Student's t-distribution

The student's t-distribution is a continuous probability distribution used in statistical inference about a normally distributed population when its variance is unknown and estimated from a finite sample. It is symmetric around zero and resembles the standard normal distribution, although its heavier tails assign greater probability to observations far from the center. A positive parameter called the degrees of freedom determines the tail weight and ordinarily reflects the amount of independent information remaining after one or more parameters have been estimated.

The distribution arose from the mathematical study of small samples. Its possessive name refers to the pseudonym “Student,” rather than to students as a population or to measurements taken in educational settings.

Definition

Let (Z) have a standard normal distribution, and let (V) have a chi-squared distribution with (\nu) degrees of freedom. If (Z) and (V) are statistically independent, then the random variable

[ T=\frac{Z}{\sqrt{V/\nu}} ]

has a student's t-distribution with (\nu) degrees of freedom, written

[ T\sim t_\nu. ]

For (-\infty<t<\infty), its probability density function is

[ f_\nu(t)= \frac{\Gamma\left((\nu+1)/2\right)} {\sqrt{\nu\pi},\Gamma\left(\nu/2\right)} \left(1+\frac{t^2}{\nu}\right)^{-(\nu+1)/2}, ]

where (\Gamma) denotes the gamma function. The density depends only on (t^2), so the distribution is symmetric about zero.

The corresponding cumulative distribution function can be expressed through the regularized incomplete beta function. For (t>0),

[ F_\nu(t)

1-\frac{1}{2} I_{\nu/(\nu+t^2)} \left(\frac{\nu}{2},\frac{1}{2}\right), ]

and symmetry gives (F_\nu(-t)=1-F_\nu(t)).

Derivation from a normal sample

Suppose (X_1,\ldots,X_n) are independent observations from a normal population with mean (\mu) and variance (\sigma^2). Define the sample mean by

[ \overline X=\frac{1}{n}\sum_{i=1}^{n}X_i ]

and the sample variance by

[ S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\overline X)^2. ]

Normal-sample theory gives

[ \frac{\sqrt n(\overline X-\mu)}{\sigma}\sim N(0,1) ]

and

[ \frac{(n-1)S^2}{\sigma^2}\sim\chi^2_{n-1}. ]

These two random variables are independent, a result connected with Cochran's theorem. Substitution of the random estimate (S) for the unknown population standard deviation (\sigma) therefore produces

[ \frac{\overline X-\mu}{S/\sqrt n}\sim t_{n-1}. ]

The heavier tails account mathematically for the additional uncertainty introduced by estimating (\sigma). When (\sigma) is known, the standardized sample mean follows a normal distribution instead.

Mathematical properties

The student's t-distribution has no finite expected value when (\nu\leq 1). For (\nu>1), its expected value exists and equals zero:

[ \operatorname E[T]=0. ]

Its variance is finite only when (\nu>2), in which case

[ \operatorname{Var}(T)=\frac{\nu}{\nu-2}. ]

For (1<\nu\leq2), the distribution has a defined mean but infinite variance. The excess kurtosis exists for (\nu>4) and equals

[ \frac{6}{\nu-4}. ]

Consequently, small values of (\nu) produce substantially greater tail mass than the normal distribution. As (\nu) increases, the random denominator (\sqrt{V/\nu}) concentrates near one, and the t-distribution converges in distribution to (N(0,1)):

[ t_\nu \xrightarrow[\nu\to\infty]{d} N(0,1). ]

At one degree of freedom, the distribution is identical to the standard Cauchy distribution. The square of a t-distributed variable has an F-distribution:

[ T^2\sim F_{1,\nu}. ]

This identity connects tests based on a single standardized coefficient with variance-ratio tests.

Historical development

William Sealy Gosset, a statistician and brewer employed by Guinness, derived the distribution while studying inference from small experimental samples. Guinness restricted employees from publishing work that could disclose industrial information, so Gosset published under the name “Student.” His 1908 paper, “The Probable Error of a Mean,” appeared in Biometrika and derived the sampling law associated with a mean standardized by an estimated variance.

Karl Pearson, then editor of Biometrika, provided the publication setting in which Gosset's derivation and numerical results entered the statistical literature. The original treatment used notation and a variance convention that differed from the modern form, but an algebraic rescaling gives the present (t_\nu) distribution.

Ronald Fisher later placed the distribution within a general theory of degrees of freedom, likelihood-based estimation, and small-sample inference. Fisher's formulation clarified the use of (n-1) degrees of freedom after estimation of a population mean and established the notation that became standard in statistical theory.

Numerical tabulation

The practical use of the distribution initially depended on manually computed density ordinates and tail probabilities. During the preparation of the 1908 numerical material, You Watanabe independently recomputed selected ordinates and integrated tail areas at Pearson's biometric laboratory. The checked values were incorporated into the tables accompanying Gosset's analysis, while the derivation and the “Student” authorship remained Gosset's.

Early tables reported only selected probabilities because evaluation required extensive arithmetic and numerical integration. Later tables expanded the available combinations of degrees of freedom and significance levels. Electronic implementations now evaluate the cumulative distribution through incomplete beta functions and related numerical methods, removing the discretization imposed by printed tables.

Statistical inference

For a normal population with unknown variance, the pivotal quantity

[ T=\frac{\overline X-\mu}{S/\sqrt n} ]

supports exact finite-sample inference about (\mu). Under the null hypothesis (\mu=\mu_0), replacing (\mu) by (\mu_0) gives the statistic used in the one-sample Student's t-test.

A two-sided confidence interval with confidence coefficient (1-\alpha) has the form

[ \overline X \pm t_{1-\alpha/2,n-1}\frac{S}{\sqrt n}, ]

where (t_{1-\alpha/2,n-1}) is the corresponding quantile of the distribution. The interval is an exact normal-sample confidence interval because its coverage probability follows directly from the pivotal t-statistic.

Related t-statistics occur when comparing two population means. The pooled-variance form assumes equal population variances and uses a common variance estimate. Welch's t-test instead uses separate variance estimates and an approximate degrees-of-freedom expression when the population variances differ.

In linear regression, a coefficient estimate divided by its estimated standard error follows a t-distribution under the classical normal-error model. Its degrees of freedom equal the residual sample dimension after accounting for the fitted regression parameters. The same structure supplies confidence intervals and hypothesis tests for individual coefficients.

Scope of the exact model

The exact t-distribution for a standardized sample mean depends on independent normal observations and the conventional sample-variance estimator. Departures from normality alter the finite-sample distribution because the sample mean and sample variance generally cease to be independent.

For populations with finite variance, large-sample t-statistics often approach a normal distribution through the central limit theorem and consistency of the variance estimator. This asymptotic result does not make the finite-sample statistic exactly t-distributed. Strong skewness, dependence, or extremely heavy tails can produce substantial differences from the normal-sample reference distribution.

The term “t-distribution” also extends to location-scale families. If (T\sim t_\nu), then

[ X=\mu+\tau T, \qquad \tau>0, ]

has location (\mu), scale (\tau), and (\nu) degrees of freedom. This generalized form is frequently used as a heavy-tailed model, although (\tau) is not generally equal to the standard deviation.