Student's t-test

Student's t-test calibrates a standardized mean contrast using a Student's t-distribution under a specified sampling model. For independent $X_1,\dots ,X_n\sim N(\mu ,{\sigma }^2)$ with $n\ge 2$ and unknown positive variance, the one-sample statistic isUnder the null hypothesis $\mu ={\mu }_0$, $T\sim t_{n-1}$ exactly.

The exact law uses two facts: $Z=\sqrt{n}(\bar{X}-{\mu }_0)/\sigma$ is standard normal, and $V=(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$ independently of $Z$. Hence $T=Z/\sqrt{V/(n-1)}$, the defining construction of the Student's t-distribution. A normally distributed numerator divided by an arbitrary estimated standard error need not have this law.

For a two-sided level-$\alpha$ test, reject when $|T|>t_{n-1,1-\alpha /2}$; a directional alternative uses the corresponding one-sided quantile. The paired t-test applies the one-sample construction to independent pairs' differences. The exact pooled two-sample t statistic uses independent normal samples with a common variance; Welch's t-test instead estimates unequal variances and uses an approximate degrees-of-freedom calibration. The selected model and statistical degrees of freedom determine which reference distribution is justified.

For one two-sided t procedure with observed estimate $\hat{\Delta }$, positive standard error $s$, null value ${\Delta }_0$ and reference Student t-distribution with $\nu$ statistical degrees of freedom, the test statistic and p-value must satisfyIts symmetric confidence interval at coverage probability $1-\alpha$ is $[\hat{\Delta }-t_{\nu ,1-\alpha /2}s,\hat{\Delta }+t_{\nu ,1-\alpha /2}s]$ and therefore has midpoint $\hat{\Delta }$. When the same standard error and reference distribution are used, excluding ${\Delta }_0$ is equivalent to $|t|>t_{\nu ,1-\alpha /2}$ and hence $p<\alpha$ for a continuous reference distribution. This proves the test–interval equivalence and supplies several independent checks on a printed output. An approximate Welch's t-test uses its same approximate statistical degrees of freedom throughout. Rounding can blur a boundary decision but cannot explain a confidence-interval midpoint far from the reported estimate.

A one-sample Student t-test on independent pairs' differences $d_i$. It tests a specified mean difference $d_0$, using the standard deviation of those differences rather than treating both measurements in a pair as independent. Exact small-sample inference assumes normally distributed differences.

For two independent samples from approximately normal distributions, Welch's t-test compares their means without assuming equal unknown variances. The displayed test statistic uses sample variances $s_X^2,s_Y^2$. Under the equal-mean null hypothesis, its reference Student t-distribution has approximate statistical degrees of freedomA p-value from this approximation addresses the specified statistical error model; it cannot distinguish a material effect from an unmeasured experimenter bias when sample and experimenter are confounded.

For two samples of sizes $n_1,n_2>1$, write $a=s_1^2/n_1$ and $b=s_2^2/n_2$, with $a+b>0$. The reference statistical degrees of freedom in Welch's t-test satisfyIndeed the Cauchy-Schwarz inequality applied to $(a/\sqrt{n_1-1},b/\sqrt{n_2-1})$ and $(\sqrt{n_1-1},\sqrt{n_2-1})$ gives the inequality after division. Thus an independent two-sample Welch's t-test on $N$ complete observations cannot have reference statistical degrees of freedom exceeding $N-2$. The exact pooled two-sample t statistic also uses $N-2$, under its stronger equal-variance assumptions.

For independent normal samples of sizes $m,n$ with a common unknown variance and equal means under the null, $Z=(\overline{X}-\overline{Y})/(\sigma \sqrt{1/m+1/n})$ is standard normal. The independent centered sample sums of squares give $V=(m+n-2)S_p^2/{\sigma }^2\sim {\chi }_{m+n-2}^2$, independently of $Z$. Thus $Z/\sqrt{V/(m+n-2)}$ has the exact Student t-distribution, giving the statistic $(\overline{X}-\overline{Y})/(S_p\sqrt{1/m+1/n})$. A lower one-sided test uses its lower-tail probability.

For an independent sample from a normal distribution, maximization of the likelihood over the unknown variance gives the likelihood ratio for a fixed mean against an unrestricted mean as displayed, where $t=\sqrt{n}(\bar{X}-{\mu }_0)/S$ and $S^2={\sum }_i(X_i-\bar{X}{)}^2/(n-1)$. It decreases with $|t|$. Under the null, the normal sample mean is independent of its centered sum of squares, so $t$ has the exact Student t-distribution with $n-1$ degrees of freedom. The resulting two-sided test requires no asymptotic chi-squared approximation.

In a fixed full-rank normal linear model, the studentized estimator of one coefficient has a noncentral t-distribution with noncentrality equal to that coefficient divided by its standard deviation. The fitted and residual orthogonal projections of the Gaussian noise are independent. Therefore the statistical power of its exact Student's t-test does not depend on the other regression coefficients, with design and variance fixed.