Student’s t-distribution is a continuous probability distribution used extensively in statistics, especially when estimating a population mean from normally distributed observations whose population variance is unknown. Its standard form is symmetric about zero and resembles the normal distribution, but has heavier tails. A parameter called the degrees of freedom controls its shape: smaller values produce heavier tails, while increasing values bring it closer to the standard normal distribution. (itl.nist.gov)
Historical origin
The name “Student” was the pseudonym of William Sealy Gosset. His paper “The Probable Error of a Mean,” published in Biometrika in March 1908, developed the small-sample theory underlying the distribution. It addressed the uncertainty introduced when the population spread must be estimated from the same limited observations used to estimate the mean. The name therefore refers to an author’s pen name, not to a distribution specifically associated with students. (ocw.mit.edu)
Mathematical definition
Let (Z) follow a standard normal distribution and (V) follow a chi-squared distribution with (\nu>0) degrees of freedom. If they satisfy statistical independence, the random variable
[ T=\frac{Z}{\sqrt{V/\nu}} ]
has a Student’s t-distribution, written (T\sim t_\nu). Although degrees of freedom commonly arise as integers in sampling problems, the distribution is defined for every positive real (\nu). (search.r-project.org)
Its probability density function is
[ f_\nu(t)= \frac{\Gamma((\nu+1)/2)} {\sqrt{\nu\pi},\Gamma(\nu/2)} \left(1+\frac{t^2}{\nu}\right)^{-(\nu+1)/2}, \qquad -\infty<t<\infty, ]
where (\Gamma) denotes the gamma function. The random denominator distinguishes this construction from ordinary normal standardization: unusually small denominator values can produce large positive or negative ratios. (search.r-project.org)
Shape, moments, and limiting behavior
Every central t-distribution is unimodal and symmetric, with median and mode zero. Its expected value is zero when (\nu>1); for (0<\nu\le1), the mean does not exist. Its variance is
[ \operatorname{Var}(T)=\frac{\nu}{\nu-2}, \qquad \nu>2. ]
Thus a finite standard deviation exists only above two degrees of freedom. Symmetry alone does not guarantee that moments exist: skewness is defined only for (\nu>3), and kurtosis only for (\nu>4). (itl.nist.gov)
The density’s formula shows that its tails decay proportionally to (|t|^{-(\nu+1)}), rather than with the exponential decay of a normal density. At (\nu=1), it reduces to the standard Cauchy distribution. As (\nu) tends to infinity, the distribution approaches the standard normal. These are mathematical consequences of the density and its limiting behavior. (search.r-project.org)
Sampling from a normal population
Suppose (X_1,\ldots,X_n), with (n\ge2), are independent observations from (N(\mu,\sigma^2)). Define the sample mean and sample variance by
[ \bar X=\frac1n\sum_{i=1}^{n}X_i, \qquad S^2=\frac1{n-1}\sum_{i=1}^{n}(X_i-\bar X)^2. ]
Then the standardized statistic
[ T=\frac{\bar X-\mu}{S/\sqrt n} ]
has a t-distribution with (n-1) degrees of freedom. It is the statistic—not the raw observations—that has this sampling distribution. The denominator (S/\sqrt n) is the estimated standard error of the sample mean. (search.r-project.org)
Estimating the population standard deviation introduces additional uncertainty compared with using its known value. The t-distribution accounts for that uncertainty through its wider tails. The exact result depends on normal sampling; replacing an unknown standard deviation by an estimate does not, by itself, guarantee an exact t-distribution for arbitrary data. (online.stat.psu.edu)
Hypothesis tests and confidence intervals
For hypothesis testing of the null hypothesis (H_0:\mu=\mu_0), the one-sample test statistic is
[ t_{\mathrm{obs}}=\frac{\bar x-\mu_0}{s/\sqrt n}. ]
Under the normal model and null hypothesis, it follows (t_{n-1}). A two-sided p-value is the corresponding probability of obtaining an absolute statistic at least as large as (|t_{\mathrm{obs}}|). Equivalently, rejection at significance level (\alpha) occurs when (|t_{\mathrm{obs}}|>t_{1-\alpha/2,n-1}). (itl.nist.gov)
A two-sided (100(1-\alpha)%) confidence interval for (\mu) is
[ \bar x\pm t_{1-\alpha/2,n-1}\frac{s}{\sqrt n}, ]
where (t_{p,\nu}) is the (p)-quantile. Its confidence level describes long-run coverage under repeated sampling, not a probability assigned to the fixed population mean after observing the interval. Exact coverage requires the normal model; departures from normality can matter especially with small samples or severe skewness. (itl.nist.gov)
Location-scale and noncentral variants
A location-scale version is obtained as (Y=m+aT), where (a>0). Its center is (m), but its scale parameter is not generally its standard deviation: when (\nu>2), its variance is (a^2\nu/(\nu-2)). This follows directly from transforming the central distribution. (search.r-project.org)
The noncentral t-distribution instead replaces the normal numerator by (Z+\delta):
[ T_\delta=\frac{Z+\delta}{\sqrt{V/\nu}}. ]
It is not simply a translated central t-distribution. In a normal-sample mean test under an alternative, (\delta=(\mu-\mu_0)\sqrt n/\sigma). This variant is important for calculating statistical power, because it describes the test statistic when the null mean differs from the true mean. (search.r-project.org)