CalcKitHome›Math & Stats›T critical value
T Critical Value Calculator
Your test setup
Need a level that is not listed? Choose “Custom” and type any confidence level below.
For a one-sample t-test or a confidence interval for one mean, df = sample size − 1.
Two-tailed is the usual choice for a confidence interval. One-tailed is used when the test only looks in one direction.
With 10 degrees of freedom, a sample mean more than 2.228 standard errors from the null value falls outside the 95% confidence interval. The matching z-value is 1.960 — the t-value is larger because a small sample carries more uncertainty.
Example: 95% confidence, two-tailed, df = 10 — t* = 2.228 (the matching z-value is 1.960).
Behind the numbers
How the t critical value is calculated
The t critical value t* is the point on the t-distribution with the chosen degrees of freedom that leaves the confidence level in the middle (two-tailed) or below it (one-tailed). For a two-tailed 95% interval, 2.5% of the area sits in each tail, so t* is the 97.5th percentile of the t-distribution. This calculator finds it exactly by inverting the t cumulative distribution function, which is built from the regularized incomplete beta function — no rounded table lookup.
where F is the CDF of the t-distribution with df degrees of freedom and α = 1 − confidence level. Worked example: 95% two-tailed with df = 10 means α = 0.05, so t* is the 97.5th percentile of the t-distribution with 10 degrees of freedom: t* = 2.228. The matching normal (z) value is 1.960, and the gap between 2.228 and 1.960 is the extra uncertainty from a sample of only 11 observations.
Worked examples
Example t critical values
| Confidence | Tails | df | t* | z-value |
|---|---|---|---|---|
| 95% | Two-tailed | 10 | 2.228 | 1.960 |
| 95% | Two-tailed | 24 | 2.064 | 1.960 |
| 99% | Two-tailed | 30 | 2.750 | 2.576 |
| 90% | Two-tailed | 10 | 1.812 | 1.645 |
| 98% | Two-tailed | 15 | 2.602 | 2.326 |
| 95% | Two-tailed | 5 | 2.571 | 1.960 |
As the degrees of freedom grow, t* moves closer to the z-value: at df = 100 the 95% two-tailed t* is 1.984, and at df = 1,000 it is 1.962 — almost the z-value of 1.960.
Using t* the right way
For a confidence interval for one mean, df = n − 1. A sample of 11 gives df = 10 and a 95% t* of 2.228 — wider than the z-interval because the sample standard deviation is itself an estimate.
Two-tailed is the default for confidence intervals and for most t-tests. Switch to one-tailed only when your test has a stated direction before you see the data.
Compare t* with the z-value in the result card: the gap between them shows how much extra uncertainty a small sample adds. Past df = 30 the gap is small, and past df = 100 it is barely noticeable.
Do not round t* early. A rounded t* of 2.23 instead of 2.228 changes a confidence interval only slightly, but rounding to 2.2 can matter in a small-sample test.
Good to know
Your t critical value questions, answered.
What is a t critical value?
A t critical value, written t*, is the cutoff on a t-distribution that leaves a chosen share of the area in the middle (two-tailed) or below it (one-tailed). For example, with 10 degrees of freedom the two-tailed 95% critical value is 2.228, so 95% of the t-distribution with 10 degrees of freedom lies between -2.228 and 2.228.
When do I use a t critical value instead of a z critical value?
Use a t critical value when the population standard deviation is unknown and you estimate it from a sample, which is the usual case in t-tests and confidence intervals for a mean. Use a z critical value when the population standard deviation is known, or when the sample is so large that the t-distribution is almost identical to the normal distribution.
What are degrees of freedom?
Degrees of freedom measure how much independent information is in your sample. For a one-sample t-test or a confidence interval for one mean, degrees of freedom equal the sample size minus 1 (df = n - 1). Smaller samples have fewer degrees of freedom and larger critical values, because there is more uncertainty.
How do I read a t-table with this calculator?
Pick your confidence level, enter the degrees of freedom and choose two-tailed or one-tailed, and the calculator gives the exact t* value instead of a rounded table entry. For instance, 95% two-tailed with 24 degrees of freedom gives t* = 2.064, and 99% two-tailed with 30 degrees of freedom gives t* = 2.750.