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For a normal sample of variance ${\sigma }^2>0$, define $S^2=(n-1{)}^{-1}{\sum }_i(X_i-\overline{X}{)}^2$. The exact chi-squared distribution identity $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$ and the central limit theorem for independent squared standard normal random variables implyThe delta method for $g(v)=\sqrt{v}$, with derivative $g^{\prime }({\sigma }^2)=1/(2\sigma )$, therefore gives the displayed limit for $S$. Exact independence of the normal sample mean and sample variance also makes $S$ independent of $\overline{X}$. The resulting joint limit supports combining location and scale estimates when a model relates their parameters.

For an independent normal sample with common mean $\mu$ and positive variance ${\sigma }^2$, its sample mean and sample variance are independent. Indeed, the mean and residual vector $(X_i-\overline{X}{)}_i$ are jointly multivariate normal, and $\mathrm{Cov}(\overline{X},X_i-\overline{X})={\sigma }^2/n-{\sigma }^2/n=0$. Vanishing cross-covariance between jointly Gaussian blocks proves independence of the mean from the entire residual vector. Since the sample variance is a measurable function of that vector, independence follows. Independence from each coordinate separately would not suffice without the joint Gaussian argument.

Assume $n\ge 2$ and ${\sigma }^2>0$. Up to a constant, the log-likelihood of the normal sample isFor each $s$, the square is minimized at $\mu =\bar{X}$; optimizing the remaining one-variable likelihood gives the maximum-likelihood estimatorsThey are the global maximum when the residual sum is positive, which occurs with probability one. On the exceptional all-equal sample the likelihood increases without bound as $s\downarrow 0$.

To derive the distributions and independence, write $Z_i=(X_i-\mu )/\sigma$ and perform an orthogonal change of coordinates whose first row is $n^{-1/2}(1,\dots ,1)$. The transformed components $Z_1^{\prime },\dots ,Z_n^{\prime }$ remain independent standard normal random variables, since their joint density depends only on the squared Euclidean norm. ThenThe first expression uses only $Z_1^{\prime }$, the second only the other components. ConsequentlyThis also exhibits why the maximum-likelihood variance uses divisor $n$, even though its chi-squared degrees of freedom are $n-1$.

The independent future observation satisfies $X_0-\bar{X}\sim N(0,{\sigma }^2(1+1/n))$. It is independent of the residual sum of squares, because both $X_0$ and $\bar{X}$ are independent of the residual vector. Combining this normal variable with the independent chi-squared variable givesThe event in the proposed prediction interval is equivalent to $|T|<\gamma \sqrt{(n-1)/n}$. By symmetry and continuity of Student's t-distribution, exact joint coverage $1-\alpha$ therefore requiresThis is the normal prediction interval with maximum-likelihood variance. The factor $\sqrt{n/(n-1)}$ cannot be dropped: the interval uses $\hat{\sigma }$, not the unbiased residual standard deviation. Coverage here is over repeated joint sampling of the old and new observations, not a claim of parameter-free conditional coverage for every realized sample.

A $100(1-\alpha )\%$ confidence interval procedure produces random endpoints $L,U$ such that ${\mathbb{P}}_{\mu }(L\le \mu \le U)=1-\alpha$, under the model. In everyday terms, repeating the same sampling and interval construction many times would give intervals containing the true mean in about $100(1-\alpha )\%$ of repetitions.

For $n\ge 2$, define the sample mean $\bar{X}=n^{-1}{\sum }_i{X}_i$ and the sample variance $S^2=(n-1{)}^{-1}{\sum }_i(X_i-\bar{X}{)}^2$. For independent observations from a normal distribution with ${\sigma }^2>0$, the standard sampling results areThe first follows from the independent normal sample mean and sample variance. Let $q=t_{n-1,1-\alpha /2}$ be the positive Student quantile. Symmetry gives the central confidence intervalIts width is $W=2qS/\sqrt{n}$, so

Write $S^2=Q/(n-1)$ and $S=\sqrt{S^2}$, with $n\ge 2$. Two exact facts about a normal sample areand independence of the normal sample mean and sample variance holds. A chi-squared distribution with $n-1$ statistical degrees of freedom is a sum of $n-1$ independent ${\chi }_1^2$ variables, each with mean one and variance two. The central limit theorem therefore givesFor $g(x)=\sqrt{x}$, the derivative at ${\mu }^2$ is $1/(2\mu )$ because $\mu >0$. The delta method gives the asymptotic normality of a normal sample standard deviation:The exact independence of $\overline{X}$ and $S$ makes their limiting normal distributions independent. Consequently,ThusThis has the same leading asymptotic variance as $\hat{\mu }$. The weights $1/3$ and $2/3$ are those in the optimal asymptotic combination of a normal sample mean and standard deviation. This does not require $S$ to be an exactly unbiased estimator of $\mu$.

For a normal sample $X_1,\dots ,X_n$ with unknown mean $\mu$, positive variance ${\sigma }^2$ and $n\ge 2$, let $S^2=(n-1{)}^{-1}{\sum }_i(X_i-\overline{X}{)}^2$. Thenand an exact $(1-\alpha )$ confidence interval for $\mu$ is $\overline{X}\pm t_{n-1,1-\alpha /2}S/\sqrt{n}$.

The proof uses independence of the normal sample mean and sample variance: $\sqrt{n}(\overline{X}-\mu )/\sigma$ is standard normal, independently of $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$. Dividing produces Student's t-distribution. The random scale increases uncertainty relative to a known-variance normal interval. Non-normal samples need an approximation argument; an independent scale estimate alone does not make an arbitrary statistic exactly Student t.

For a normal sample of size $n\ge 2$, the central Student t confidence interval has width $W=2t_{n-1,1-\alpha /2}S/\sqrt{n}$. Its squared width is therefore a fixed multiple of the sample variance. The exact sampling identity $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$ gives the displayed law. It quantifies the random variability in interval precision even when the sample size and coverage level are held fixed.

William Sealy Gosset published statistical research under the name Student while working as a brewer for Guinness. His practical problems involved variation in ingredients and brewing conditions, where obtaining a large sample was often difficult. This motivated methods that account for uncertainty in estimating variability from a small sample.^[ref]

The Student's t-distribution describes the standardized mean of an independent normal sample when its unknown standard deviation is replaced by the sample standard deviation. Its heavier tails, compared with a Standard normal distribution, reflect that extra uncertainty. Gosset's work helped turn the limitations of a small experiment into an explicit part of inference rather than treating a variance estimate as known.

Biography: www.dib.ie/biography/gosset-william-sealy-student-a3543