How to use the t, chi-square and f distribution calculator
- Choose the distribution: Student t, chi-square or F.
- Enter the degrees of freedom. The F distribution needs two numbers, one for the numerator and one for the denominator.
- Choose Value for probability to find a critical value, or Probability at a value to find the area below a value you already have.
- Read the value, the probability below and above it, the density at that point and, for the t and F cases, the two sided tail probability.
Formula
Student t with v degrees of freedom: density = Gamma((v+1)/2) / (sqrt(v x pi) x Gamma(v/2)) x (1 + x^2/v)^(-(v+1)/2)
Chi-square with k degrees of freedom: density = x^(k/2-1) x e^(-x/2) / (2^(k/2) x Gamma(k/2))
F with d1 and d2 degrees of freedom: probability below x = I(d1 x / (d1 x + d2); d1/2, d2/2), the regularized incomplete beta function
A value for a probability is found by solving the cumulative probability for x with a bracketing search
The cumulative probabilities come from the regularized incomplete beta and gamma functions, which are evaluated with continued fractions and series. The search for a value stops when the probability matches to well beyond the digits shown.
The 97.5th percentile of t with 10 degrees of freedom
Choose Student t, find the value for a probability, enter 10 degrees of freedom and a probability of 0.975. The page returns 2.22814. That is the familiar critical value for a two sided test at the 5 percent level with 10 degrees of freedom, because 2.5 percent is left in each tail. The density at that point is 0.042385 and the two sided tail probability is 0.05.
For chi-square with 10 degrees of freedom the 0.95 point is 18.307, and for F with 5 and 10 degrees of freedom the 0.95 point is 3.32583. Going the other way, a t value of 2.1 with 10 degrees of freedom has 0.96896138 of the distribution below it, which leaves 0.031038622 above and a two sided p value of 0.062077244.
Which distribution to pick
The t distribution describes a sample mean standardized with an estimated standard deviation, so it is used for small sample tests and intervals for a mean. The chi-square distribution describes a sum of squared standard normal values, which is why it appears in tests of counts and in the variance of a sample. The F distribution is a ratio of two scaled chi-square values and appears in analysis of variance and in comparing two variances.
If you have a test statistic and want its p value, the p value calculator works from the statistic directly. For a table of many critical values at once, use the statistical tables generator.
Reading the tails
A one sided test uses the probability above the value. A two sided test for a t statistic doubles the tail beyond the absolute value, which is the two sided tail probability on the page. Chi-square and F tests in analysis of variance and in counts are normally upper tail only, so read the probability above the value.
For a worked test on counts see the chi-square calculator, and for comparing several means see the ANOVA calculator.
Assumptions and limits
- Degrees of freedom are positive numbers. Whole numbers are the normal case, and fractional values are accepted for cases such as Welch tests.
- A probability for a value search must be strictly between 0 and 1.
- Results are the exact distribution functions, not normal approximations.
- The page does not pick a test or a significance level for you. You choose them.
Frequently asked questions
How do I find a t critical value?
Choose Student t, find the value for a probability, and enter the degrees of freedom and one minus half of your significance level. For a 5 percent two sided test with 10 degrees of freedom enter 0.975 and read 2.22814.
What is the difference between this and a statistics table?
A table lists a few rounded values. This page solves for any probability and any degrees of freedom, and it gives the density and both tails.
Why does F need two degrees of freedom?
An F value is the ratio of two independent variance estimates, and each has its own degrees of freedom. The first is for the numerator and the second is for the denominator.
Can I enter a fractional degrees of freedom?
Yes. Welch tests and some pooled procedures produce fractional values, and the distribution functions are defined for them.