tdistribution

The t-distribution, also known as the Student's t-distribution, is a family of symmetric probability distributions that arise when estimating the mean of a normally distributed population in situations where the population standard deviation is unknown. The family is indexed by the degrees of freedom ν (nu). It has heavier tails than the standard normal distribution, which makes it more appropriate for small samples. As the degrees of freedom increase, the t-distribution converges to the standard normal distribution.

The probability density function of the t-distribution with ν degrees of freedom is

f(t; ν) = Γ((ν+1)/2) / (√(νπ) Γ(ν/2)) · [1 + t²/ν]^(-(ν+1)/2),

for all real t. The distribution is symmetric about zero.

Key moments and properties include: the mean is 0 for ν > 1; the variance is ν/(ν−2) for ν > 2 (and is infinite or undefined for smaller ν). The tails are heavier than those of the normal distribution, which reflects increased uncertainty when the population standard deviation is unknown and the sample size is small. As ν → ∞, the t-distribution tends to the standard normal distribution.

Uses and interpretation: the t-distribution is central to t-tests (one-sample, paired, and two-sample) and to confidence intervals for a population mean when σ is unknown and the sample size is small. The test statistic T = (X̄ − μ) / (S/√n) follows a t-distribution with ν = n−1 under the null hypothesis. In practice, larger ν yields results closer to those from the normal distribution.