Student t Distribution CDF Calculator

Enter any finite t statistic and positive degrees of freedom to obtain a deterministic cumulative probability between zero and one. This is useful for checking test statistics, translating t values into tail areas, validating statistical software, and exploring how the heavier tails of the t distribution change as degrees of freedom increase.

What the cumulative probability means

A cumulative distribution function answers a precise question: if a random variable follows the selected distribution, what probability lies at or below a particular value? Here, the selected distribution is Student's t distribution, the boundary is the supplied t statistic, and the result is a number from zero through one. A negative t value produces a probability below one half, zero produces exactly one half, and a positive t value produces a probability above one half. The distribution is symmetric, so results at opposite t values complement one another. This calculator evaluates a single point rather than conducting a complete hypothesis test. It does not decide whether an effect is important, select a significance threshold, or determine whether a one-sided or two-sided test is appropriate. Those choices belong to the study design. Use the returned cumulative probability as a reliable building block for those later calculations, or as a direct check of a probability reported by statistical software, a textbook, or an analysis pipeline.

Choose t and degrees of freedom correctly

Supply t as a finite numeric statistic, including its sign. Do not enter an absolute value unless your intended calculation specifically discards direction, because the sign determines which side of the distribution the cumulative probability describes. Degrees of freedom control the shape of the Student t distribution and must be positive. In a familiar one-sample t test, degrees of freedom are commonly the sample size minus one, but other models use different formulas. Regression, paired tests, unequal-variance comparisons, and more elaborate designs can each derive degrees of freedom differently, including non-integer values. Take the value from the statistical method you are actually using rather than guessing it from the number of observations. Smaller degrees of freedom produce heavier tails, while larger values make the distribution approach the standard normal distribution. The calculator accepts positive fractional degrees of freedom because valid approximations, such as the Welch procedure, can yield them. It rejects zero, negative, missing, infinite, and nonnumeric degrees of freedom instead of silently returning a misleading result.

Turn the CDF into tail probabilities

The output is the lower-tail probability P(T ≤ t). For a right-tail probability at the same t value, subtract the returned result from one. For a left-tailed test, the CDF itself is normally the relevant tail area. For a two-sided test based on a symmetric t distribution, compute twice the smaller of the lower-tail probability and its complement. This produces the probability associated with outcomes at least as far from zero in either direction, but it should only be called a p-value when the assumptions and test definition justify that interpretation. Floating-point arithmetic can make probabilities in extremely distant tails round to zero or one, so treat those endpoints as values beyond ordinary machine precision rather than proof that an event is logically impossible. The implementation uses the regularized incomplete beta function and a bounded continued fraction, with no network requests, random values, or current-time dependency. Browser and API executions therefore apply the same deterministic calculation. Automated API requests cost $0.002 each, while the browser calculation is available directly on the page.