Standard Deviation

Standard deviation is the positive square root of the variance. It is denoted by s.
 
The standard deviation of the observations x₁, x₂, x₃,..., x_n =   σ = $\sqrt{\frac{\text{1}}{\text{n}}\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{)}}^{\text{2}}}$
 
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.
 
=  $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}$ $\text{2}{\text{(}\stackrel{\_}{\text{x}}\text{)}}^{\text{2}}\text{+}{\text{(}\stackrel{\_}{\text{x}}\text{)}}^{\text{2}}$                                            {∵  $\stackrel{\_}{\text{x}}$ = $$ \frac{\text{(}{\text{x}}_1\text{+}{\text{x}}_2\text{+ ... +}{\text{x}}_{\text{n}}\text{)}}{\text{n}}$$}

=  $\frac{\text{(}{{\text{x}}_{\text{1}}}^{\text{2}}\text{+}{{\text{x}}_{\text{2}}}^{\text{2}}\text{+ .... +}{{\text{x}}_{\text{n}}}^{\text{2}}\text{)}}{\text{n}}\text{-}$ ${\text{(}\stackrel{\_}{\text{x}}\text{)}}^{\text{2}}$         

σ² = $$ \frac{\sum _{\mathrm{\text{i = 1}}}^{\text{n}}{{\text{x}}_{\text{i}}}^2}{\text{n}}\mathrm{\text{-}}{\left(\frac{\sum _{\mathrm{\text{i = 1}}}^{\text{n}}{\text{x}}_{\text{i}}}{\text{n}}\right)}^2$$

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 = $\frac{\text{1}}{\text{N}}$ $\sum _{\text{i = 1}}^{\text{n}}{\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{)}}^{\text{2}}$    
x_i’s are the n observations.

 f_i’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 = $\sqrt{\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{)}}^{\text{2}}}$

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 = $$ {\frac{1}{\text{N}}\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}{{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\mathrm{\text{-}}\stackrel{\_}{\text{x}}\text{)}}^2$$

x_i’s are the mid-values of the class intervals.

f_i’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 = $\sqrt{\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\text{-}\stackrel{\_}{\text{x}}\text{)}}^{\text{2}}}$

Alternative formula for the standard deviation of a continuous frequency distribution:
 
Variance of a continuous distribution = σ²

= $$ {\frac{1}{\text{N}}\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}{{\text{f}}_{\text{i}}\text{(}{\text{x}}_{\text{i}}\mathrm{\text{-}}\stackrel{\_}{\text{x}}\text{)}}^2$$

= $\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{f}}_{\text{i}}\text{(}{{\text{x}}_{\text{i}}}^{\text{2}}\text{- 2}{\text{x}}_{\text{i}}\stackrel{\_}{\text{x}}\text{+}{\stackrel{\_}{\text{x}}}^{\text{2}}\text{)}$

= $\frac{\text{1}}{\text{N}}\text{[}\sum _{\text{i = 1}}^{\text{n}}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^{\text{2}}\text{-}\sum _{\text{i = 1}}^{\text{n}}\text{2}\stackrel{\_}{\text{x}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}\text{+}\underset{\text{i = 1}}{\overset{\text{n}}{\sum {\text{f}}_{\text{i}}}}{\stackrel{\_}{\text{x}}}^{\text{2}}\text{]}$

=$$ \frac{1}{\text{N}}\mathrm{\text{[}}\sum _{\mathrm{\text{i = 1}}}^{\text{n}}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^2\mathrm{\text{-}}\text{2}\stackrel{\_}{\text{x}}\sum _{\mathrm{\text{i = 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}\mathrm{\text{+}}{\sum {\text{f}}_{\text{i}}}_{\mathrm{\text{i = 1}}}^{\text{n}}{\stackrel{\_}{\text{x}}}^2\text{]}$$

= $$ \frac{1}{\text{N}}\mathrm{\text{[}}\sum _{\mathrm{\text{i = 1}}}^{\text{n}}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^2\mathrm{\text{-}}\text{2}\stackrel{\_}{\text{x}}\text{N}\stackrel{\_}{\text{x}}\mathrm{\text{+}}{\stackrel{\_}{\text{x}}}^{\text{2}}\text{N]} $$(Since, $$ {\sum }_{\mathrm{\text{i = 1}}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}$$  = N $\stackrel{\_}{\text{x}}$ )

=  $$ \underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{n}}\sum }}\text{(}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^{\text{2}}\text{-}{\stackrel{\_}{\text{x}}}^{\text{2}}\text{N)}$$

= $\underset{\text{i = 1}}{\overset{\text{n}}{\frac{\text{1}}{\text{N}}\sum }}{\text{f}}_{\text{i}}{{\text{x}}_{\text{i}}}^{\text{2}}\text{-}{\left(\frac{\sum _{\text{i = 1}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}}{\text{N}}\right)}^{\text{2}}\text{N (Since,}\frac{\sum _{\text{i = 1}}^{\text{n}}{\text{f}}_{\text{i}}{\text{x}}_{\text{i}}}{\text{N}}\text{=}\stackrel{\_}{\text{x}}\text{)}$

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,....,x_n is σ and k is some constant, then:
  • The standard deviation of the observations x₁ ± k, x₂ ± k, x₃ ± k,..., x_n ± k is σ.
  • The standard deviation of the observations kx₁, kx₂, kx₃,....,kx_n is kσ.
  • The standard deviation of the observations x₁/k, x₂/k, x₃/k,....,x_n/k is σ/k.