p and c Charts for Defects

An attribute chart is simply a control chart for counts, like the fraction of boards that fail inspection each day or the number of flaws per roll of film. The limits follow the same "center $\pm$ 3 standard deviations" rule as the last lesson; only the standard deviation changes. For a fraction defective, it's $\sqrt{\bar p(1 - \bar p)/n}$, the binomial standard error you already know. For a count of flaws, it's $\sqrt{\bar c}$, because a Poisson count's variance equals its mean. Same idea, two familiar formulas.

Counting instead of measuring. Some quality checks give no number on a gauge. A circuit board passes automated optical inspection or it doesn't; a painted door has 0, 1, 2 or more blemishes. These attribute data are cheap to collect, and they get their own charts.

Defectives versus defects. A defective (nonconforming) unit fails the spec; a defect is a single flaw, and one unit can carry several. Count defectives in samples of $n$ units and the count is binomial, so the fraction defective $\hat p$ goes on a p chart. Count defects per unit (per door, per roll, per square meter) and there are many opportunities, each one rare, so the count is Poisson and goes on a c chart. It's the same split as in Counting Defects.

Why these limits. The recipe never changes: center line ± 3 standard deviations of whatever is plotted.

• p chart. The number of defectives $X$ in a sample is binomial with $\text{Var}(X) = np(1 - p)$, so $\hat p = X/n$ has $\text{Var}(\hat p) = \text{Var}(X)/n^2 = p(1 - p)/n$: the same standard error as a confidence interval for a proportion. Estimate $p$ by $\bar p$, total defectives over total inspected.
• np chart. Plot the count itself, with limits $n\bar p \pm 3\sqrt{n\bar p(1 - \bar p)}$. Handy when $n$ never changes.
• c chart. A Poisson count's variance equals its mean, so $\sigma = \sqrt{\bar c}$ and the limits are $\bar c \pm 3\sqrt{\bar c}$.
• u chart. When the amount inspected varies (rolls of different lengths, batches of different sizes), plot defects per unit $u_i = c_i/n_i$. A Poisson count with mean $un_i$, divided by $n_i$, has variance $u/n_i$, so the limits are $\bar u \pm 3\sqrt{\bar u/n_i}$ and tighten for bigger samples.

A negative lower limit is set to 0. When sample sizes vary, a p chart's limits change from sample to sample as well.

A point below the LCL is news. Either the process got better (find out why, and keep it) or the inspection got worse (a lamp burned out, a new inspector). Investigate either way. But a chart can only show an improvement if its LCL is above 0:

At $\bar p = 2\%$ that means more than 441 units per sample. That's the price of attribute data: a pass-or-fail result carries far less information than a measurement, so attribute charts need big samples, and they react only once parts are already bad, while an X̄ chart can see a drift before parts leave the spec.

The limits are approximate. The ±3σ recipe borrows the normal curve. When $n\bar p$ or $\bar c$ is small, binomial and Poisson distributions lean right, so more than the normal curve's 0.135% of in-control points land above the UCL, and the LCL vanishes. The second picture shows this for the c chart. Run rules still help, read with the same caution.

Electronics assembly lines put the fraction of boards failing automated optical inspection on p charts, and paint shops and textile mills chart blemishes per unit on c and u charts. Hospitals use p charts for infection and readmission rates, call centers for the fraction of abandoned calls, and software teams track defects per thousand lines of code on u charts.