t-Distribution Calculator

Enter your degrees of freedom and a t-score, then choose a probability type (left-tail, right-tail, or two-tailed) to get the corresponding p-value from the Student's t-distribution. You can also work in reverse — enter a probability to find the critical t-value. Perfect for hypothesis testing, confidence intervals, and statistical inference. Also try the Normal Distribution Calculator.

Results

p-value

Left-Tail P(X ≤ t)

Right-Tail P(X ≥ t)

Two-Tailed P(|X| ≥ |t|)

Between P(−|t| ≤ X ≤ |t|)

What is a t-score?

A t-score (or t-statistic) measures how many standard errors a sample mean is from the population mean. It is calculated as t = (x̄ − μ) / (s / √n), where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. Larger absolute t-scores indicate greater evidence against the null hypothesis. See also our calculate Probability Density Function PDF Value f(x).

What are degrees of freedom?

Degrees of freedom (ν or df) represent the number of independent values that can vary in a statistical calculation. For a one-sample t-test, df = n − 1, where n is the sample size. For a two-sample t-test, the formula is more complex. Degrees of freedom determine the exact shape of the t-distribution — as df increases, the distribution approaches the standard normal distribution.

What is the difference between a left-tail, right-tail, and two-tailed probability?

A left-tail probability P(X ≤ t) gives the chance of observing a t-score at or below your value. A right-tail probability P(X ≥ t) gives the chance of observing a t-score at or above your value. A two-tailed probability P(|X| ≥ |t|) combines both extremes and is used when your alternative hypothesis is non-directional (simply 'not equal to').

What is a p-value?

A p-value is the probability of observing a test statistic as extreme as (or more extreme than) your result, assuming the null hypothesis is true. A small p-value (typically below 0.05) suggests the result is statistically significant and provides evidence to reject the null hypothesis. The p-value alone does not measure effect size or practical importance. You might also find our find Result with Standard Normal Distribution Calculator useful.

How does the t-distribution differ from the normal distribution?

The t-distribution has heavier tails than the standard normal distribution, which accounts for the additional uncertainty that comes from estimating the population standard deviation from a small sample. As the degrees of freedom increase, the t-distribution converges toward the standard normal (Z) distribution. For df greater than 30, the two are very similar in practice.

When should I use the t-distribution instead of the normal distribution?

Use the t-distribution when your sample size is small (typically n < 30) and the population standard deviation is unknown — which is almost always the case in practice. Use the normal distribution only when the population standard deviation is known or when the sample size is very large, making the t-distribution essentially identical to the normal.

How do I calculate degrees of freedom for a two-sample t-test?

For a two-sample t-test with equal variances, df = n₁ + n₂ − 2. For a Welch's t-test (unequal variances), the degrees of freedom are estimated using the Welch–Satterthwaite equation: df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]. Most statistical software computes this automatically.

What is a standard deviation in the context of the t-distribution?

In this context, the standard deviation (s) is the sample standard deviation — a measure of how spread out the data values are around the sample mean. It is used in the denominator of the t-statistic formula along with the square root of the sample size to form the standard error. A larger standard deviation relative to the sample size results in a smaller t-score.