DF Race: Degrees of Freedom for t-Tests

Quick answer: degrees of freedom pick the row of the t-table. Use df = n − 1 for a one-sample t-test or a confidence interval for one mean, df = pairs − 1 for a paired t-test, df = n₁ + n₂ − 2 for a pooled two-sample t-test, and df = n − 2 for a regression slope. For n = 16, df = 15 and the two-tailed 0.05 critical value is t* = 2.131.

This game drills those rules against the clock: 10 scenarios, a score, a streak and a timer, and the rule plus the t-table row after every answer.

Play: DF Race

Ten timed scenarios: one-sample and paired t-tests, pooled two-sample t-tests, confidence intervals and regression slopes. Tap the degrees of freedom, or type the number. After each answer you see the rule and the t-table row it points to.

Worked example

Question: 12 students take a test before and after a course (24 scores). You run a paired t-test. What is df, and what critical t do you use at α = 0.05, two-tailed?

Step 1. A paired test works on the 12 differences, so df = 12 − 1 = 11, not 24 − 1 or 24 − 2.

Step 2. In the two-tailed t-table, row df = 11, column α = 0.05: t* = 2.201.

Compare: if the 24 scores came from two separate groups of 12 (pooled test), df = 12 + 12 − 2 = 22 instead.

FAQ

How do you find degrees of freedom for a t-test?

It depends on the test. One-sample t-test or a confidence interval for one mean: df = n − 1. Paired t-test: df = number of pairs − 1. Pooled two-sample t-test: df = n₁ + n₂ − 2. Slope of a simple linear regression: df = n − 2.

What df does a paired t-test use?

The number of pairs minus 1, not the number of measurements. 12 people measured before and after give 24 numbers but only 12 differences, so df = 11.

What df does a two-sample t-test use?

The pooled (equal-variance) test uses df = n₁ + n₂ − 2, for example 10 + 8 − 2 = 16. Welch's test (unequal variances) uses the Welch–Satterthwaite formula instead, which is usually not a whole number; software computes it, and a conservative hand rule is the smaller n − 1.

Why does df matter for the t-table?

Each df is a separate row of the t-table. Smaller df means heavier tails and a larger critical value: two-tailed α = 0.05 gives t* = 2.131 at df = 15 but 2.064 at df = 24.

How is the race scored?

You get 10 scenarios. Each correct df adds a point and the clock runs from the first question to the last. Get all 10 right to set a best time, saved only in your own browser.

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