c Chart and u Chart: Controlling Defects per Unit, With Example

What the c chart and u chart are

The c chart tracks the number of defects per inspected unit when the unit size is constant; the u chart tracks defects per unit when the sample size changes from one sample to the next. Both are attribute control charts for defects, meaning every individual flaw found, even when one part has several.

This is different from the p chart, which counts defective parts (the part either passes or fails). These charts belong to SPC and start from a well-defined check sheet.

Visual map: when to use c, u, p or np

The map shows how to choose among the four attribute charts and carries the formulas and numbers of the example.

Defect or defective: which chart to use

A defective part is rejected as a whole. A defect is each individual flaw: a board with 3 cold solder joints has 3 defects but is 1 defective board. The choice of chart depends on what you count and on whether the sample is fixed or variable.

The c and u charts assume defects occur at random and independently within a fixed area or volume of opportunity, which matches the Poisson distribution. In that distribution the variance equals the mean, which is where the square root in the limit formula comes from.

c chart: formula and example

c-bar = Σ c ÷ k  |  UCL = c-bar + 3 × √c-bar  |  LCL = c-bar − 3 × √c-bar

Fictional example: on a paint line, one panel of the same area is inspected every hour and finish defects are counted (runs, dust, orange peel). Over 20 panels the defects were:

1. Total: Σ c = 99 defects in 20 panels.
2. Mean: c-bar = 99 ÷ 20 = 4.95 defects per panel.
3. Spread: √4.95 = 2.225; 3 × 2.225 = 6.67.
4. Limits: UCL = 4.95 + 6.67 = 11.62; LCL = 4.95 − 6.67 = -1.72, which is set to 0 because a count cannot be negative.
5. Reading: the largest value is 9 (panel 14), below 11.62. The process is stable, averaging about 5 defects per panel.

If the next panel shows 13 defects, the point exceeds 11.62 and signals a special cause: check the paint lot, the booth and the spray gun before continuing. With a low mean the LCL sits at zero, so the chart cannot flag improvement through points below it; use the other rules, such as a run of points below the center line.

u chart: limits that vary with sample size

u = c ÷ n  |  u-bar = Σ c ÷ Σ n  |  limits = u-bar ± 3 × √(u-bar ÷ n)

Fictional example: in final inspection of electronic boards, the number of boards checked per lot varies from 40 to 65, and solder defects are counted. Over 20 lots, 1,010 boards were inspected with 588 defects.

1. Mean: u-bar = 588 ÷ 1,010 = 0.582 defects per board.
2. One lot's value: lot 6 (n = 40, 19 defects): u = 19 ÷ 40 = 0.475. Lot 8 (n = 65, 41 defects): u = 41 ÷ 65 = 0.631.
3. Limits for lot 6: 3 × √(0.582 ÷ 40) = 0.362, so LCL = 0.220 and UCL = 0.944.
4. Limits for lot 8: for n = 65, LCL = 0.298 and UCL = 0.866.

The larger the sample, the tighter the limits. Across the 20 lots every u value falls inside the limits for its own n (the lowest is 0.444 and the highest 0.667), so the process is stable. If n varies little (say, within ±25% of the average), many practitioners use fixed limits based on the average n; with larger variation, compute limits point by point.

Step by step

1. Define what counts as a defect and the inspection unit (board, panel, meter of fabric), with a written criterion shared by all inspectors.
2. Choose between c and u: is the unit constant or variable?
3. Collect 20 to 25 samples and record the number of defects.
4. Compute the mean and limits and plot the points.
5. Investigate points beyond the limits, fix the cause and recalculate the limits without those points.
6. Prioritize the most frequent defect types with a Pareto diagram and log them on a check sheet.

For measured data such as diameter, the chart is the X-bar and R chart.

Common mistakes

• Using a c chart with a variable sample size, which creates false alarms.
• Confusing a defect with a defective part and picking the wrong chart.
• Defining the defect subjectively, with each inspector counting differently.
• Lumping very different defect types into one count, which hides the cause.
• Using the chart with very low means (near 1) without checking whether the normal approximation holds.
• Treating a negative LCL as a real value instead of setting it to zero.

When a point goes beyond a limit, handle it through corrective action (CAPA).

Frequently asked questions

What is the difference between a c chart and a u chart?

The c chart uses the number of defects with a constant unit size; the u chart uses defects per unit when the sample size varies.

What is the difference between a c chart and a p chart?

The c chart counts defects (several flaws per part); the p chart counts defective parts (pass or fail).

What is the c chart limit formula?

c-bar ± 3 × √c-bar. If the LCL comes out negative, set it to zero.

Why do the u chart limits vary?

Because they depend on n: limits = u-bar ± 3 × √(u-bar ÷ n). Larger samples give tighter limits.

When should I use an np chart?

When you count defective parts and the sample size is always the same.

How many samples do I need?

Twenty to 25 samples for the initial limits.

About the author

Vagner Soares

Lean Manufacturing & Behavioral Management Specialist

Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.

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