Definition · Plain-language
t-test
A t test is a statistical test that compares means — between a sample and a known value, between two groups, or between paired observations — to judge whether the difference is larger than chance would predict.
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The three types of t test
The one-sample t test compares the mean of a single sample with a known or hypothesised value — for example, testing whether a class’s average score differs from a national benchmark. The independent-samples (two-sample) t test compares the means of two separate, unrelated groups, such as a treatment group and a control group. The paired (dependent-samples) t test compares two measurements taken from the same subjects, such as scores before and after an intervention. Choosing the right form depends on how the data were collected and whether the observations are related.
How a t test works
A t test computes a t-statistic: the difference between means divided by the standard error of that difference, which scales the gap by the variability in the data. The larger the t-statistic relative to its degrees of freedom, the less plausible it is that the difference arose by chance. This value is compared against the t-distribution to produce a p-value. A significant result indicates the means differ more than sampling variation alone would predict; it does not by itself say how large or important the difference is, which is why an accompanying effect size such as Cohen’s d should be reported.
Assumptions and when to use it
A t test assumes the outcome is measured on a continuous scale, that observations are independent (except the paired design, where pairs are linked), and that the data are approximately normally distributed. The independent-samples test additionally assumes the two groups have roughly equal variances, though a Welch correction relaxes this. The t test is designed for comparing one or two means; when three or more group means must be compared, repeated t tests inflate the error rate, so analysis of variance (ANOVA) is the correct choice instead.
Key facts
At a glance
- Definition: a parametric test that compares means
- One-sample: sample mean vs a known or hypothesised value
- Independent: two separate, unrelated groups
- Paired: same subjects measured twice (e.g. before/after)
- Assumptions: approximate normality; similar variances for independent test
- Use ANOVA when: comparing three or more group means
Common misconceptions
What people often get wrong
Often heard: A t test can compare the means of any number of groups at once.
Actually: A t test compares only one or two means. Running many t tests across three or more groups inflates the false-positive rate; analysis of variance (ANOVA) is the correct test for comparing three or more group means.
Often heard: A significant t test proves the difference between the means is large and important.
Actually: It shows the difference is unlikely to be chance, not that it is large. Magnitude is given by an effect size such as Cohen’s d, which should be reported alongside the t test and its p-value.
Often heard: The paired and independent t tests are interchangeable.
Actually: They suit different designs. The paired test is for two linked measurements on the same subjects; the independent test is for two separate groups. Using the wrong one mismodels the data and gives misleading results.
Going deeper
