p / np Chart (Fraction Defective)

Attribute control chart for the fraction of defective units per sample — binomial-distributed.

Also known as: p chart, np chart, fraction defective chart

The p chart tracks the fraction of defective units $p_{i} = d_{i} / n_{i}$ ; the np chart the number of defective units $n p_{i}$ at constant $n$ . Both assume the binomial distribution (each unit is either defective or not).

UCL_{p} = \overset{p}{ˉ} + 3 \frac{p ˉ ( 1 - p ˉ )}{n}

Constant n → np chart is easier to read (absolute counts)
Variable n → only p chart is meaningful; limits recomputed per point
Prerequisite $n \overset{p}{ˉ} \geq 5$ — otherwise the binomial approximation breaks and limits become asymmetric

Used in

Control Chart (Attribute)

In the Algorithm Lab

control-charts/attribute-control-chart

Sources

Shewhart, W. A. — Economic Control of Quality of Manufactured Product, 1931
Montgomery, D. C. — Introduction to Statistical Quality Control, 8th Edition, Chapter 7
Laney, D. B. — Improved Control Charts for Attributes, Quality Engineering, 14(4), 2002