t-Distribution Calculator (PDF, CDF, p-value)

Compute the Student's t-distribution probability density (PDF), lower cumulative probability (CDF), upper tail, and two-sided probability (p-value) from a t value and degrees of freedom, with PDF and CDF graphs.

Input

Enter a t value and the degrees of freedom nu to compute the Student''s t-distribution probability density, cumulative probability, upper tail, and two-sided probability (p-value).

t value

The t value, such as a test statistic. Negative values are allowed.

Degrees of freedom (nu)

Enter a positive value. Larger values approach the normal distribution.

Result

Two-sided probability (p-value) at t = 2, nu = 10

0.07338803

Probability density (PDF)

0.06114577

Lower cumulative probability (CDF)

0.96330598

Upper tail probability

0.03669402

Mean

Variance

1.25

Probability density function (PDF) graph

Cumulative distribution function (CDF) graph

How it works

The t-distribution density is f(t) = Gamma((nu+1)/2) divided by ( sqrt(nu pi) Gamma(nu/2) ) times (1 + t squared over nu) raised to -(nu+1)/2, where nu is the degrees of freedom.
The cumulative distribution (lower tail) uses the regularized incomplete beta function I_x(a, b) with x = nu divided by (nu + t squared). When t is at least 0 it equals 1 minus I_x(nu/2, 1/2) over 2, and when t is below 0 it equals I_x(nu/2, 1/2) over 2.
The upper tail probability is 1 minus the lower tail, and the two-sided probability (p-value) sums both tails as P(absolute T is at least absolute t).
The mean is 0 when nu is greater than 1, and the variance is nu divided by (nu minus 2) when nu is greater than 2. Otherwise they are undefined.
As the degrees of freedom nu increase, the t-distribution approaches the standard normal distribution.
Calculations use a continued-fraction expansion of the regularized incomplete beta function and a Lanczos approximation of the gamma function, so the precision follows the formulas directly without lookup tables.

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