# p chart (proportion defective)

Operations · `p chart` · verification: source checked

The p chart is an attributes control chart that plots the proportion of nonconforming items in each sample. It is used when items are classified as conforming or nonconforming, and its statistical basis is the binomial distribution. Control limits are placed at three standard deviations of the sample proportion around the process fraction nonconforming p.

## Specification

| Property | Value |
|---|---|
| Kind | tool |

## Formula

```
UCL = p + 3*sqrt(p(1-p)/n); Center Line = p; LCL = p - 3*sqrt(p(1-p)/n)
```

- `p`: true fraction nonconforming of the process
- `n`: sample size, number of items inspected per sample

## Study notes

- The number of defectives D in a sample of size n follows the binomial distribution: P(D=x) = C(n,x) p^x (1-p)^(n-x) for x = 0, 1, ..., n.
- An item is classified as nonconforming if at least one of the quality characteristics inspected on it does not conform to standard.
- The sample fraction nonconforming is p^ = D/n, with mean mu = p and variance sigma_p^2 = p(1-p)/n.
- When m samples of size n are available, the process fraction is estimated by p-bar = sum(Di)/(mn) = sum(p^i)/m.
- The proportion may be expressed as a decimal or, multiplied by 100, as a percent.
- Worked example in the document: chips on 30 wafers, 50 chips measured per wafer, a defective recorded when horizontal and/or vertical misregistration from the centre is found; the 30 sample fractions range from 0.08 to 0.48 and are plotted on the chart.

## Sources

1. [6.3.3.2. Proportions Control Charts](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm) (accessed 2026-10-25)

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