T Table (Critical Values)

How to read this t-table

The table gives critical values of Student's t distribution: the values a t statistic must beat to be declared significant. Columns are one-tailed significance levels (upper tail). Rows are degrees of freedom. For a two-tailed test, halve your alpha before entering the table: a two-tailed 5% test uses the 0.025 column.

Say your one-sample t test on 8 observations returns t = 2.679. Degrees of freedom are 7. The 0.025 column (two-tailed 5%) at df 7 reads 2.365. Your 2.679 exceeds it, so the result is significant at the two-tailed 5% level. It falls short of the 0.005 column (3.499), so it is not significant at the two-tailed 1% level. That bracket, p between 0.01 and 0.05, is what a table can tell you; the calculator gives the exact p (0.032 here).

Why the rows shrink toward the bottom row

With few degrees of freedom the t distribution has heavy tails: small samples produce wild t values by chance, so the bar for significance sits high (6.314 at df 1 for a one-tailed 5%). As df grows, the sample standard deviation becomes trustworthy, the tails thin, and the critical values settle toward the bottom row: the normal distribution's own quantiles (1.645 at 0.05). Past df 120 the t and normal values agree to two decimals for most purposes.

Which degrees of freedom?

One-sample and paired tests: n − 1. Two-sample pooled: n₁ + n₂ − 2. Welch's test: a fractional value from the Welch-Satterthwaite formula; round down to the table row below for a conservative read, or use the calculator, which keeps the fraction. Regression slope: n − 2. Each design spends degrees of freedom on the quantities it estimates.

Frequently asked questions