T-Table (Student's t Distribution)

How to Use the T-Table (Student's t Distribution)

1. 1. Find your degrees of freedom. n − 1 for a one-sample or paired t-test, n₁ + n₂ − 2 for a pooled two-sample test, n − k − 1 for a regression coefficient with k predictors.
2. 2. Choose the column. Read the one-tail α row for a one-tailed test and the two-tail row for a two-tailed test. One column serves both, because a two-tailed test splits α into two tails of α/2.
3. 3. Read the critical value. It sits where your df row meets the α column. If your |t| is at least that large, the result is significant at that level.
4. 4. Or use the lookup. Enter any df and α to get the critical value to four decimals — including df the table skips, such as 35, and the fractional df of Welch's test.

How to Read the T-Table

Each entry is the value t* that leaves an area of α in the upper tail of the t distribution with the given degrees of freedom. The columns carry two labels because one number serves two tests: a one-tailed test at α = .025 and a two-tailed test at α = .05 share a critical value, since the two-tailed test puts α/2 in each tail. The third header row gives the matching confidence level for an interval.

If your df is not printed — 35, say — the conservative textbook rule is to use the next smaller df in the table, 30, which gives 2.042 and makes the test slightly harder to pass. The lookup box avoids the issue: it gives 2.0301 for df = 35, and 2.1059 for a Welch test with 17.43 df.

Degrees of Freedom and the Shape of t

The t distribution is the standard normal's heavier-tailed cousin: it allows for the extra uncertainty that comes from estimating the standard deviation from the sample itself. With few degrees of freedom the tails are far heavier — the two-tailed .05 critical value is 12.706 at df = 1 — and as df grows the curve closes in on the normal: 2.228 at df = 10, 2.086 at df = 20, 1.980 at df = 120, and 1.960, the z value, at ∞.

Common mistakes

• Reading the two-tailed row for a one-tailed test, or the reverse. It doubles or halves your real α without any warning.
• Using n instead of n − 1. For small samples it changes the answer: df = 10 gives 2.228, df = 11 gives 2.201.
• Comparing a negative t with a positive critical value. For a two-tailed test compare |t|; for a lower-tail test the critical value is negative, −t*.

Frequently Asked Questions

How do I read a t-table?

Find the row for your degrees of freedom and the column for your significance level — using the one-tail row for a one-tailed test and the two-tail row for a two-tailed test. The entry is the critical value. For df = 10 and α = .05 two-tailed it is 2.228: reject the null hypothesis if |t| ≥ 2.228.

What is the critical t-value for a 95% confidence interval?

It depends on the degrees of freedom: 2.228 for df = 10, 2.064 for df = 24, 2.042 for df = 30, and 1.960 as df grows without limit. Use the column headed 95% — one tail .025, two tails .05.

What is the difference between one-tailed and two-tailed critical values?

A two-tailed test splits α between both tails, so its critical values cut off α/2 on each side: ±2.228 at α = .05 with df = 10. A one-tailed test puts all of α in one tail and needs a smaller cut-off, 1.812. Decide which you need before seeing the data.

What if my degrees of freedom are not in the table?

Use the lookup box for the exact value, or follow the textbook convention of taking the next smaller df that is printed, which errs on the side of not rejecting. Interpolating between the two neighbouring rows also gives a close answer.

Why does the t-table end with ∞?

As the degrees of freedom increase, the t distribution converges to the standard normal, so the ∞ row holds z critical values: 1.645, 1.960, 2.326 and 2.576 for one-tail areas of .05, .025, .01 and .005. Even at df = 1000 the t values differ from z only in the third decimal.

How do I calculate the degrees of freedom for a t-test?

One-sample and paired tests use n − 1, counting pairs for paired data. Two independent groups with a pooled variance use n₁ + n₂ − 2. Welch's test uses the Welch–Satterthwaite formula, which usually gives a fractional df, and a regression coefficient uses n − k − 1.