t-Distribution Percent Point (Critical Value)

Find the percent point (critical value t) of the Student t-distribution from a probability p and degrees of freedom. Supports lower, upper, and two-sided tails for building t-test and confidence interval tables.

Input

Enter a probability p, a tail mode, and degrees of freedom nu to compute the percent point (critical value t) of the Student t-distribution.

Tail mode

The t value whose lower probability equals p, that is the probability that T is at most t equals p.

Probability p

A value strictly between 0 and 1. Example: 0.975

Degrees of freedom nu

A positive value, need not be an integer. Example: 10

Result

t value for Lower, p=0.975, df nu=10

2.22813885

Degrees of freedom nu

Probability p

0.975

Lower cumulative P(T ≤ t)

0.975

Upper probability P(T greater than t)

0.025

Density f(t)

0.04238498

Mean

Variance

1.25

Probability density function PDF

Cumulative distribution function CDF

How it works

The cumulative distribution function of the Student t-distribution can be written with the regularized incomplete beta function I_x(a,b). For degrees of freedom nu, let x = nu/(nu + t^2). When t is at least 0, P(T ≤ t) = 1 - 0.5 I_x(nu/2, 1/2); when t is below 0, P(T ≤ t) = 0.5 I_x(nu/2, 1/2).
The percent point is found by forming the target lower probability for the chosen mode and inverting the monotonically increasing CDF by bisection. Lower solves P(T ≤ t)=p, upper solves P(T ≤ t)=1-p, and two-sided solves P(T ≤ t)=(1+p)/2.
The probability density is f(t) = Gamma((nu+1)/2) / (sqrt(nu pi) Gamma(nu/2)) times (1 + t^2/nu)^(-(nu+1)/2). The mean is 0 when degrees of freedom exceed 1, and the variance is nu/(nu-2) when degrees of freedom exceed 2.
The log gamma function uses the Lanczos approximation and the regularized incomplete beta function uses a Lentz continued fraction. Enter a probability p strictly between 0 and 1, and a positive value for the degrees of freedom nu.

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