How to do a t-test (two groups, with a worked example)

How to · Updated 2026-11-10

Paste two columns of numbers, let the tool pick Welch's or Student's t-test, then read t, the degrees of freedom, the p-value, the confidence interval and Cohen's d.

A two-sample t-test asks whether the gap between two group means is bigger than random noise would usually give: divide the difference in means by its standard error, then compare that ratio with a t distribution to get a p-value. Paste both groups into the hypothesis tests page and it chooses the right version, runs it and spells out what the answer means.

What you need

• Two groups of measured numbers, such as reaction times, marks or plant heights, from different people or things. If the same people were measured twice, you need the paired t-test instead (the same page has it).
• At least two values per group. Eight or more each gives the test a fair chance of seeing a real difference; the sample size and power calculator tells you how many you really need.
• A decision made before you look: are you testing for any difference (two-sided) or only for one direction (one-sided)?

Step by step

1. Open hypothesis tests. Under What did you measure? keep Numbers, and under What are you comparing? keep Two separate groups.
2. Leave Roughly bell-shaped (normal)? on Not sure: check my data. The page then runs a Shapiro-Wilk normality check on each group. If either group is clearly skewed and has fewer than 30 values, it switches to the Mann-Whitney U test, which works on ranks.
3. Paste your data into the box: one column per group, with the group names in the top row. Copying two columns straight from a spreadsheet works.
4. Check Alternative (two-sided unless you planned otherwise), the Confidence level for the interval, and Significance level α, which only changes the wording of the verdict.
5. Read the large line at the top of the results, the plain-English paragraph under it, the list of numbers and the Assumptions checklist. The strip chart shows every value with each group’s mean and confidence interval.
6. Save the results as a CSV, or as an image, PDF or Word report.

The test behind it is Welch’s t-test, the safe default because it does not assume both groups are equally spread out:

• t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂)
• degrees of freedom ν = (s₁²/n₁ + s₂²/n₂)² ÷ [ (s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1) ]

Here x̄ is a group mean, s² its sample variance (dividing by n − 1) and n its size. Student’s version pools the two variances and uses n₁ + n₂ − 2 degrees of freedom; pick it from the Test list only if you have a good reason to think the spreads match.

A worked example

A tutor tries two ways of revising for a quiz. Eight students use spaced practice and score 68, 74, 71, 79, 65, 72, 77 and 70 %. Eight others reread their notes and score 61, 66, 70, 63, 59, 68, 64 and 62 %.

By hand:

• Means: 576 ÷ 8 = 72 % and 513 ÷ 8 = 64.125 %, a difference of 7.875 points.
• Sample variances: 21.14 and 13.55, so s²/n is 2.643 and 1.694.
• Standard error: √(2.643 + 1.694) = √4.337 = 2.083.
• t = 7.875 ÷ 2.083 = 3.781.
• Welch’s degrees of freedom: 4.337² ÷ (2.643²/7 + 1.694²/7) = 13.36.

Pasted into the page, the Shapiro-Wilk checks find nothing odd (p = 0.98 and 0.95), so Welch’s test is chosen. The headline reads t(13.36) = 3.781, p = 0.002. The 95 % confidence interval for the difference runs from 3.388 to 12.36 points, and Cohen’s d is 1.891, a large effect by the usual guide (0.2 small, 0.5 medium, 0.8 large).

In words: if both methods really worked equally well, a gap this big would turn up only about 2 times in 1,000 experiments like this one. The tutor has good evidence that spaced practice scored higher, probably by somewhere between 3 and 12 points.

Check your answer

• The sign. t is positive because the first column has the bigger mean. Swap the columns and t, the difference and its interval flip sign, but p stays the same.
• The interval and p agree. A 95 % interval that does not include 0 always goes with a two-sided p below 0.05. Here 0 lies well outside 3.4 to 12.4.
• Degrees of freedom. Welch’s ν always lies between the smaller n − 1 (7) and n₁ + n₂ − 2 (14). Something outside that range means a typing slip.
• Rough size. t is about the difference over the standard error. A gap of about 8 with a standard error of about 2 should give t near 4, and it does.

Questions people ask

What does “statistically significant” actually mean here? Only that p fell below the α you chose, usually 0.05. It says nothing about whether the difference matters in practice; look at the interval and the effect size for that, and convert d to other measures with the effect size converter.

Can I run a t-test in Excel instead? Yes, with the T.TEST function, but it gives only the p-value. You would still need to work out the interval, the effect size and the normality checks separately.

My p-value is 0.08. Is there no difference? Not proven either way. A small study can miss a real effect. Report the interval, which shows the range of differences the data still allow.