T-Distribution Calculator

A t-distribution calculator finds areas, p-values and critical t-values for Student’s t-distribution. Enter your t value and degrees of freedom to get the cumulative probability and the one- or two-tailed p-value, or enter a confidence level to get the critical t (t*). The chart compares the t curve with the standard normal curve, showing how its heavier tails shrink as df grows.

How to use the t-distribution calculator

Use Area / P-Value mode when you already have a test statistic and want a probability. Use Critical t (t*) mode when you are building a confidence interval or a rejection region and need the cut-off value first.

1. Pick a mode. Choose “Area / P-Value” to work from a t value, or “Critical t (t*)” to work from a confidence level.
2. Enter the t value and df (area mode), or the confidence level and df (critical-t mode). For a one-sample test, df = n − 1.
3. Choose the tail. Pick two-tailed for a standard p-value, or one-tailed when your alternative hypothesis points in one direction only.
4. Read the headline answer, then check the results grid and the calculation ledger, which shows every step, including the incomplete beta value used for the CDF.

Try it: t = 2.0 with df = 10 gives a two-tailed p-value of 0.0734, and the 95% two-tailed critical t at df = 20 is 2.086. If you started from a z-score instead, use our p-value calculator from a z-score.

What is the t-distribution?

The t-distribution, also called Student’s t-distribution, is a family of symmetric, bell-shaped distributions used when you are working with small samples or when the population standard deviation (σ) is unknown and estimated from the sample. It was introduced by William Gosset, writing as “Student”, for exactly that situation: quality-control samples too small for the normal distribution to be trusted.

Each member of the family is set by its degrees of freedom (df). With few degrees of freedom the curve is shorter and wider, with heavier tails than the normal curve, so extreme values are more likely. As df grows, the curve rises in the middle, its tails thin out, and it converges on the standard normal distribution. At df = 100 the difference is already tiny: the 95% two-tailed critical t is 1.984, against z = 1.960.

The distribution is the basis of the one-sample and two-sample t-tests and of confidence intervals for a mean when σ is unknown. A two-sample test needs a combined spread first, which is where a pooled standard deviation calculator comes in, and the resulting interval can be built with a confidence interval calculator for a mean and standard deviation.

When to use a t-distribution instead of a z-distribution

Use the t-distribution when the population standard deviation is unknown and estimated from your sample, especially with small samples. Use the z-distribution when σ is known, or when the sample is large enough (usually n ≥ 30) that the sample standard deviation is a reliable stand-in. That is the short version of the t vs z decision, and it matches how the two tests are taught:

If you are unsure which statistic you are working with, start by learning how to calculate a z-score, then compare the cut-offs: a critical z-value calculator gives 1.960 for 95% two-tailed, while the t cut-off is always a little larger at the same confidence level. For a test about a proportion rather than a mean, the right tool is a one-proportion z-test calculator, which also works on the normal distribution.

How degrees of freedom change the t-distribution

Degrees of freedom set the shape of the t curve. At df = 1 the distribution is very flat with extremely heavy tails; at df = 5 it is visibly wider than the normal curve; by df = 30 the two are hard to tell apart by eye; and at df = 100 the curves almost coincide. That is why the critical t values below shrink toward 1.960 as df grows — the t-distribution is converging on the standard normal.

The calculator above reproduces every value in this table: switch to Critical t mode, keep 95% two-tailed, and type in the df. The table itself is taken from the NIST/SEMATECH e-Handbook of Statistical Methods, which publishes the critical values of the Student’s t-distribution.

Worked examples

P-value from a t value

Example 1. t = 2.0, df = 10, two-tailed. The area in both tails beyond ±2.0 is 0.0734, so the p-value is 0.0734 — not significant at the 5% level, but close. The CDF is P(T ≤ 2.0) = 0.9633, so the one-tailed right probability is 1 − 0.9633 = 0.0367.

Example 2. t = 1.0, df = 5, right-tailed. P(T > 1.0) = 0.1816. Because the distribution is symmetric, the left-tailed area P(T < −1.0) is the same, 0.1816.

Example 3 (symmetry check). For any df, P(T < 0) = 0.500 exactly: the t-distribution is symmetric about zero, so half its area lies on each side. Enter t = 0 in the calculator and the CDF reads 0.5000 for every df.

Critical t from a confidence level

Example 4. 95% confidence, two-tailed, df = 20: t* = 2.086. A confidence interval built with this value is a little wider than one built with z = 1.960, because the sample standard deviation adds uncertainty.

Example 5. 95% confidence, one-tailed, df = 15: t* = 1.753. One-tailed cut-offs are smaller because all of α sits in one tail.

Example 6 (convergence). 95% confidence, two-tailed, df = 100: t* = 1.984, only 0.024 above the normal value of 1.960. As df grows further, t* keeps approaching 1.960 — the t and normal distributions become indistinguishable for practical purposes once df passes about 100.

Reading a p-value always raises the question of what a p-value is, and choosing between 95% and 99% confidence is a question of margin of error vs confidence interval — a higher confidence level means a wider interval and a larger margin of error.

Frequently asked questions

Source: the critical values quoted on this page are from the NIST/SEMATECH e-Handbook of Statistical Methods, Critical Values of the Student’s t Distribution. All examples on this page were checked against that table.

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