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Let the columns of a centered design matrix $X\in {\mathbb{R}}^{n\times p}$ be variables and write $X^{T}X=V\Lambda V^T$ by the spectral theorem for real symmetric matrices. The columns $u_i$ of $U=XV$ are the sample principal-component score vectors. They satisfy $u_i^T{u}_j={\Lambda }_{ij}$, so distinct score vectors have zero sample covariance and $u_i$ has sample variance ${\Lambda }_{ii}/n$ under the divisor-$n$ convention.

The sample standard deviation is the nonnegative square root of the usual sample variance for $n\ge 2$. The displayed convention uses divisor $n-1$. Although the sample variance is an unbiased estimator of the population variance under independent identically distributed sampling with finite variance, taking its square root does not in general give an unbiased estimator of the population standard deviation.

Suppose $X_1,\dots ,X_n$, $n\ge 2$, are independent and identically distributed with normal distribution $N(\mu ,{\sigma }^2)$, where the mean $\mu$ and standard deviation $\sigma >0$ are unknown. Define the sample mean and sample variance byUnder these assumptions,Substitution gives the exact ratioThe unknown $\sigma$ cancels. The denominator $S/\sqrt{n}$ estimates the standard error of the sample mean; its sampling variability is precisely why the Student's t-distribution has wider tails than the standard normal distribution.

The crucial independence has a geometric explanation. Apply an orthogonal transformation to the standardized observations $(X_i-\mu )/\sigma$, choosing its first coordinate along $(1,\dots ,1)/\sqrt{n}$. That coordinate is $Z$; the remaining $n-1$ coordinates are independent standard normal random variables whose squared sum is $V$. Their joint probability density factorizes because the transformation preserves the squared length. This also explains the loss of one statistical degree of freedom when estimating the mean. Uncorrelatedness alone would not establish the required independence for arbitrary data.

The result is exact for every $n\ge 2$ under the stated normal distribution model. A small sample size by itself does not justify it: a sample from an arbitrary probability distribution need not have the displayed chi-squared distribution or the necessary independence.

For independent and identically distributed observations with finite positive variance, the central limit theorem, consistency of the sample variance and Slutsky's theorem give $T_n\stackrel{d}{\to }N(0,1)$. Under an independent normal distribution sample it instead has the exact finite-sample Student t-distribution with $n-1$ degrees of freedom, as described by the studentized normal sample mean. The asymptotic statement does not require normal individual observations.

The symmetric kernel $h(x,y)=(x-y{)}^2/2$ has expectation equal to the population variance for independent identically distributed observations with finite second moment. Its U-statistic reduces to the displayed unbiased sample variance, using ${\sum }_{i<j}(X_i-X_j{)}^2=n{\sum }_i(X_i-\overline{X}{)}^2$.

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.