t distribution

Let X and Y be random variablesMathworldPlanetmath such that

  1. 1.

    X and Y are independentPlanetmathPlanetmath

  2. 2.

    XN(0,1), the standard normal distributionMathworldPlanetmath (with mean 0 and varianceMathworldPlanetmath 1)

  3. 3.

    Yχ2(n), the chi-square distributionDlmfPlanetmathPlanetmath with n degrees of freedom

Define a new random variable Z by


Then the distribution of Z is called the t distribution with n degrees of freedom, denoted by Zt(n).

By transformation of the random variables X and Y, one can show that the probability density function of the t distribution of Z has the form:


where B(α,β) is the beta functionDlmfDlmfMathworldPlanetmath.

Below are graphs of some t probability density functions for various degrees (d) of freedom.

d=1, d=2, d=3, d=500


  • the t distribution is also known as the Student’s t distribution. The name Student came from the 19th Century research chemist William Sealy Gossett, who was employed by the brewing company Guinness to improve the yield of crops used to produce its beer. Gossett conducted agricultural experiments and used random numbers to help determine the sampling distribution of the data he collected. Because the brewing company wanted to keep the research results confidential for competitve reasons, Gossett had to use a pen name to publish his findings. Student was his pen name and the distribution he found turned out to be the t distribution.

  • t(1)=Cauchy(0,1), the Cauchy distributionMathworldPlanetmath with parametersMathworldPlanetmath 0 and 1.

  • t(n) is asymptotically (as n) N(0,1), the standard normal distribution with mean 0 and variance 1.

  • If Xt(n), E[|X|k] exists iff k<n. Therefore, a t distribution has no mean if it has one degree of freedom. For n>1, E[X]=0. For n>2, Var[X]=nn-2.

  • If X1,,Xn are random samples from a normal distribution with mean μ and variance σ2. Let μ^ be the sample meanMathworldPlanetmath and σ^2 the sample variance, then


    Please note that the statisticMathworldMathworld U does not depend on σ2, and thus is used often in testing hypotheses involving comparison of the sample mean to the true mean, given a set of random samples that are normally distributed with an unknown mean and an unknown variance. This is an example of a t test.

Title t distribution
Canonical name TDistribution
Date of creation 2013-03-22 14:26:50
Last modified on 2013-03-22 14:26:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 34
Author CWoo (3771)
Entry type Definition
Classification msc 62A01
Synonym Student’s t
Synonym t-distribution