The two sample t-test is used to compare the means of two independent samples. For the null hypothesis, the observed t-statistic is equal to the difference between the two sample means divided by the standard error of the difference between the sample means. If the two population variances can be assumed equal, the standard error of the difference is estimated from the weighted variance about the means. If the variances cannot be assumed equal, then the standard error of the difference between means is taken as the square root of the sum of the individual variances divided by their sample size. In the latter case the estimated t-statistic must either be tested with modified degrees of freedom, or it can be tested against different critical values. A weighted t-test must be used if the unit of analysis comprises percentages or means based on different sample sizes.

The two-sample t-test is probably the most widely used (and misused) statistical test. Comparing means based on convenience sampling or non-random allocation is meaningless. If, for any reason, one is forced to use haphazard rather than probability sampling, then every effort must be made to minimize selection bias.

## Unpaired and Overlapping Two-Sample T-Tests

Two-sample t-tests for a difference in mean involve independent samples, paired samples and overlapping samples. Paired t-tests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared. In a different context, paired t-tests can be used to reduce the effects of confounding factors in an observational study.

### Independent Samples

The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomize 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the t-test .

### Overlapping Samples

An overlapping samples t-test is used when there are paired samples with data missing in one or the other samples (e.g., due to selection of "I don't know" options in questionnaires, or because respondents are randomly assigned to a subset question). These tests are widely used in commercial survey research (e.g., by polling companies) and are available in many standard crosstab software packages.