Pooled variance

In statistics, pooled variance (also known as combined, composite, or overall variance) is a method for estimating variance of several different populations when the mean of each population may be different, but one may assume that the variance of each population is the same.

Under the assumption of equal population variances, the pooled sample variance provides a higher precision estimate of variance than the individual sample variances. This higher precision can lead to increased statistical power when used in statistical tests that compare the populations, such as the t-test.

The square-root of a pooled variance estimator is known as a pooled standard deviation (also known as combined, composite, or overall standard deviation).

In statistics, many times, data are collected for a dependent variable, y, over a range of values for the independent variable, x. For example, the observation of fuel consumption might be studied as a function of engine speed while the engine load is held constant. If, in order to achieve a small variance in y, numerous repeated tests are required at each value of x, the expense of testing may become prohibitive. Reasonable estimates of variance can be determined by using the principle of pooled variance after repeating each test at a particular x only a few times.

If the populations are indexed $i=1,\ldots ,k$ $i=1,\ldots ,k$ , then the pooled variance $s_{p}^{2}$ $s_{p}^{2}$ (or $s_{c}^{2}$ $s_{c}^{2}$ ) can be estimated by the weighted average:

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