Pooled standard deviation is a combined estimate of variability calculated from two or more independent samples. It assumes that the groups have similar variances and hypothesis tests often use it to compare group means, such as a two-sample t-test. The pooled standard deviation provides a weighted average of the individual sample standard deviations.
Some statistical tests, such as the two-sample t-test and ANOVA, assume equal variances across groups and use the pooled standard deviation to increase statistical power. This pooled statistic is a more precise estimate of the common variance than the individual sample variances, which helps improve the test’s ability to detect true differences between group means. However, there are alternative versions of these tests that do not require equal variances, such as Welch’s t-test and Welch’s ANOVA, and they do not use the pooled standard deviation.
The formula for the pooled standard deviation is:
Sp = √ [ ((n₁ − 1)s₁² + (n₂ − 1)s₂²) / (n₁ + n₂ − 2) ]
Where:
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Sp = pooled standard deviation
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n₁, n₂ = sample sizes of group 1 and group 2
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s₁, s₂ = standard deviations of group 1 and group 2
This formula calculates a weighted average of the variances, giving more influence to larger samples, and then takes the square root to return to the standard deviation scale.
For example, when comparing the test scores of two classrooms, the pooled standard deviation can summarize the overall variability across both groups, helping assess whether their mean scores differ significantly.
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