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Pooled variance

Definition

Pooled variance is an estimate formed by combining variance information from several samples. In the common equal-variance two-group setting, it is the weighted average of the sample variances, with weights equal to their degrees of freedom: s_p^2 = [(n_1-1)s_1^2 + (n_2-1)s_2^2]/(n_1+n_2-2). It estimates a shared within-group population variance under that model. It is not generally the variance obtained by ignoring group labels and calculating variance across all observations together.

What it means

The purpose is to combine evidence about variation around each group's own mean. Groups can have different means while sharing a common variance. Pooling their within-group sums of squares uses both samples to estimate that common spread. The result differs from the spread of the merged dataset, which also reflects separation between group means. Whether pooling is substantively appropriate depends on the design and variance assumptions. A weighted calculation can always be performed, but the interpretation as one common population variance requires justification.

How it works

Calculate each sample variance using its own sample mean and denominator n_j minus one. Multiply that variance by its degrees of freedom to recover the within-group sum of squares. Add these sums and divide by the sum of the degrees of freedom. For k groups, the denominator becomes N minus k under the usual independent-group model. Take the square root only if a pooled standard deviation is wanted. Inference using a pooled variance must also account for the sampling design and the assumptions of the intended test or interval; equal-variance arithmetic alone does not validate the whole procedure.

Three practical examples

Two-group comparison: estimate the common error variance for a pooled two-sample t-test when its assumptions are appropriate.
Analysis of variance: combine within-group variation to obtain the error mean square in a standard one-way model.
Measurement studies: summarize repeatability across groups that plausibly share a common measurement variance, while distinguishing within-group noise from differences in group-level offsets.

Think of it this way

Think of measuring how much each group spreads around its own center, then combining those measurements according to how much information each provides. Moving an entire group to a different location does not change its internal spread. It can, however, greatly change the spread of all observations after the groups are mixed together.

Worked example

For an original example, group A contains 1, 3 and 5, with mean 3 and sample variance 4. Group B contains 10, 12, 14, 16 and 18, with mean 14 and sample variance 10. The within-group sums of squares are 2 times 4 = 8 and 4 times 10 = 40. The pooled variance is (8+40)/(2+4) = 8, and the pooled standard deviation is sqrt(8), approximately 2.8284. The simple unweighted average of the two variances would be 7, which is not the degrees-of-freedom-weighted result. If the common-variance model is used for the difference of means, its estimated variance is 8(1/3 + 1/5) = 64/15, giving a standard error of approximately 2.0656. By contrast, merging all eight observations and calculating ordinary sample variance gives approximately 39.2679, because the two means are far apart.

Illustrative case study

In a fictional case, Ridge Instruments compares repeated measurements from two setups. Analysts initially calculate the variance of all readings together and describe it as measurement noise. A review shows that the setups have different average offsets. The team separates within-setup variability from between-setup differences and considers whether a shared variance model is reasonable. It then reports the pooled estimate only with that interpretation, while preserving the separate setup variances for diagnosis.

Common misconceptions and mistakes

Pooled variance is not obtained by averaging standard deviations and squaring the answer. Degrees-of-freedom weights are not generally the same as equal weights or raw sample-size weights. Equal population variances do not require identical observed sample variances, since samples fluctuate. Conversely, a non-significant variance comparison does not prove equality. Pooling can be misleading when heterogeneity is substantial or group structures are ignored. For a comparison of two means with unequal variances, Welch's method estimates uncertainty differently and does not rely on a common pooled variance in the same way.

Business use

Businesses should name the quantity being summarized before pooling. Combining datasets from different devices, periods or regions can conceal useful variation if a common spread is assumed without investigation. Show group sizes and group variances so readers can understand the combined value. If the objective is the variability of a mixed population, include differences between group means rather than reporting only a within-group estimate. For testing or forecasting, choose the method that matches the design and heterogeneity instead of selecting pooling simply because it yields a convenient single number.

Check your understanding

Adding 100 to every observation in one group leaves the other group unchanged. Does the usual pooled within-group variance change?

Answer: No. Each group's deviations from its own mean stay the same, although the merged-data variance can change greatly.

Related terms

Sample variance
Degrees of freedom
Welch's t-test
Analysis of variance
within-group variation
pooled standard deviation

Learn next

Learn the decomposition of total variation into within-group and between-group components, then compare pooled and unequal-variance methods for estimating the uncertainty of a mean difference.

Total variation

Sources

NIST Dataplot, T TEST, pooled and unequal-variance formulas: https://itl.nist.gov/div898/software/dataplot/refman1/auxillar/t_test.htm Penn State, STAT 500, Comparing Two Population Parameters: https://online.stat.psu.edu/stat500/Lesson07

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