Tfordelingen

Tfordelingen, commonly known as the t-distribution or Student's t-distribution, is a probability distribution used in statistics to make inferences about a population mean when the sample size is small and the population variance is unknown. It was introduced by William Sealy Gosset under the pseudonym Student in 1908. The distribution depends on a parameter called degrees of freedom, denoted ν (nu).

Definition and formula: If Z is a standard normal variable and U is a chi-square variable with ν degrees of freedom, and Z and U are independent, then the random variable T = Z / sqrt(U/ν) follows a t-distribution with ν degrees of freedom. Its probability density function is f(t) = Γ((ν+1)/2) / [√(νπ) Γ(ν/2)] × (1 + t²/ν)^(-(ν+1)/2). The shape is determined by ν, with larger ν producing a distribution closer to the standard normal.

Properties: The t-distribution is symmetric around zero and has heavier tails than the standard normal distribution, especially for small ν. As ν increases, the t-distribution converges to N(0,1). The mean is 0 for ν > 1; the variance is ν/(ν−2) for ν > 2. With small ν, tail probabilities are larger, reflecting greater uncertainty in the estimate of the standard deviation from the sample.

Applications: It is widely used in hypothesis testing and confidence interval estimation for a population mean when the population variance is unknown. Test statistics in one-sample and two-sample t-tests follow a t-distribution with appropriate degrees of freedom. Degrees of freedom depend on the sampling scenario: for a single sample, df = n−1; for two-sample tests with equal variances, df = n1 + n2 − 2; Welch's method uses an approximate df to handle unequal variances.