Find Critical t-Value
What the table reports
This reference gives positive upper-tail critical values c such that P(T_df > c) = a. The printed column headings show that upper-tail probability a. The interactive lookup lets you specify the total significance level and whether the procedure is one- or two-tailed.
For a two-sided test at α = 0.05, split α between the two tails and use the printed 0.025 column. The cutoffs are −c and +c. For a one-sided upper-tail test at α = 0.05, use the 0.05 column.
Critical values by degrees of freedom
| df | Upper tail 0.10 | Upper tail 0.05 | Upper tail 0.025 | Upper tail 0.01 | Upper tail 0.005 |
|---|---|---|---|---|---|
| 1 | 3.0777 | 6.3138 | 12.7062 | 31.8205 | 63.6567 |
| 2 | 1.8856 | 2.9200 | 4.3027 | 6.9646 | 9.9248 |
| 3 | 1.6377 | 2.3534 | 3.1824 | 4.5407 | 5.8409 |
| 4 | 1.5332 | 2.1318 | 2.7764 | 3.7469 | 4.6041 |
| 5 | 1.4759 | 2.0150 | 2.5706 | 3.3649 | 4.0321 |
| 6 | 1.4398 | 1.9432 | 2.4469 | 3.1427 | 3.7074 |
| 7 | 1.4149 | 1.8946 | 2.3646 | 2.9980 | 3.4995 |
| 8 | 1.3968 | 1.8595 | 2.3060 | 2.8965 | 3.3554 |
| 9 | 1.3830 | 1.8331 | 2.2622 | 2.8214 | 3.2498 |
| 10 | 1.3722 | 1.8125 | 2.2281 | 2.7638 | 3.1693 |
| 12 | 1.3562 | 1.7823 | 2.1788 | 2.6810 | 3.0545 |
| 15 | 1.3406 | 1.7531 | 2.1314 | 2.6025 | 2.9467 |
| 20 | 1.3253 | 1.7247 | 2.0860 | 2.5280 | 2.8453 |
| 24 | 1.3178 | 1.7109 | 2.0639 | 2.4922 | 2.7969 |
| 25 | 1.3163 | 1.7081 | 2.0595 | 2.4851 | 2.7874 |
| 30 | 1.3104 | 1.6973 | 2.0423 | 2.4573 | 2.7500 |
| 40 | 1.3031 | 1.6839 | 2.0211 | 2.4233 | 2.7045 |
| 60 | 1.2958 | 1.6706 | 2.0003 | 2.3901 | 2.6603 |
| 100 | 1.2901 | 1.6602 | 1.9840 | 2.3642 | 2.6259 |
| Normal limit | 1.2816 | 1.6449 | 1.9600 | 2.3263 | 2.5758 |
Values are rounded to four decimals. The normal-limit row is a reference, not a rule to use z whenever n is above a particular cutoff.
Match df to the analysis
| Procedure | Degrees of freedom |
|---|---|
| One-sample t-test or mean interval | n − 1 |
| Paired t-test | Number of complete independent pairs − 1 |
| Pooled two-sample t-test | n₁ + n₂ − 2, with equal-variance justification |
| Welch two-sample t-test | Welch–Satterthwaite df; can be fractional |
| Simple-regression slope test | n − 2 |
If a printed table lacks your df, the next lower available df gives a conservative positive cutoff. Software can use the actual df, including a fractional value; arbitrary rounding is unnecessary.
Worked example: a two-sided mean test
A sample of 16 observations has mean 498 g and sample SD 4 g. Test H₀: μ = 500 g at α = 0.05, two-sided.
- SE = 4/√16 = 1 g.
- t = (498 − 500)/1 = −2.
- df = 15.
- Choose the upper-tail 0.025 column: c = 2.1314.
- Since |−2| < 2.1314, fail to reject H₀.
The exact two-sided p-value is about 0.0639. A table may only bracket a p-value; a critical value is not itself a p-value.
Use the same cutoff for a matching interval
For a 95% mean interval with unknown population SD, use x̄ ± c × s/√n. In the example, this gives [495.8686, 500.1314] g. The null value is included, consistent with the corresponding two-sided test.
If instead the predeclared alternative is μ < 500, the lower-tail 5% cutoff is −1.7531. Its decision differs because it is a different question. Do not choose the tail after seeing the result.
When t is appropriate
Use t for mean inference when population variability is estimated from the observations and the design and distributional conditions fit the procedure. Even with a large sample, estimating σ does not make it a known population value; t and z merely become numerically similar at large df.
For small-sample exact one-sample inference, the model assumes independent normal observations. For paired inference, assess the differences. Larger samples can improve robustness to some departures but cannot repair dependence, selection bias, or extreme influence automatically.
Common errors
- Counting two measurements per person as two independent units in a paired test.
- Using the total two-sided α in a printed upper-tail column without halving it.
- Substituting n for df.
- Treating a nonsignificant result as proof of equality.
- Assuming a sample mean alone is enough to calculate t; you also need variability and sample size.
Practice
Twelve independent pairs give a paired t statistic of 2.1. At α = 0.05, two-sided, should you reject?
Show the explanation
df = 11 and the positive cutoff is about 2.2010. Since 2.1 < 2.2010, fail to reject at this threshold. Use the actual paired differences and inspect assumptions before drawing a substantive conclusion.
Reference and related tools
NIST’s t-distribution reference describes the distribution. Read the t-test lesson, use the t-test calculator for raw observations, or use the confidence interval calculator for mean/SD/n inputs.