np Chart

An np chart monitors the number of nonconforming units in samples of constant size. It is the count-scale companion to the p chart.^[1]

Prerequisites

Prerequisites: binomial distribution and p chart logic.

Process Context

Use an np chart when each sample has the same inspected size $n$ and the audience should see counts rather than proportions.

Definition

For sample $i$ , plot $D_{i}$ , the number of nonconforming units. Under stable conditions, $D_{i}$ is modeled as binomial with parameters $n$ and $p$ .

Assumptions / Requirements

Fixed sample size $n$ .
Each unit has two possible classifications.
Unit classifications are independent.
The probability $p$ is stable when only common causes are present.

Notation

Symbol	Meaning
$D_{i}$	Nonconforming units in sample $i$
$n$	Fixed inspected sample size
$\bar{p}$	Pooled estimate of fraction nonconforming

Control Limits / Formula

C L_{n p} = n \bar{p}

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

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

Interpretation Rules

A high point signals more nonconforming units than expected from stable variation.
Low points can signal improvement or changed inspection/classification.
If sample size varies, use a p chart instead.

Worked Example

Five samples of $n = 100$ units have nonconforming counts $4, 7, 5, 6, 8$ . The pooled estimate is $\bar{p} = 0.06$ , so $C L = 6$ .

U C L = 6 + 3 \sqrt{100 (0.06) (0.94)} = 13.13

L C L = max (0, 6 - 7.13) = 0.

All five counts are within the chart limits.

Common Mistakes

Using an np chart when sample sizes vary.
Plotting defects instead of nonconforming units.
Forgetting that one nonconforming unit may contain many defects.
Replacing control limits with accept/reject specifications.

Connections

Related note	Use
p chart	Variable sample size or proportion scale
c chart	Defect counts
u chart	Defects per unit
Control charts	Chart taxonomy

References

Douglas C. Montgomery, Introduction to Statistical Quality Control, 8th ed., Wiley, ISBN 978-1-119-39930-8. ↩︎