Computes the complement of the cdf of the Student’s t distribution.


p = cdfTc(x, df)
  • x (NxK matrix) – values at which to evaluate the cumulative distribution function for the Student’s t distribution. \(−\infty \leq x \leq \infty\).
  • df (LxM matrix) – ExE conformable with x. Degrees of freedom. \(df > 1\).

p (matrix, max(N,L) by max(K,M)) – Each element in p is the complement of the cumulative distribution function of the Student’s t distribution evaluated at the corresponding element in x.


// Values
x = { .1, .2, .3, .4 };

// Degrees of freedom
df = 3;

p = cdfTc(x, df);

After running above code,

p =


A -1 is returned for those elements with invalid inputs.

This equals:

\[1 − F(x,df)\]

where F is the t cdf with df degrees of freedom. Thus, to get the t cdf, subtract cdfTc(x, df) from 1. The complement of the cdf is computed because this is what is most commonly needed in statistical applications, and because it can be computed with fewer problems of roundoff error.

Technical Notes

For results greater than 0.5e-30, the absolute error is approx. ±1e-14 and the relative error is approx. ±1e-12. If you multiply the relative error by the result, then take the minimum of that and the absolute error, you have the maximum actual error for any result. Thus, the actual error is approx. ±1e-14 for results greater than 0.01. For results less than 0.01, the actual error will be less. For example, for a result of 0.5e-30, the actual error is only ±0.5e-42.


  1. Abramowitz, M. and I.A. Stegun, eds. Handbook of Mathematical Functions. 7th ed. Dover, New York, 1970. ISBN 0-486-61272-4.
  2. Hill, G.W. ‘’Algorithm 395 Student’s t-Distribution.’’ Comm. ACM. Vol. 13, No. 10, Oct. 1970.
  3. Hill, G.W. ‘’Reference Table: Student’s t-Distribution Quantiles to 20D.’’ Division of Mathematical Statistics Technical Paper No. 35. Commonwealth Scientific and Industrial Research Organization, Australia, 1972.

See also

Functions cdfTci()