Pooled Variance

Estimate of Population Variance: One Sample Case

In usual research settings, researchers do not know the exact population variance. When there is only one sample from a population, researchers generally use the variance of the sample as an estimate of the population variance. In the case of one sample, the fomula to calculate the sample variance is

where Xis are the observed scores in the sample, n is the sample size, and M and SD are the sample mean and standard deviation, respectively. This sample variance is an unbiased estimate of the population variance. In other words, the mean of variances of all possible random samples of the same size drawn from the population is equal to the population variance.

Pooled Variance

In many statistical procedures involving multiple groups, there are multiple sample variances that are independent estimates of the same population variance. For example, when samples from the same population are randomly assigned to two or more experimental groups, each group's variance is an independent estimate of the same population variance. In such a condition, the pooled variance is a more precise estimate of the population variance than an estimate based on only one sample's variance. Thus, the variances of all samples are aggregated to obtain an efficient estimate of the population variance.

In the case of k samples whose variances are independent estimates of the same population variance, the fomula to calculate the pooled estimate of the population variance is

where n1, n2,…, nk are the sample sizes and SD21, SD22, …, SD2k are the sample variances. To obtain the pooled estimate of the population variance, each sample variance is weighted by its degrees of freedom. Thus, pooled variance can be thought of as the weighted average of the sample variances based on their degrees of freedom.

Homogeneity of Variance

A common application of pooled variance is in statistical procedures where samples are drawn from different populations/subpopulations whose variances are assumed to be similar. In such conditions, each sample's variance is an independent estimate of the same population variance. When variances of samples that belong to different (sub)populations are pooled to yield an estimate of the (sub)population variance, an important assumption is that the (sub)populations have equal variances (σ21 = σ22 = … = σ2k). This assumption is known as the homogeneity of variance assumption.

There are statistical tests to evaluate the tenability of the homogeneity of variance assumption, such as Levene's test or Bartlett's test. For these tests, the null hypothesis is that the population variances are equal. It is important to note that the null hypothesis refers to the population parameters. If the null hypothesis is not rejected (i.e., pcalculated value > α), the homogeneity of variance assumption holds, and the pooled estimate of the population variance can be used.

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