c-Chart vs u-Chart: Picking the Right Attribute Chart for Defect Counts
Forty coated panels come off the line and you log the surface defects: three scratches on one, a pinhole on another, nothing on most. You want a control chart for the defect rate. Then tomorrow's run is sixty panels, and next week's inspection covers square meters of laminate instead of discrete panels. The chart you reach for — c-chart or u-chart — turns entirely on one question: does the size of your inspection unit stay constant?
Both charts count defects (nonconformities) and both assume those defects follow a Poisson distribution. That shared foundation is why they are easy to confuse. The split is narrow but consequential: the c-chart needs a constant inspection unit, the u-chart handles a varying one, and choosing wrong gives you control limits that are either falsely tight or falsely loose. This post walks the prerequisite distinction, the two formulas with worked numbers, and a verdict by inspection scenario.
First: defects, not defectives
The most common error here is not c-versus-u — it is using either chart on the wrong kind of data. The c-chart and u-chart count defects: a single unit can carry several (three scratches on one panel is a count of three). The p-chart and np-chart count defective units: a panel either passes or fails, contributing at most one. If a unit can have more than one flaw and you care about the flaw rate, you want a c- or u-chart. If a unit is simply conforming or nonconforming, you want a p- or np-chart. (ASQ's control chart resource lays out the full attribute-versus-variable family if you need the wider map.)
The c-chart: a constant inspection unit
Use a c-chart when every inspection unit is the same size — the same panel, the same fixed area, the same number of items per sample. Because the unit is constant, the expected defect count is constant, and the control limits are a single pair of horizontal lines built on the Poisson property that the variance equals the mean:
The standard deviation is \(\sqrt{\bar{c}}\) — that is the Poisson assumption doing the work. The LCL is floored at zero because you cannot have a negative defect count. The NIST/SEMATECH e-Handbook section on counts control charts derives both the c- and u-chart limits from the Poisson model.
The 20 plotted counts sum to 60, confirming the center line of 3.0; all fall under the UCL of 8.2, so this process is in statistical control for defect rate. (The limits are Poisson-based, ̅c ± 3√̅c — not the sample standard deviation of the points.)
The u-chart: a varying inspection unit
Use a u-chart when the inspection unit size changes from sample to sample — rolls of fabric of different lengths, circuit boards with different component counts, daily output where the number of items inspected fluctuates. You normalize to defects per unit and the control limits recompute for each sample's size \(n_i\):
Because \(n_i\) sits under the radical, the limits tighten as the sample grows and widen as it shrinks — the chart shows a stair-step boundary rather than two flat lines.
c-chart vs u-chart, side by side
| Criterion | c-chart | u-chart |
|---|---|---|
| What it plots | Count of defects per inspection unit | Defects per unit (rate) |
| Inspection unit size | Constant | Varies sample to sample |
| Control limits | Two flat lines (̅c ± 3√̅c) | Stair-step; recomputed per ni |
| Underlying distribution | Poisson | Poisson |
| Counts defects or defectives? | Defects (multiple per unit) | Defects (multiple per unit) |
| Operator readability | Higher — fixed limits | Lower — moving limits |
| Simplification | — | Use c-chart if n varies < 25% |
The verdict, by inspection scenario
- If your inspection unit is genuinely fixed — one assembly, one board, a constant sampled quantity every time — use a c-chart. The flat limits are easier for operators and the math is simpler.
- If the area, length, or item count you inspect changes each sample: use a u-chart. Forcing a c-chart on variable samples gives limits that are too tight for small samples (false alarms) and too loose for large ones (missed signals).
- If the unit varies only slightly (under ~25%): a c-chart with the average sample size is the pragmatic choice — cleaner to read, negligible loss of accuracy.
- If you are counting pass/fail units, not flaws: neither chart applies — reach for a p-chart or np-chart instead. Confirm you are counting defects (a unit can have several) before either of these.
If you are still mapping data types to chart types across all of variable and attribute data, start with the control chart selection guide, which places c, u, p, and np within the full decision tree. Once you know which chart fits, the control chart tool computes the Poisson-based limits — including the per-sample u-chart limits — from your uploaded counts, so you do not recalculate the radical for every changing sample size by hand.