To pool, or not to pool?

Let's start with the most familiar context for pooling data (at least to students in introductory courses in statistics), viz., the pooling of sample variances in a t test of the significance of the difference between two independent sample means. The null hypothesis to be tested is that the means of two populations are equal (the populations from which the respective samples have been randomly sampled). We almost never know what the population variances are (if we did we'd undoubtedly also know what the populations means are, and there would be no need to test the hypothesis), but we often assume that they are equal, so we need to have some way of estimating from the sample data the variance that the two populations have in common. I won't bore you with the formula (you can look it up in almost any statistics texxtbook), but it involves, not surprisingly, the two sample variances and the two sample sizes. You should also test the "poolability" of the sample variances before doing the pooling, by using Bartlett's test or Levene's test, but almost nobody does; neither test has much power.