T-Table (Student's t Distribution)
Pick the degrees of freedom and α (or a confidence level) to get the exact critical t-value for a one- or two-tailed test, or enter a t-score to get its p-value. The shaded curve shows the tail area, and the full t-table below highlights the matching cell and can be printed or downloaded.
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One-sample or paired test: df = n − 1. Type ∞ or inf for the normal (z) limit. Decimals (Welch) are fine.
Full t-table (critical values)
Each cell is the t-value with the given area in the upper tail. Find your df in the left column and your α in the header (use the one-tail row for one-sided tests, the two-tail row for two-sided). Click a cell to load it above.
How to use the T-Table (Student's t Distribution)
- Enter the degrees of freedom, usually n − 1 for a one-sample or paired t-test.
- Choose the test type (two-tailed, right or left) and type α such as 0.05, or a confidence level such as 95%.
- Read the critical value at the top, with the shaded rejection region on the t curve.
- For a p-value, switch to the second tab, type your t-score and choose the alternative hypothesis.
- Use the full table below: find your row and column, click a cell to use it, or print / download the table.
What the t-table shows
The t-table lists critical values of Student's t distribution: for each number of degrees of freedom (df) and upper-tail area α, the value tα,df such that P(T > tα,df) = α. It is used for t-tests and confidence intervals when the population standard deviation is unknown and estimated from the sample.
- Right-tailed test: reject H₀ if t > tα,df.
- Left-tailed test: reject H₀ if t < −tα,df (the curve is symmetric).
- Two-tailed test: reject H₀ if |t| > tα/2,df.
- Confidence interval: x̄ ± tα/2,df · s/√n, with confidence level 1 − α.
Worked example
A sample of n = 16 bags has mean weight 498.2 g and standard deviation 4 g. Is the mean below 500 g at α = 0.05? The test statistic is t = (498.2 − 500) / (4/√16) = −1.8 with df = 15. The left-tailed critical value is −t0.05,15 = −1.753. Since −1.8 < −1.753, reject H₀. The exact p-value is about 0.046 (choose "p-value from a t-score" above to check).
Most used critical values
| df | 90% (two-tail .10) | 95% (two-tail .05) | 98% (two-tail .02) | 99% (two-tail .01) |
|---|---|---|---|---|
| 1 | 6.314 | 12.706 | 31.821 | 63.657 |
| 5 | 2.015 | 2.571 | 3.365 | 4.032 |
| 10 | 1.812 | 2.228 | 2.764 | 3.169 |
| 20 | 1.725 | 2.086 | 2.528 | 2.845 |
| 30 | 1.697 | 2.042 | 2.457 | 2.750 |
| 60 | 1.671 | 2.000 | 2.390 | 2.660 |
| ∞ (z) | 1.645 | 1.960 | 2.326 | 2.576 |
Tips
- Some tables label columns by the two-tail area or by confidence level instead of the one-tail area; this table shows all three headers so you can match your textbook.
- With df above about 100, t and z critical values differ by less than 2%, which is why large-sample tests often use z.
- For a p-value from a printed table you can only bracket it between two columns; the calculator gives the exact value.
Frequently asked questions
How do you read a t-table?
Find the row for your degrees of freedom and the column for your significance level. Use the one-tail header for a one-sided test and the two-tail header for a two-sided test. The cell where they meet is the critical t-value; for df = 10 and a two-tailed α = 0.05 (one tail 0.025) it is 2.228.
What is the t critical value for a 95% confidence interval?
It depends on the degrees of freedom: 12.706 for df = 1, 2.776 for df = 4, 2.228 for df = 10, 2.086 for df = 20, 2.042 for df = 30 and 1.984 for df = 100. As df grows it approaches the z-value 1.960.
How do I find degrees of freedom?
For a one-sample t-test or paired t-test, df = n − 1. For a two-sample test with equal variances, df = n₁ + n₂ − 2. For Welch's unequal-variance test, df comes from the Welch–Satterthwaite formula and is usually not a whole number; this tool accepts decimal df.
What if my degrees of freedom are not in the table?
Printed tables skip many df values above 30. The traditional conservative choice is the next smaller df listed, which gives a slightly larger critical value. This calculator computes the exact value for any df, so you can skip the approximation.
What is the difference between one-tailed and two-tailed critical values?
A two-tailed test splits α between both tails, so each tail holds α/2 and the critical value is larger: for df = 10, α = 0.05 gives ±2.228 two-tailed but 1.812 one-tailed. Use one-tailed only when the alternative hypothesis specifies a direction in advance.
Why does the t distribution have fatter tails than the normal?
The t distribution accounts for the extra uncertainty from estimating the standard deviation from the sample. With few degrees of freedom the tails are much heavier, so larger critical values are needed. As df increases the t distribution converges to the standard normal.
How accurate are the values?
Tail areas are computed from the regularized incomplete beta function and critical values by solving for the exact quantile, so they match published tables to every printed digit (for example 636.619 for df = 1 and one-tail α = 0.0005).
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