p Chart

A p chart monitors sample proportions of nonconforming units.
It charts units classified conforming or nonconforming.^[1]

Prerequisites

Prerequisites: binomial distribution and control chart basics.

Use p charts when each inspected unit is counted once as conforming or nonconforming.
Sample sizes may vary.

For sample $i$ , let $D_{i}$ be the number of nonconforming units and $n_{i}$ the number inspected. The plotted statistic is:

{\hat{p}}_{i} = \frac{D_{i}}{n_{i}} .

Each unit is classified into one of two outcomes.
Units are independent within and between samples.
The probability of nonconformance is stable when only common causes are present.
Sample sizes are large enough for normal-approximation limits to be useful, or exact/binomial methods are considered.

Symbol	Meaning
$D_{i}$	Nonconforming units in sample $i$
$n_{i}$	Inspected units in sample $i$
${\hat{p}}_{i}$	Sample proportion nonconforming
$\bar{p}$	Pooled estimate of process fraction nonconforming

Estimate:

\bar{p} = \frac{\sum_{i} D_{i}}{\sum_{i} n_{i}} .

For sample $i$ :

U C L_{i} = \bar{p} + 3 \sqrt{\frac{\bar{p} (1 - \bar{p})}{n_{i}}}

C L = \bar{p}

L C L_{i} = max (0, \bar{p} - 3 \sqrt{\frac{\bar{p} (1 - \bar{p})}{n_{i}}}) .

Points above the UCL suggest a worse-than-stable nonconforming rate.
Points below the LCL can signal improvement, inspection changes, or classification changes.
Variable $n_{i}$ means variable limits.
If $n_{i}$ is fixed and counts are easier to communicate, an np chart is equivalent.

Five samples of 100 units have nonconforming counts $4, 7, 5, 6, 8$ . Then $\bar{p} = 30 / 500 = 0.06$ .

U C L = 0.06 + 3 \sqrt{\frac{0.06 (0.94)}{100}} = 0.1313

L C L = max (0, 0.06 - 0.0713) = 0.

The observed proportions $0.04, 0.07, 0.05, 0.06, 0.08$ are inside the limits.

Related note	Use
np chart	Fixed sample-size count of nonconforming units
c chart	Defect counts, constant opportunity
u chart	Defect rates, variable opportunity
Control charts	Attribute chart taxonomy
Control Limits and Specification Limits	Limit distinction

NIST/SEMATECH, e-Handbook of Statistical Methods, "Proportions Control Charts", https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm ↩︎