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}}}

Here

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

The combined 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  63 dollars and 9 dollars respectively. For a group of 40 female workers these values are 54 dollars and 6 dollars 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} = \frac{{{n_1}{{\overline X }_1} + {n_2}{{\overline X }_2}}}{{{n_1} + {n_2}}}
 = {X_c} = \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