T Distribution Calculator

Common questions

How do I calculate a p value from a known t statistic?

Select Probability and enter the t statistic and degrees of freedom from your test. Read as selects Lower, Upper, or Two-sided. At 10 degrees of freedom and t = 2.228139, the upper tail is 0.025000 and the two-sided p value is 0.050000. This converts an existing statistic; it does not calculate a t statistic from sample data or perform the test for you.

Is my p value one-sided or two-tailed?

Lower means P(T <= t), the area to the left of t. Upper means P(T >= t), the area to the right. Two-sided, also called two-tailed, means P(|T| >= |t|), both tails beyond the magnitude of t. That is twice the upper tail at |t|, including when the entered t is negative.

How do I find a percentile or inverse t value?

Select Critical value, choose Lower, and enter the cumulative probability as a decimal or percentage. A 95th percentile uses 0.95 or 95%. At 10 degrees of freedom it gives t = 1.812461. Upper instead takes the area to the right of the cutoff, while Two-sided takes the combined area in both outer tails.

What is the critical t value for a 95% confidence interval?

Select Critical value, choose Two-sided, and enter 0.05, the area outside a 95% interval. At 10 degrees of freedom, the positive cutoff is 2.228139 and the negative cutoff has the opposite sign. This finds the critical multiplier; it does not calculate confidence interval endpoints from a sample mean, standard deviation, and sample size.

What degrees of freedom can I enter?

The calculator accepts whole or fractional degrees of freedom from 0.0001 through 1000000. For a one-sample t statistic, degrees of freedom are n - 1. For a pooled two-sample statistic they are n1 + n2 - 2. A Welch test can supply a fractional value such as 7.5. Enter the degrees of freedom already calculated for your statistic; this page does not derive them from sample inputs.

What do the graph, CDF, and density show?

The CDF is the lower cumulative probability, the area to the left of t. The density is the height f(t), not a probability at that exact point. The graph shades the selected lower, upper, or two-sided area. It displays at most -12 to 12; an off-scale cutoff is described in the graph label without a shaded region. It does not accept two separate interval bounds.

How do I read the t distribution table?

Each table column gives a positive critical t value for the stated one-sided tail probability. Double that probability to get the combined two-sided tail area. For a degree of freedom or probability absent from the table, the Critical value controls calculate it directly.

How accurate are the displayed probabilities and quantiles?

The numerical accuracy was measured against a 120-digit reference over 294 cases, with degrees of freedom from 0.5 to 1000000. Measured relative errors were within 1e-14 for tails, densities, and ordinary quantiles, and within 1e-13 for the deepest quantiles. Displayed results are rounded; this does not establish the same error bound for every possible input.

Can a very small tail probability still be shown?

Yes. Probabilities smaller than a double can hold, about 5e-324, are displayed from their logarithms in scientific notation. At 10 degrees of freedom, t = 1e40 still has a nonzero tail displayed. Inverse inputs must be representable and strictly between 0 and 1; an unrepresentable quantile or two-sided half-tail produces an error.

Can I copy the result?

The Copy button copies the answer and working, including full-precision numbers, to the clipboard when the browser permits it. Each case also has a Copy button. If clipboard access is blocked, a message explains how to copy the displayed text by hand. There is no file export or PDF download in this calculator.