Student’s t distribution replaces the normal curve whenever a standard deviation is estimated from the sample. Its critical values mark the cutoffs for t-tests and the multipliers for t confidence intervals. Instead of hunting through a printed table and interpolating, enter the degrees of freedom and α (or a confidence level) to get the exact critical value, the matching one- and two-tailed values, the normal comparison and a complete t-table.
How to use the t critical value calculator
- Enter the degrees of freedom. Fractional values from Welch’s test are fine.
- Choose whether to specify a significance level α or a confidence level.
- With α, pick two-tailed, right-tailed or left-tailed. A confidence level is always treated as two-sided.
- Read the critical value on the tape. The curve shades the rejection region, and the table below lists critical values for 44 common df values plus the normal limit, at eight tail areas, with your df highlighted.
How the critical value is defined
For a right-tail area a, the critical value ta, df satisfies
where T follows a t distribution with df degrees of freedom. Two-tailed tests use a = α/2, so that a total of α lies beyond ±t. A confidence level C uses α = 1 − C. The calculator inverts the exact cumulative distribution
where I is the regularized incomplete beta function, then checks the answer by plugging it back in. Results match standard tables: t₀.₀₂₅,₁₀ = 2.228, t₀.₀₅,₁ = 6.314 and t₀.₀₀₀₅,₅ = 6.869.
Worked example
You have 11 measurements and want a 95% confidence interval for the mean, so df = 11 − 1 = 10 and α = 0.05 split into two tails of 0.025.
- Find t with P(T > t) = 0.025 at df = 10: t* = 2.2281.
- The normal value would be 1.96, so the t multiplier is about 13.7% larger. That extra width is the price of estimating σ from only 11 values.
- If the sample mean is 50 and s = 4, the margin of error is 2.2281 × 4 ÷ √11 = 2.687, giving an interval of 47.31 to 52.69.
For a right-tailed test at α = 0.05 with the same df, all of α goes into one tail and the critical value is 1.8125.
How t critical values change with df
| df | Two-tailed α = 0.10 | α = 0.05 | α = 0.01 |
|---|---|---|---|
| 1 | 6.314 | 12.706 | 63.657 |
| 5 | 2.015 | 2.571 | 4.032 |
| 10 | 1.812 | 2.228 | 3.169 |
| 20 | 1.725 | 2.086 | 2.845 |
| 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 |
Two patterns stand out. With very few degrees of freedom the values are huge, because a standard deviation estimated from two or three points is unreliable. Past about df = 30 the values level off near the normal ones, which is why some older texts switch to z for large samples.
Using the table columns
The table on this page is indexed by the upper-tail area. For a two-tailed test at α = 0.05, read the 0.025 column; for a one-tailed test at α = 0.05, read the 0.05 column. Mixing these up is the most common t-table error.
Related tools
To run the full test from your data, use the t-test calculator. To go the other way, from a t statistic to a p-value, use the p-value calculator. For an interval built around the critical value, see the confidence interval calculator.
Frequently asked questions
What is the t critical value for a 95% confidence interval?
It depends on the degrees of freedom. With df = 10 it is 2.228; with df = 30 it is 2.042; with df = 100 it is 1.984; and as df grows without limit it approaches the normal value 1.960. Use df = n − 1 for a one-sample interval.
How do I find the degrees of freedom?
For a one-sample t-test or t interval, df = n − 1. For a paired test, df = number of pairs − 1. For a pooled two-sample test, df = n₁ + n₂ − 2. Welch's two-sample test uses a formula that usually gives a non-whole number, which this calculator accepts.
What is the difference between one-tailed and two-tailed critical values?
A two-tailed test splits α between both ends, so at α = 0.05 each tail holds 0.025. A one-tailed test puts all 0.05 in one end, so its critical value is smaller: 1.812 instead of 2.228 at df = 10.
Why is the t value larger than the z value?
Using the sample standard deviation adds uncertainty, which the t distribution reflects with heavier tails. The gap is large for tiny samples (12.71 versus 1.96 at df = 1) and shrinks quickly; by df = 30 the difference is about 4%.
What if my df is not in a printed table?
Printed tables skip values such as df = 37 or 250 and force you to interpolate. This calculator solves for the exact value at any df, including fractional ones, so no interpolation is needed.