Write up this t-test
Get an APA-style results paragraph built only from the numbers above.
Uses an AI model via Vercel AI Gateway. It only sees what you send here; check anything you publish.
How the t-test works
The t statistic is a signal-to-noise ratio: the difference you observed divided by its standard error. If H₀ is true, t follows a t distribution with the degrees of freedom shown; the shaded area in the plot is the p-value, the probability of a t at least this extreme if there were no real difference.
- Independent (Welch): t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂), with Welch–Satterthwaite df.
- Independent (pooled): t = (x̄₁ − x̄₂) / (sp√(1/n₁ + 1/n₂)), df = n₁ + n₂ − 2.
- Paired: t = d̄ / (sd/√n) on the pairwise differences, df = n − 1.
- One-sample: t = (x̄ − μ₀) / (s/√n), df = n − 1.
Cohen's d uses the pooled SD for independent groups. For paired data, dz uses the SD of the differences; it is the value to enter in a power analysis for a paired design.
Assumptions to check
- Observations are independent (within groups, and between groups for the independent test).
- The outcome is roughly continuous and, for small samples, approximately normal (or the differences are, for paired data).
- No extreme outliers driving the result. Plot your data first.
Frequently asked questions
Which t-test should I use?
One-sample: compare one mean with a known value. Independent (two-sample): two separate groups of people. Paired: the same people measured twice, or matched pairs. Choosing independent when the data are paired throws away power.
Welch's or Student's t-test?
Use Welch's by default. It does not assume equal variances, and when variances are equal it loses almost nothing (Delacre, Lakens & Leys, 2017). Student's pooled test can give wrong p-values when group sizes and variances differ.
How do I report a t-test in APA style?
Give the group means and SDs, then t(df) = value, p = value, and an effect size with its CI, e.g. “t(38) = 2.45, p = .019, d = 0.77.” Welch df are usually reported to one or two decimals.
What if my data are not normal?
With 30 or more per group the t-test is robust to moderate non-normality. For small, skewed samples consider the Mann–Whitney U test (independent) or Wilcoxon signed-rank test (paired).