p and c control charts

Control charts for count data: the p chart tracks the proportion defective in samples (binomial), the c chart the number of defects per constant inspection unit (Poisson).

p \text{ chart: } \bar p \pm 3\sqrt{\frac{\bar p(1-\bar p)}{n_i}}, \qquad \bar p = \frac{\sum d_i}{\sum n_i}

p chart (limits vary with each sample size)

$d_i$: number of defective items in sample $i$
$n_i$: size of sample $i$
$\bar p$: overall proportion defective (centre line)

c \text{ chart: } \bar c \pm 3\sqrt{\bar c}

c chart (constant area of opportunity)

$\bar c$: average number of defects per inspection unit

When to use

p chart (or np chart for constant $n$ ): items classified as defective / not defective.
c chart: counts of defects (flaws, errors, complaints) in units of the same size; use a u chart ( $\bar u \pm 3\sqrt{\bar u/n_i}$ ) when the inspected amount varies.

Assumptions

p chart: binomial — items independent with a common defect probability when in control.
c chart: Poisson — defects occur independently at a constant rate (variance equals mean). Overdispersed counts produce too many false alarms.
Lower limits below 0 are set to 0, and p-chart upper limits are capped at 1.

How to interpret

Points above the upper limit signal deterioration; points below the lower limit (when it is above 0) may signal improvement — or a measurement problem.
With a lower limit of 0, decreases cannot be signalled by rule 1.

Common mistakes

Using a c chart when the inspection area changes (use u).
Using a p chart for defects per item (an item can have several defects).
Recomputing limits continually so that every shift is absorbed into the centre line.

Example

p chart: defectives 8, 12, 6, 10, 9, 15 in samples of 200. $\bar p$ = 60/1 200 = 0.05; limits 0.0038 to 0.0962. The largest proportion, 15/200 = 0.075, is inside: in control.
c chart: defects 4, 7, 3, 5, 6, 2, 13. $\bar c$ = 5.71; UCL = $5.71 + 3\sqrt{5.71}$ = 12.89, LCL = 0 (negative, truncated). The 7th unit (13 defects) is beyond the UCL.