Mathematics and Statistics · Ch 11 — Measures of Dispersion

Combined Standard Deviation

Sometimes two groups are described separately (each by its size, mean and standard deviation) and we need the mean and standard deviation of the two groups taken together, without going back to the individual values.

First the combined mean, which is just the weighted average of the two means:

Note

Combined Mean

$\bar{x}_{12} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$

where group 1 has $n_1$ items with mean $\bar{x}_1$ , and group 2 has $n_2$ items with mean $\bar{x}_2$ .

The combined standard deviation must account for two things: the spread within each group ( $\sigma_1, \sigma_2$ ) and how far each group's mean sits from the overall combined mean.

Note

Combined Standard Deviation

$\sigma_{12} = \sqrt{\dfrac{n_1(\sigma_1^{2} + d_1^{2}) + n_2(\sigma_2^{2} + d_2^{2})}{n_1 + n_2}}$

where $d_1 = \bar{x}_1 - \bar{x}_{12}$ and $d_2 = \bar{x}_2 - \bar{x}_{12}$ are the deviations of each group mean from the combined mean.

The $d_1^2$ and $d_2^2$ terms are essential: even if both groups had zero internal spread, placing two groups with different means side by side still creates spread in the combined set. The formula extends naturally to three or more groups by adding an $n_i(\sigma_i^2 + d_i^2)$ term for each. …

Definition 15Combined mean

$\bar{x}_{12} = \dfrac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$ , the weighted mean of two …

Definition 16Combined standard deviation

$\sigma_{12} = \sqrt{\dfrac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}}$ , where $d_1 = \bar{x}_1 - \bar{x}_{12}$ , $d_2 = \bar{x}_2 - \bar{x}_{12}$ ; the standard deviat …

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