Histogram, fitted normal curve and a normality test

The bin count changes the story

A histogram is not one picture of your data, it is a family of them, and the bin count picks which member you see. Too few bins and two populations merge into one smooth hump. Too many and ordinary sampling noise looks like structure. Neither is more honest than the other.

Two rules are in common use and this tool reports both:

• Sturges — ⌈log₂ n⌉ + 1. Simple, and what most textbooks and most software default to. It assumes data that is roughly normal and it under-bins large samples noticeably.
• Freedman-Diaconis — bin width 2 × IQR / ∛n. Uses the interquartile range instead of the count, so it is far more robust on skewed data and on samples with a long tail.

Freedman-Diaconis is usually the better default, which is why it is the recommendation here — but change the bin count and watch what happens. If the story changes, that is worth knowing before you tell it to anyone.

A normal curve drawn over a histogram is not a test

It is a picture, and eyeballing it fails in both directions.

Small samples look non-normal when they are fine. Thirty readings from a genuinely normal process routinely produce a lumpy, lopsided histogram, because thirty readings is not many. People look at it and conclude their process is skewed.

Large samples make trivial departures look dramatic. At five thousand readings a slight asymmetry that has no practical consequence whatsoever is plainly visible, and a normality test will reject decisively. Statistical significance is not practical significance, and at that sample size the test is answering a question nobody needed asked.

This tool runs the Anderson-Darling test, which is more sensitive in the tails than the alternatives — and the tails are what matter for capability. But read what it says carefully: a non-significant result is a failure to reject, not evidence of normality. No test can demonstrate that data is normal. At thirty readings the test can barely reject anything at all, so "not rejected" is a cheap statement, and the tool says so rather than letting you read it as a pass.

For a visual check, a normal probability plot beats a fitted curve substantially: departures show up as curvature away from a straight line, which the eye judges far better than it judges the fit of a bell.

When normality actually matters — and when it does not

People routinely worry about the wrong one of these.

It matters a great deal for turning a capability index into a defect rate. Converting a Cpk into parts per million is a normal-curve tail area and nothing else. If the distribution is skewed — and anything bounded at zero, like a flatness, a roundness or an impurity level, is skewed by its own physics — the PPM figure is wrong, usually optimistically on the bounded side. Enter a tolerance above and the tool shows you the observed proportion outside it next to the proportion the normal fit predicts. When those two disagree badly, any Cpk-derived PPM from this data is not to be trusted.

It matters much less for whether a control chart works. Shewhart charts are famously robust to non-normality, and subgrouped charts especially so — the central limit theorem works on the subgroup averages regardless of what the individual readings do. Shewhart's three-sigma limits were never a normality argument; they were an economic choice about false alarms. An individuals chart is more exposed than a subgrouped one, but even there the usual outcome of skewed data is somewhat more upper-side signals than the nominal rate, not a chart that has stopped working.