t Table: Critical t Values, Made Interactive
A t table turns your degrees of freedom and significance level into the critical t value a test statistic must beat to count as significant. Set the three inputs below to light up the exact cell, read the value off a shaded curve, or flip to reverse mode to turn a t statistic back into a p-value. Every number is generated live and matches R's qt().
I want to find the critical t value
Two-tailed splits α across both tails, so the tool looks up qt(1 − α/2, df). One-tailed puts it all in one side: qt(1 − α, df).
t* = 2.086
Critical value for df = 20.
Full t distribution table
The complete critical-value table below is rendered as plain HTML, so it prints, copies and is fully indexable. Each row is a value of degrees of freedom; each column is an upper-tail area (with the matching two-tailed α and confidence level under it). Your current selection highlights the exact cell.
| df | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0005 |
|---|---|---|---|---|---|---|---|
| two-tailed α | 0.2 | 0.1 | 0.05 | 0.02 | 0.01 | 0.002 | 0.001 |
| confidence | 80% | 90% | 95% | 98% | 99% | 99.8% | 99.9% |
| 1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | 318.309 | 636.619 |
| 2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.327 | 31.599 |
| 3 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.215 | 12.924 |
| 4 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 |
| 5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.893 | 6.869 |
| 6 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |
| 7 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |
| 8 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |
| 9 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 | 4.781 |
| 10 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 |
| 11 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 |
| 12 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 | 4.318 |
| 13 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 | 4.221 |
| 14 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.140 |
| 15 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 |
| 16 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 | 4.015 |
| 17 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 | 3.965 |
| 18 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 |
| 19 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |
| 20 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 |
| 21 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 |
| 22 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 |
| 23 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 | 3.768 |
| 24 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.745 |
| 25 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 | 3.725 |
| 26 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.707 |
| 27 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.690 |
| 28 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.674 |
| 29 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.659 |
| 30 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 |
| 35 | 1.306 | 1.690 | 2.030 | 2.438 | 2.724 | 3.340 | 3.591 |
| 40 | 1.303 | 1.684 | 2.021 | 2.423 | 2.704 | 3.307 | 3.551 |
| 45 | 1.301 | 1.679 | 2.014 | 2.412 | 2.690 | 3.281 | 3.520 |
| 50 | 1.299 | 1.676 | 2.009 | 2.403 | 2.678 | 3.261 | 3.496 |
| 60 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.232 | 3.460 |
| 70 | 1.294 | 1.667 | 1.994 | 2.381 | 2.648 | 3.211 | 3.435 |
| 80 | 1.292 | 1.664 | 1.990 | 2.374 | 2.639 | 3.195 | 3.416 |
| 90 | 1.291 | 1.662 | 1.987 | 2.368 | 2.632 | 3.183 | 3.402 |
| 100 | 1.290 | 1.660 | 1.984 | 2.364 | 2.626 | 3.174 | 3.390 |
| 120 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.160 | 3.373 |
| 150 | 1.287 | 1.655 | 1.976 | 2.351 | 2.609 | 3.145 | 3.357 |
| 200 | 1.286 | 1.653 | 1.972 | 2.345 | 2.601 | 3.131 | 3.340 |
| ∞ (z) | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 3.090 | 3.291 |
The last row (∞) is the normal limit: with infinite df the critical t value equals the z value. R: qt(p, df) reproduces any cell.
How this value is computed
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When you reach for a t table
Three everyday jobs, all answered by the same table of critical values.
A 95% interval for a mean is the estimate plus or minus t* × SE, where t* is the two-tailed 0.05 critical value for your df. Smaller samples get a wider t*.
Compare your test statistic to the critical value for your α and tails. If |t| exceeds it, reject the null hypothesis. The reverse mode turns the statistic straight into a p-value.
With few degrees of freedom the tails are heavy, so the critical value is large. As df climbs, the t distribution tightens toward the normal and t* settles near the familiar z cutoffs.
One-tailed vs two-tailed, and how the columns line up
| You have | Tail area to look up | R call | df = 20 |
|---|---|---|---|
| 95% CI / two-tailed α = .05 | 0.025 in each tail | qt(0.975, 20) | 2.086 |
| 99% CI / two-tailed α = .01 | 0.005 in each tail | qt(0.995, 20) | 2.845 |
| one-tailed α = .05 | 0.05 in one tail | qt(0.95, 20) | 1.725 |
| one-tailed α = .01 | 0.01 in one tail | qt(0.99, 20) | 2.528 |
A two-tailed test asks only whether two things differ, so it guards both directions and needs a larger cutoff. A one-tailed test commits to a direction in advance and spends all of α on that side, which lowers the bar. Pick the tail count before you see the data, not after.
t table questions
How do I read a t table?
Find the row for your degrees of freedom and the column for your significance level, then read the cell where they meet: that is the critical t value. Use the two-tailed α header for a two-sided test and the one-tailed area for a one-sided test. If your observed statistic is more extreme than the cell, the result is significant at that α.
What are degrees of freedom?
Degrees of freedom count the independent pieces of information behind your variance estimate. A one-sample or paired t test has df = n − 1; a Welch two-sample test has a fractional df from its variance-weighting formula. This tool accepts any df, whole or decimal, so a Welch df like 17.4 works directly.
What is the critical t value for a 95% confidence interval?
A 95% interval leaves 0.05 split across the two tails, so you want the value with 0.025 above it: qt(0.975, df). It is 2.086 at df = 20 and 2.009 at df = 50, and it approaches the z value 1.960 as the sample grows.
What are degrees of freedom in a t test?
Degrees of freedom (df) are the number of independent pieces of information used to estimate variability. For a one-sample or paired t test, df equals the sample size minus one. For a two-sample test, R uses the Welch approximation, which can be fractional. This calculator accepts any df, including decimals.
What is the difference between one-tailed and two-tailed critical values?
A two-tailed test splits alpha across both tails, so it uses qt(1 - alpha/2, df). A one-tailed test puts all of alpha in one tail, so it uses qt(1 - alpha, df) and gives a smaller critical value. Choose one-tailed only when your hypothesis specifies a direction in advance.
Does this t table match R?
Yes. Every printed value and every interactive result is generated by the same routine and checked against R's qt() and pt() functions across more than 2500 cases to better than seven significant figures.
Why is the critical value bigger for small samples?
With few degrees of freedom the t distribution has heavier tails than the normal, reflecting the extra uncertainty in estimating the standard deviation from a small sample. A larger critical value is the price of that uncertainty; it shrinks toward the z value as df grows.
Does this match R and printed textbook tables?
Yes. The printed table and the live result share one routine, checked against R's qt() and pt() across more than 2500 cases to better than seven significant figures. Small last-digit gaps versus an old textbook table are rounding in the book, not here.