Combined Variance

Like combined mean, the combined variance or standard deviation can be calculated for different sets of data. Suppose we have two sets of data containing {n_1} and {n_2} observations with means {\overline X _1}and {\overline X _2}, and variances {S_1}^2 and {S_2}^2. If {\overline X _c}is the combined mean and {S_c}^2 is the combined variance of {n_1} + {n_2}observations, then combined variance is given by

{S_c}^2 =  \frac{{{n_1}{S_1}^2 + {n_2}{S_2}^2 + {n_1}{{\left( {{{\overline X }_1} -  {{\overline X }_c}} \right)}^2} + {n_2}{{\left( {{{\overline X }_2} -  {{\overline X }_c}} \right)}^2}}}{{{n_1} + {n_2}}}


It can be written as

{S_c}^2 =  \frac{{{n_1}\left[ {{S_1}^2 + {{\left( {{{\overline X }_1} - {{\overline X  }_c}} \right)}^2}} \right] + {n_2}\left[ {{S_2}^2 + {{\left( {{{\overline X  }_2} - {{\overline X }_c}} \right)}^2}} \right]}}{{{n_1} + {n_2}}}

Where

{X_c}^2 =  \frac{{{n_1}{{\overline X }_1} + {n_2}{{\overline X }_2}}}{{{n_1} + {n_2}}}


The combine standard deviation {S_c}can be calculated by taking the square root of{S_c}^2.

Example:

For a group of 50 male workers the mean and standard deviation of their daily wages are Dollar 63 and Dollar 9 respectively. For a group of 40 female workers these values are Dollar 54 and Dollar 6 respectively. Find the mean and variance of the combined group of 90 workers.

Solution:
            Here {n_1} = 50, {\overline X _1} = 63, {S_1}^2 = 81
                    {n_2}  = 40, {\overline X _2} = 54, {S_2}^2 = 36
            Combined Arithmetic Mean  = {X_c}^2 =  \frac{{{n_1}{{\overline X }_1} + {n_2}{{\overline X }_2}}}{{{n_1} + {n_2}}}
                                                           =  {X_c}^2 = \frac{{50\left( {63} \right) + 40\left( {54} \right)}}{{50 + 40}} =  \frac{{5310}}{{90}} = 59
            Combined Variance {S_c}^2 = \frac{{{n_1}\left[  {{S_1}^2 + {{\left( {{{\overline X }_1} - {{\overline X }_c}} \right)}^2}}  \right] + {n_2}\left[ {{S_2}^2 + {{\left( {{{\overline X }_2} - {{\overline X  }_c}} \right)}^2}} \right]}}{{{n_1} + {n_2}}}
                                                     =  \frac{{50\left[ {81 + {{\left( {63 - 59} \right)}^2}} \right] + 40\left[ {36 +  {{\left( {54 - 59} \right)}^2}} \right]}}{{50 + 40}}
                                                     =  \frac{{4850 + 2440}}{{90}} = \frac{{7290}}{{90}} = 81

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