k, x2 ± k, x3 ± k,..., xn ± k is σ. The standard deviation of the observations kx1, kx2, kx3,....,kxn is kσ. The standard deviation of the observations x1/k, x2/k, x3/k,....,xn/k is σ/k.

#### Summary

Standard deviation is the positive square root of the variance. It is denoted by s.   The standard deviation of the observations x1, x2, x3,..., xn =   σ = $\sqrt{\frac{\text{1}}{\text{n}}\text{}\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}$   Variance =  σ2 = $\frac{\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$                       = $\frac{{\text{(}{\text{x}}_{\text{1}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}\text{+}{\text{(}{\text{x}}_{\text{2}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}\text{+ .... +}{\text{(}{\text{x}}_{\text{n}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$                       = $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{) - 2}\stackrel{_}{\text{x}}\text{(}{\text{x}}_{\text{1}}\text{+}{\text{x}}_{\text{2}}\text{+ .... +}{\text{x}}_{\text{n}}\text{) + n}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$                       = $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}\frac{\text{2}\stackrel{_}{\text{x}}\text{(}{\text{x}}_{\text{1}}\text{+}{\text{x}}_{\text{2}}\text{+ .... +}{\text{x}}_{\text{n}}\text{)}}{\text{n}}\text{+}\frac{\text{n}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}{\text{n}}$   Sum of the observations divided by the number of observations is the mean of the observations.   =  $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}$ $\text{2}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}\text{+}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}$                                            {∵  $\stackrel{_}{\text{x}}$ = $\frac{\text{(}{\text{x}}_{1}\text{+}{\text{x}}_{2}\text{+ ... +}\text{}{\text{x}}_{\text{n}}\text{)}}{\text{n}}$} =  $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}$ $\text{}{\text{(}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}$          σ2 = $\frac{\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{{\text{x}}_{\text{i}}}^{2}}{\text{n}}\mathrm{\text{-}}\text{}{\left(\frac{\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\text{x}}_{\text{i}}}{\text{n}}\right)}^{2}$ σ = $\sqrt{\frac{\sum _{\text{i = 1}}^{\text{n}}{{\text{x}}_{\text{i}}}^{\text{2}}}{\text{n}}\text{-}{\left(\frac{\sum _{\text{i = 1}}^{\text{n}}{\text{x}}_{\text{i}}}{\text{n}}\right)}^{\text{2}}}$   Therefore, standard deviation of an ungrouped data can also be calculated by using this formula.   Variance of a discrete frequency distribution: σ2 = $\frac{\text{1}}{\text{N}}$ $\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}$     xi’s are the n observations.  fi’s are the frequencies of the observations. N is the total frequency. $\stackrel{_}{\text{x}}$ is the mean of the n observations. Standard deviation of a discrete frequency distribution: σ = $\sqrt{\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}$ Alternate formula: σ = $\frac{\text{1}}{\text{N}}$ $\sqrt{\text{N}\sum {{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}}^{\text{2}}\text{-}{\text{(}\sum {{\text{f}}_{\text{i}}\text{x}}_{\text{i}}\text{)}}^{\text{2}}}$   Variance of a continuous frequency distribution: σ2 = ${\frac{1}{\text{N}}\mathrm{\text{}}\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}\mathrm{\text{}}{\mathrm{\text{}}\mathrm{\text{}}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\mathrm{\text{}}\mathrm{\text{-}}\mathrm{\text{}}\stackrel{_}{\text{x}}\text{)}}^{2}$ xi’s are the mid-values of the class intervals. fi’s are the frequencies of the class intervals. $\stackrel{_}{\text{x}}$ is the mean of the frequency distribution observation. N is the total frequency. Standard deviation of a continuous frequency distribution: σ = $\sqrt{\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{_}{\text{x}}\text{)}}^{\text{2}}}$ Alternative formula for the standard deviation of a continuous frequency distribution:   Variance of a continuous distribution = σ2 = ${\frac{1}{\text{N}}\mathrm{\text{}}\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}\mathrm{\text{}}{\mathrm{\text{}}\mathrm{\text{}}{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\mathrm{\text{}}\mathrm{\text{-}}\mathrm{\text{}}\stackrel{_}{\text{x}}\text{)}}^{2}$ = $\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{f}}_{\text{i}}\text{(}{{\text{x}}_{\text{i}}}^{\text{2}}\text{- 2}{\text{x}}_{\text{i}}\stackrel{_}{\text{x}}\text{+}{\stackrel{_}{\text{x}}}^{\text{2}}\text{)}$ = $\frac{\text{1}}{\text{N}}\text{[}\sum _{\text{i = 1}}^{\text{n}}\text{}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^{\text{2}}\text{-}\sum _{\text{i = 1}}^{\text{n}}\text{2}\stackrel{_}{\text{x}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}\text{+}\underset{\text{i = 1}}{\overset{\text{n}}{\sum {\text{f}}_{\text{i}}}}{\stackrel{_}{\text{x}}}^{\text{2}}\text{]}$ =$\frac{1}{\text{N}}\mathrm{\text{[}}\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\text{f}}_{\text{i}}\text{}{{\text{x}}_{\text{i}}}^{2}\mathrm{\text{-}}\text{2}\stackrel{_}{\text{x}}\text{}\sum _{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\text{f}}_{\text{i}}\text{}{\text{x}}_{\text{i}}\mathrm{\text{+}}{\sum \text{}{\text{f}}_{\text{i}}}_{\mathrm{\text{i = 1}}}^{\text{n}}\text{}{\stackrel{_}{\text{x}}\text{}}^{2}\text{]}$ = (Since, ${\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}\mathrm{\text{}}\mathrm{\text{}}\mathrm{\text{}}{\text{f}}_{\text{i}}\mathrm{\text{}}{\text{x}}_{\text{i}}\mathrm{\text{}}$  = N $\stackrel{_}{\text{x}}$ ) =  $\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{n}}\sum }}\text{(}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^{\text{2}}\text{-}{\stackrel{_}{\text{x}}}^{\text{2}}\text{N)}\text{}$ = σ2 = $\frac{1}{{\text{N}}^{2}}\left[\text{N}\sum _{\mathrm{\text{i= 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}^{2}\mathrm{\text{-}}{\left(\sum _{\text{i = 1}}^{\text{n}}{{\text{f}}_{\text{i}}\text{x}}_{\text{i}}\right)}^{\text{2}}\right]$ Standard derivation (σ) = $\frac{1}{\text{N}}\sqrt{\left[\text{N}\sum _{\mathrm{\text{i= 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}^{2}\mathrm{\text{-}}{\left(\sum _{\mathrm{\text{i = 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}\right)}^{2}\right]}$ This is the alternative formula for the standard deviation. Note that there is no need to find the mean and deviations of the frequency distribution.   Note: The standard deviation of an arithmetic series a, a+d, a+2d, a+3d,.....,a+(n-1)d is $\sqrt{\left(\frac{{\text{n}}^{2}\mathrm{\text{-}}1}{12}\right)}$ where c.d. is the common difference of the series and n is the number of terms in the series. If the standard deviation of the observations, x1, x2, x3,....,xn is σ and k is some constant, then: The standard deviation of the observations x1 ± k, x2 ± k, x3 ± k,..., xn ± k is σ. The standard deviation of the observations kx1, kx2, kx3,....,kxn is kσ. The standard deviation of the observations x1/k, x2/k, x3/k,....,xn/k is σ/k.

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