Business Mathematics and Statistics · Ch 9 — Measures of Dispersion

Combined Standard Deviation of Two Series

A business often needs to combine two separately-recorded groups of data into one overall picture — the combined output of two factory shifts, or the combined test scores of two class sections — without going back to the original raw observations. Given only the size, mean and standard deviation of each of the two groups, the combined standard deviation of the merged group can be computed directly.

The combined mean is found first, as a size-weighted average of the two means:

Note

Combined Mean

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

The combined variance then needs each group's own variance PLUS a correction term accounting for how far each group's own mean sits from the new 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 $d_2 = \bar{x}_2 - \bar{x}_{12}$ . …

Definition 1Combined Mean

The size-weighted average of two groups' individual means, $\bar{x}_{12}=(n_1\bar{x}_1+n_2\bar{ …

Definition 2Combined Standard Deviation

The SD of two merged groups, computed from each group's own size, mean and SD without needing the raw data: $\sigma_{12}=\sqrt{[n_1(\sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)]/(n_1+n_2)}$ , where $d_1,d_2$ are each group's …

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