Statistics How To

Pooled Standard Deviation

Effect Size > Pooled Standard Deviation

What is a Pooled Standard Deviation?

The Pooled Standard Deviation is a weighted average of standard deviations for two or more groups. The individual standard deviations are averaged, with more “weight” given to larger sample sizes.

Once the pooled standard deviation has been calculated, SDpooled is used in place of SD1 and SD2 in the formula for standard error. Along with an updated degrees of freedom formula (df = n1 + n2 – 2), the idea is that you would be able to get a better model for the sampling distribution of the sample mean.

Pooled standard deviations are used in many areas in statistics, including: effect size calculations, t-tests, and ANOVAs. They are also used in lab-based sciences like biology and chemistry, where they can be an indication for repeatability of an experiment.

How to Calculate the Pooled Standard Deviation

Cohen (1988) offers a couple of options for calculating the pooled standard deviation. The simplest is:
standard-deviation-pooled
Where:

I’m including Cohen’s alternative formula here for reference, although there’s no clear benefit to using this one rather than the simpler formula above:
pooled standard deviation-2
Where:

For three or more groups, change the “2” in the denominator to reflect the number of samples, k:
three-or-more

For equal sample sizes, the formula can be simplified to:
three-or-more-2

You can only use the above formulas if the standard deviations for the two groups are the same (this is because it would otherwise be violating the assumption of homogeneity of variances. If the standard deviations are different, run Hedge’s g or Glass’s Delta instead.

References:
Cohen, J. (1988), Statistical Power Analysis for the Behavioral Sciences, 2nd Edition. Hillsdale: Lawrence Erlbaum.
Hedges L. V., Olkin I. (1985). Statistical methods for meta-analysis. San Diego, CA: Academic Press

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Pooled Standard Deviation was last modified: October 15th, 2017 by Andale