Intervals for a Mean and for Variability

An interval for σ is simply a range of plausible values for how much a process really scatters, built from the scatter you saw in a sample. You already know intervals for a mean; this is the same idea for spread. The key fact looks dense, but it's one ratio: $(n - 1)s^2/\sigma^2$, which is your sample variance $s^2$ compared with the true variance $\sigma^2$, scaled by $n - 1$ (the degrees of freedom). That ratio follows a known curve, the chi-square, so you look up two cutoffs, flip the inequality around, and take square roots. Bear with the algebra; there's only one idea in it.

Why spread gets its own interval. A grinding cell drifts off target and an operator nudges an offset; nothing nudges σ. Whether the cell can hold ±12 µm is a question about its spread, so capability studies, gauge studies and machine acceptance tests all need an interval for $\sigma$.

The mean, briefly. For roughly normal data (or large $n$), $\bar x \pm t_{\alpha/2,n-1}s/\sqrt n$ captures $\mu$ with confidence $1 - \alpha$, as in t intervals. Notation for this whole course: $z_\alpha$, $t_{\alpha,\nu}$ and $\chi^2_{\alpha,\nu}$ leave area $\alpha$ to their right.

Where the χ² comes from. For normal data each $(x_i - \mu)/\sigma$ is standard normal, and a sum of $n$ independent squared standard normals is, by definition, $\chi^2$ with $n$ degrees of freedom. Swapping $\mu$ for $\bar x$ costs one: the deviations $x_i - \bar x$ always sum to zero, so only $n - 1$ of them are free. That gives the pivot

A $\chi^2_\nu$ variable has mean $\nu$ and variance $2\nu$, so $E(s^2) = \sigma^2$ and $\operatorname{Var}(s^2) = 2\sigma^4/(n-1)$.

Turning it around. The middle $1 - \alpha$ of $\chi^2_{n-1}$ lies between its two critical values:

Solve each inequality for $\sigma^2$. Taking reciprocals flips them, so the larger critical value gives the lower limit. Square roots keep the order, so the same limits, rooted, bracket $\sigma$.

Lopsided on purpose. χ² is skewed right, so the interval is not $s^2$ plus or minus a margin: it reaches much farther above $s$ than below. With ten parts, $\sigma$ could be anywhere from $0.69s$ to $1.83s$ (95%). Small samples say very little about spread.

One-sided bounds are what a spec usually needs. To show a machine is good enough, bound $\sigma$ from above with all of α in one tail: $\sigma \le s\sqrt{(n-1)/\chi^2_{1-\alpha,n-1}}$.

Fragile. The t interval survives skewed data because the CLT normalizes $\bar x$. The χ² interval doesn't: it's built from squared deviations of individual values, and heavy tails make it far too narrow. Check a probability plot first.

Common slips: swapping the two critical values, using $n$ instead of $n - 1$, and forgetting to square $s$ or to take the final square root.

Where it leads: an upper bound on $\sigma$ is a lower bound on capability $C_p = (USL - LSL)/(6\sigma)$ (process capability), it's half of a tolerance interval, and the same pivot becomes the χ² test for variance.

Before a plant buys a grinder, the supplier runs a capability trial: 30 or 50 parts, an upper confidence bound on σ, and a pass only if the bound beats the spec. The same calculation qualifies gauges (is the measurement σ small enough?), decides how often a scale must be recalibrated, and turns a capability index into a statement with honest uncertainty, as in the challenge. Instrument datasheets that quote a "repeatability" are reporting σ estimates that deserve exactly this interval.