Statistics · step 10 of 13

t-tests & comparing two means

Three t-tests cover most comparisons, and choosing correctly is most of the mark. A one-sample test compares a mean to a fixed value; a PAIRED test compares two measurements on the same subjects by analysing the differences; a two-sample test compares independent groups. Using the two-sample test on paired data throws away the pairing and badly loses power.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Identify the designAre the two samples independent, or matched pairs (before/after, twins, same subject)? This decides the test.
Paired: analyse the differencesCompute dᵢ = after − before and run a one-sample t-test on them, with df = n − 1.
Two-sample: use the appropriate standard errorWelch’s test uses √(s₁²/n₁ + s₂²/n₂) and does not assume equal variances — the safer default.
State the df and conclude in contextPaired uses n − 1; Welch uses a computed approximation. Always report which test you ran.

Worked example

8 subjects are measured before and after a training programme. Which test applies, and what are the hypotheses?

The same subjects appear in both conditions, so the observations are paired.
Compute the 8 differences d = after − before.
H₀: μ_d = 0 against H₁: μ_d ≠ 0 (or > 0 if improvement was predicted in advance).
Test statistic: t = d̄/(s_d/√8) with df = 7.

Answer. A paired t-test on the 8 differences, with df = 7 — not a two-sample test, which would ignore the pairing and inflate the standard error.

Where marks get dropped

These are the specific errors that cost credit on t-tests & comparing two means questions — QED's rubric penalises each of them separately.

Running a two-sample test on paired data. The pairing removes between-subject variability, and ignoring it wastes that advantage.
Assuming equal variances without checking. Welch’s t-test avoids the assumption at almost no cost and should be the default.
Getting the degrees of freedom wrong, which changes the critical value and can flip the conclusion for small samples.

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t-tests & comparing two means — frequently asked questions

When should I use a paired t-test?

Whenever each observation in one group corresponds to a specific observation in the other — before/after, matched subjects, or two measurements on the same item.

What are the assumptions?

Independence between units, and approximate normality of the differences (paired) or of each group (two-sample). The t-test is fairly robust to mild non-normality for moderate n.

What if the data are badly non-normal?

Use a non-parametric alternative: Wilcoxon signed-rank for paired data, Mann–Whitney U for independent groups.

The rest of Statistics

Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.

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