The T-Test › Summary

The t-test is the fundamental tool for comparing means when population variance is unknown, which is nearly always. Key takeaways:

The t-distribution:

• Accounts for uncertainty from estimating variance
• Has heavier tails than the normal distribution
• Converges to normal as degrees of freedom increase
• Requires larger critical values for small samples

One-sample t-test:

• Compares sample mean to hypothesized value
• Test statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
• Degrees of freedom: $n - 1$

Two-sample t-test:

• Compares means from two independent groups
• Pooled variant assumes equal variances
• Welch's variant (recommended default) handles unequal variances
• Degrees of freedom differ between variants

Paired t-test:

• For matched pairs or repeated measures
• Analyzes within-pair differences
• Often more powerful than independent tests by eliminating between-subject variability

Assumptions:

• Independence (critical, design-based)
• Normality (less restrictive for large samples)
• Equal variances (pooled test only; avoid by using Welch's)

When in doubt: Use Welch's t-test for two-sample comparisons. You sacrifice minimal power when variances are equal but avoid the equal-variance assumption when they differ.