u Chart

A u chart monitors nonconformities per unit when the number of inspected units or the opportunity for defects changes by sample. It is the rate-scale companion to the c chart.^[1]

Prerequisites

Prerequisites: Poisson distribution, c chart logic.

Process Context

Use a u chart when defect counts are collected over samples with different sizes, areas, lengths, forms, or exposure.

Definition

For sample $i$ , let $c_{i}$ be the number of defects and $n_{i}$ the number of inspection units. Plot:

u_{i} = \frac{c_{i}}{n_{i}} .

Assumptions / Requirements

Defects are countable events.
Inspection unit size $n_{i}$ is known for each sample.
Defects follow an approximately Poisson count model under stable conditions.
Differences in opportunity are represented by $n_{i}$ .

Notation

Symbol	Meaning
$c_{i}$	Defects in sample $i$
$n_{i}$	Inspection units in sample $i$
$u_{i}$	Defects per unit in sample $i$
$\bar{u}$	Pooled defects per unit

Control Limits / Formula

\bar{u} = \frac{\sum_{i} c_{i}}{\sum_{i} n_{i}}

For sample $i$ :

U C L_{i} = \bar{u} + 3 \sqrt{\frac{\bar{u}}{n_{i}}}

C L = \bar{u}

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

Interpretation Rules

Larger $n_{i}$ gives narrower limits because the rate is estimated more precisely.
High points signal unusual defect rates, not high raw counts.
If $n_{i}$ is constant, a c chart may communicate more simply.

Worked Example

Three lots inspect $5, 4, 6$ units and find $4, 5, 3$ defects. Then $\bar{u} = 12 / 15 = 0.8$ . For the first lot:

U C L_{1} = 0.8 + 3 \sqrt{\frac{0.8}{5}} = 2.0

L C L_{1} = max (0, 0.8 - 1.2) = 0.

The first plotted value is $u_{1} = 4 / 5 = 0.8$ , exactly on the center line.

Common Mistakes

Plotting raw defect counts when opportunity changes.
Using a u chart for nonconforming units instead of defects.
Forgetting that limits differ by sample size.
Ignoring changes in inspection intensity or defect definition.

Connections

Related note	Use
c chart	Constant inspection unit
p chart	Nonconforming proportions
np chart	Nonconforming counts
Control charts	Attribute chart taxonomy

References

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