Histogram and normality test (Anderson-Darling): theory, guide and case study

Try it while you read

Open the free Histogram and Normality Test tool and click Load sample: it reproduces the case study below.

Theory: shape before numbers

A histogram groups the values into bins and shows how often each range occurs. Before any statistic is calculated, the shape tells a lot: a single symmetric peak suggests a stable, normal process; two peaks suggest two populations (two machines, cavities, operators); a sharp edge suggests sorted or reworked parts; a long tail suggests a physical limit such as zero for flatness.

The Anderson-Darling test

The test measures how far the cumulative distribution of the data is from the fitted normal distribution, with extra weight on the tails, which is exactly where the defect rate is decided.

A² = −n − (1/n) · Σ (2i − 1) · [ln F(x₍ᵢ₎) + ln(1 − F(x₍ₙ₊₁₋ᵢ₎))]

A small A² means a good fit. The p-value answers: “if the data really were normal, how likely is a deviation this large?” Above 0.05 normality is not rejected; below 0.05 the normal model is questionable.

Probability plot

The sorted values are plotted against the quantiles of the fitted distribution. If the points follow the straight line, the model fits. Curvature at the ends shows skewness, an S-shape shows heavier or lighter tails.

Distribution fitting

The tool fits normal, lognormal, Weibull, gamma and exponential distributions by maximum likelihood and ranks them by A². The best fit can be used for the capability study with the ISO 22514-2 percentile method.

Key point

A non-normal result is a question, not a verdict: “why is it not normal?” is often more valuable than the test itself.

Step by step

Paste the values into the column, at least 50.
Add LSL / USL to see how the distribution sits in the tolerance.
Look at the histogram shape and at the descriptive statistics (skewness, kurtosis, min / max).
Read the test and the plot.
Pick the distribution with the lowest A² that also makes physical sense.

Case study: 150 measurements of a 50 mm length

150 parts are measured (the Load sample data). Mean 49.954, standard deviation 1.105, skewness −0.06, excess kurtosis 0.03.

Anderson-Darling A² = 0.328, p = 0.516: normality is not rejected.
The probability plot follows the line in both tails.
Fit ranking by A²: lognormal 0.325, gamma 0.325, normal 0.328, Weibull 1.68, exponential 65.9.

Lognormal and gamma fit marginally better than normal, but the differences are negligible and the data are symmetric. The engineer keeps the normal distribution, which makes the standard Cpk and Ppk valid.

Result

The data can be analysed with the normal model. Choosing a more exotic distribution only because its A² is 0.003 lower would add complexity without benefit.

Common mistakes

Testing normality on mixed data from several machines.
Too few bins or too many: the tool chooses a sensible default.
Transforming data (Box-Cox) without asking why the shape is skewed.

Frequently asked questions

My p-value is 0.03, is my data useless?

No. It means the normal distribution is not a good model. Check first for mixtures (two machines, two batches) and outliers; if the shape is naturally skewed, use a fitted distribution in the capability study.

Why does the test reject normality for large data sets?

With several hundred values the test detects tiny deviations that have no practical effect. Look at the probability plot: if the points follow the line in the tails, the normal model is fine for practical purposes.