T Distribution Table
Critical t values for every degrees of freedom, switchable between one- and two-tailed.
| df | α = 0.10 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 |
|---|---|---|---|---|---|
| conf. | 90% | 95% | 97.5% | 99% | 99.5% |
| 1 | 6.314 | 12.706 | 25.452 | 63.657 | 127.321 |
| 2 | 2.920 | 4.303 | 6.205 | 9.925 | 14.089 |
| 3 | 2.353 | 3.182 | 4.177 | 5.841 | 7.453 |
| 4 | 2.132 | 2.776 | 3.495 | 4.604 | 5.598 |
| 5 | 2.015 | 2.571 | 3.163 | 4.032 | 4.773 |
| 6 | 1.943 | 2.447 | 2.969 | 3.707 | 4.317 |
| 7 | 1.895 | 2.365 | 2.841 | 3.499 | 4.029 |
| 8 | 1.860 | 2.306 | 2.752 | 3.355 | 3.833 |
| 9 | 1.833 | 2.262 | 2.685 | 3.250 | 3.690 |
| 10 | 1.812 | 2.228 | 2.634 | 3.169 | 3.581 |
| 11 | 1.796 | 2.201 | 2.593 | 3.106 | 3.497 |
| 12 | 1.782 | 2.179 | 2.560 | 3.055 | 3.428 |
| 13 | 1.771 | 2.160 | 2.533 | 3.012 | 3.372 |
| 14 | 1.761 | 2.145 | 2.510 | 2.977 | 3.326 |
| 15 | 1.753 | 2.131 | 2.490 | 2.947 | 3.286 |
| 16 | 1.746 | 2.120 | 2.473 | 2.921 | 3.252 |
| 17 | 1.740 | 2.110 | 2.458 | 2.898 | 3.222 |
| 18 | 1.734 | 2.101 | 2.445 | 2.878 | 3.197 |
| 19 | 1.729 | 2.093 | 2.433 | 2.861 | 3.174 |
| 20 | 1.725 | 2.086 | 2.423 | 2.845 | 3.153 |
| 21 | 1.721 | 2.080 | 2.414 | 2.831 | 3.135 |
| 22 | 1.717 | 2.074 | 2.405 | 2.819 | 3.119 |
| 23 | 1.714 | 2.069 | 2.398 | 2.807 | 3.104 |
| 24 | 1.711 | 2.064 | 2.391 | 2.797 | 3.091 |
| 25 | 1.708 | 2.060 | 2.385 | 2.787 | 3.078 |
| 26 | 1.706 | 2.056 | 2.379 | 2.779 | 3.067 |
| 27 | 1.703 | 2.052 | 2.373 | 2.771 | 3.057 |
| 28 | 1.701 | 2.048 | 2.368 | 2.763 | 3.047 |
| 29 | 1.699 | 2.045 | 2.364 | 2.756 | 3.038 |
| 30 | 1.697 | 2.042 | 2.360 | 2.750 | 3.030 |
| 35 | 1.690 | 2.030 | 2.342 | 2.724 | 2.996 |
| 40 | 1.684 | 2.021 | 2.329 | 2.704 | 2.971 |
| 45 | 1.679 | 2.014 | 2.319 | 2.690 | 2.952 |
| 50 | 1.676 | 2.009 | 2.311 | 2.678 | 2.937 |
| 60 | 1.671 | 2.000 | 2.299 | 2.660 | 2.915 |
| 80 | 1.664 | 1.990 | 2.284 | 2.639 | 2.887 |
| 100 | 1.660 | 1.984 | 2.276 | 2.626 | 2.871 |
| 120 | 1.658 | 1.980 | 2.270 | 2.617 | 2.860 |
| ∞ | 1.645 | 1.960 | 2.241 | 2.576 | 2.807 |
Reject the null hypothesis when your calculated |t| is larger than the critical value in the row for your degrees of freedom. The confidence row underneath each α is the interval that column builds: α = 0.025 two-tailed is the 95% confidence level.
A t-table gives the cut-off your test statistic must beat to be significant. Find your degrees of freedom down the side and your significance level across the top: at df = 10 and a two-tailed α = 0.05 the critical value is 2.228, so any |t| above 2.228 rejects the null hypothesis.
Why the t-distribution exists
When you know the population standard deviation you can use the normal curve and its fixed 1.96 cut-off. You almost never know it. Estimating σ from the sample adds a second source of uncertainty, and the t-distribution is the shape that accounts for it: same bell, but heavier tails, so the critical values sit further out and the conclusions stay honest.
How much heavier depends on the sample. That is what degrees of freedom measure — for a one-sample test, df = n − 1. With df = 1 the two-tailed 5% critical value is a remote 12.706; by df = 10 it has fallen to 2.228, by df = 30 to 2.042, and as df grows it converges on the normal value of 1.960. A t-table is really a stack of normal-ish curves, one per sample size.
Reading the significance level
Every column heading is an area in the tail. A one-tailed α = 0.05 puts all 5% at one end; a two-tailed α = 0.05 splits it into 2.5% at each end, which pushes the cut-off further out. This is the single biggest source of wrong answers with a t-table, so the switch above relabels the whole table rather than asking you to halve anything yourself.
x̄ the sample mean, μ₀ the hypothesised mean, s the sample standard deviation, n the sample size
- 1 Count the degrees of freedom. A one-sample test on 15 observations has df = 15 − 1 = 14.
- 2 Decide one-tailed or two-tailed. Testing only whether the mean is larger is one-tailed; testing whether it differs in either direction is two-tailed.
- 3 Pick the significance level. The usual choice is α = 0.05, which for a two-tailed test leaves 0.025 in each tail.
- 4 Read the cell where the row and column meet. Row df = 14 and a two-tailed α = 0.05 give a critical value of 2.145.
- 5 Compare it with your test statistic. A calculated t of 2.62 exceeds 2.145, so the result is significant at the 5% level; a t of 1.80 would not be.
Critical t values at the common levels
Two-tailed values are the ones used for 95% and 99% confidence intervals. The ∞ row is the normal distribution.
| df | One-tailed α = 0.05 | Two-tailed α = 0.05 | Two-tailed α = 0.01 |
|---|---|---|---|
| 1 | 6.314 | 12.706 | 63.657 |
| 5 | 2.015 | 2.571 | 4.032 |
| 10 | 1.812 | 2.228 | 3.169 |
| 15 | 1.753 | 2.131 | 2.947 |
| 20 | 1.725 | 2.086 | 2.845 |
| 25 | 1.708 | 2.060 | 2.787 |
| 30 | 1.697 | 2.042 | 2.750 |
| 60 | 1.671 | 2.000 | 2.660 |
| 120 | 1.658 | 1.980 | 2.617 |
| ∞ | 1.645 | 1.960 | 2.576 |
Beyond the one-sample test
The same table serves several tests, only the degrees of freedom change. A paired t-test uses the number of pairs minus one. An independent two-sample test with equal variances uses n₁ + n₂ − 2. A confidence interval for a mean uses the two-tailed value at the matching confidence level — the 2.145 above is exactly the multiplier in a 95% interval built from 15 observations.
If your degrees of freedom fall between printed rows on a paper table, the conservative habit is to round down to the next row, which gives a slightly larger critical value and makes significance slightly harder to claim. The calculator above sidesteps that entirely: it computes the exact value for whatever df you type, so there is nothing to interpolate.