Calculating the Exact Pooled Variance

An exact method of calculating the variance of a pooled data set is presented. Its major advantages over the many other methods are that it is simple, is easily derived and remembered, and requires no assumptions. The result can be concisely summarized as follows: “The Exact pooled variance is the mean of the variances plus the variance of the means of the component data sets.” The proof is so simple that it has certainly been done many times before, but it is absent in the textbooks. Its practical significance is discussed.

💡 Research Summary

The paper introduces a concise, assumption‑free formula for calculating the variance of a pooled data set composed of several independent sub‑samples. The author begins by noting that many textbooks and applied guides present pooled‑variance methods that either rely on degrees‑of‑freedom weighting, assume normality, or require the raw observations from each group. In practice, however, analysts often have only the summary statistics—means and variances—of each component, and the existing formulas can be cumbersome or biased when sample sizes differ markedly.

The core contribution is a derivation that shows the total variance of the combined sample can be expressed as the sum of two intuitive components:

1. Mean of the component variances – a weighted average of each group’s internal variability, where the weight is the group’s sample size.
2. Variance of the component means – the variability among the group means themselves, also weighted by the same sample sizes.

Mathematically, if there are (k) groups with sizes (n_i), means (\bar X_i) and variances (s_i^2), the pooled variance (s_{pooled}^2) is

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