# 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 ${S_c}^2 = \frac{{{n_1}\left[ {{S_1}^2 + {{\left( {{{\overline X }_1} – {{\overline X }_c}} \right)}^2}} \right] + {n_2}\left[ {{S_2}^2 + {{\left( {{{\overline X }_2} – {{\overline X }_c}} \right)}^2}} \right]}}{{{n_1} + {n_2}}}$
$= \frac{{50\left[ {81 + {{\left( {63 – 59} \right)}^2}} \right] + 40\left[ {36 + {{\left( {54 – 59} \right)}^2}} \right]}}{{50 + 40}}$
$= \frac{{4850 + 2440}}{{90}} = \frac{{7290}}{{90}} = 81$