U-Chart: Defects per Unit, Varying Sample Size

A u-chart is an attribute control chart plotting the average defects per unit over time, used when the inspection unit (area of opportunity) varies between samples. Because its limits depend on each sample's size, they are recomputed per subgroup, giving a stepped look. The single statistic behind it is u-bar, the average defects per unit across the baseline, and every sample is judged against its own limits.

The u-chart is the tool for counting flaws when the amount you inspect keeps changing. One shift you check a 40-square-meter batch of coated sheet; the next, a 65-square-meter batch. You cannot compare raw defect counts across those two, because the bigger batch has more room for defects to land. Dividing by the inspected size fixes that, and the u-chart is the control chart built around that division. This guide covers when a u-chart fits, why its limits breathe in and out with sample size, how to build and read one, and where it sits next to the c-chart.

What is a u-chart used for?

A u-chart monitors defects per unit when the inspection unit is not constant, so you can compare a small sample and a large one on the same scale. You count the total defects in each sample, divide by that sample's size, and plot the rate. The rate is what makes samples of different sizes comparable: three defects in a 10-square-meter roll (0.3 per square meter) is a worse result than three defects in a 30-square-meter roll (0.1 per square meter), and the u-chart shows that plainly where a raw count would hide it.

The "unit" is whatever you decide the area of opportunity is: a square meter of fabric, a linear meter of extrusion, one wiring harness, one hour of production, a thousand solder joints. The sample is some number of those units, and that number is n. On a c-chart, n is locked; on a u-chart, n is allowed to float. Everything else about the two charts is the same family: both count defects, both rest on the Poisson distribution, both use Shewhart's three-sigma logic.

Why do u-chart control limits change for every sample?

The limits move because the standard error of a rate depends on how much you inspected: a rate estimated from a big sample is more precise, so its limits sit closer to the center line, and a rate from a small sample is noisier, so its limits spread wider. That is the whole reason a u-chart looks stepped instead of flat.

The math follows from the Poisson distribution, where variance equals the mean. Let u-bar be the average defects per unit across your baseline and n be the size of a given sample. Then:

• Center line (CL): u-bar, the total defects divided by the total units inspected across the baseline, Σc ÷ Σn.
• Upper control limit for sample i: UCL_i = u-bar + 3√(u-bar ÷ n_i).
• Lower control limit for sample i: LCL_i = u-bar − 3√(u-bar ÷ n_i), floored at 0 whenever the formula goes negative.

The center line stays put; only the √(u-bar/n) term changes. When n is large, that term is small and the limits pinch toward the center; when n is small, the term grows and the limits flare out. If your sample sizes only wander a little (say within 25 percent of the average), many teams simplify by using the average sample size to draw one flat pair of limits, then recompute exact limits only for the points that land near the boundary. When sizes vary a lot, compute the limits for every sample.

How do you build a u-chart?

Building a u-chart is a short routine once you have defined the inspection unit and decided how you will measure each sample's size.

1. Define the inspection unit and the size measure. State exactly what one unit is (one square meter, one harness, one hour) and how you will record n for each sample. The unit is your area of opportunity, and n counts how many of those units a sample contains.
2. Collect a baseline of counts and sizes. For each of roughly 20 to 25 samples taken while the process runs normally, record both the defect count c and the sample size n. Count every defect, not just whether the sample had any.
3. Compute the plotted rate for each sample. Divide each count by its size: u_i = c_i ÷ n_i. These rates are the points you will plot.
4. Compute the center line. Add all the defects and divide by all the units: u-bar = Σc ÷ Σn. Do not average the individual rates; pool the totals, so bigger samples carry proportional weight.
5. Compute per-sample limits. For each sample, UCL_i = u-bar + 3√(u-bar/n_i) and LCL_i = u-bar − 3√(u-bar/n_i), floored at zero. These give the stepped boundaries.
6. Plot in time order and extend forward. Draw the center line, plot each rate against its own limits, then keep plotting new samples against the frozen u-bar. Investigate any point outside its limits or any non-random pattern.

A worked example keeps it concrete. Say you inspect batches of coated sheet and count surface defects, with the unit being one square meter. Over the baseline you inspect 500 square meters total and find 75 defects, so u-bar = 75 ÷ 500 = 0.15 defects per square meter. For a 40-square-meter batch, √(0.15/40) = √0.00375 ≈ 0.061, so UCL = 0.15 + 3(0.061) = 0.33 and LCL = 0.15 − 0.18 = −0.03, floored to 0. For a 65-square-meter batch, √(0.15/65) ≈ 0.048, so UCL = 0.15 + 0.14 = 0.29, a tighter limit, because the larger batch estimates the rate more precisely. A 40-square-meter batch with 15 defects plots at 0.375, above its 0.33 limit, and signals.

How do you read a u-chart once it is running?

Read a u-chart the way you read any Shewhart chart, but always against each point's own limits. A process in control shows random scatter around the flat center line, every point inside its own stepped boundary, with no runs or trends. A point above its upper limit is the clear signal, a defect rate higher than the process normally produces, already adjusted for how much you inspected.

Three cautions are specific to the u-chart. First, judge each point against its own limits, not a single line; a point that would clear a flat limit can still breach the tighter limit that a large sample earns. Second, the run-based patterns from the Western Electric rules get awkward when limits change every sample, because the 1-, 2-, and 3-sigma zones move too; if you rely on zone tests, use the standardized u-chart, which plots each point's Z-score against fixed lines at 0 and ±3 so the zones hold still. Third, keep u-bar reasonably supported: if your average rate times sample size stays very small, the Poisson approximation is skewed and the three-sigma limits are rough, so enlarge the unit or aggregate more area per sample.

When should you use a u-chart instead of a c-chart?

Use a u-chart when the inspection unit varies and a c-chart when it is constant; both count defects and both are Poisson-based, so the only question is whether n is fixed. If every sample is one door, one board, or one fixed-length roll, the c-chart's flat limits are simpler and correct. The moment sample size floats, variable batch areas, variable production hours, variable numbers of assemblies per lot, the u-chart is the honest choice.

The Poisson math and where it comes from

The u-chart is standardized and well documented; the facts worth keeping straight are few.

• The counts-of-defects charts and their three-sigma limits are documented in the NIST/SEMATECH e-Handbook of Statistical Methods which gives the c- and u-chart formulas for constant and varying areas of opportunity (NIST/SEMATECH e-Handbook, 6.3.3.2).
• Control charts originate with Walter Shewhart at Bell Telephone Laboratories in 1924, and the three-sigma limit is his economic balance between false alarms and missed signals (ASQ, Control Chart).
• For a Poisson variable, variance equals the mean, so the standard error of a defects-per-unit rate is √(u-bar/n), the term that makes the u-chart's limits depend on sample size (ASQ).

Where the u-chart fits your quality system

The u-chart earns its keep on count-based inspections where lot size or inspected area refuses to hold still: coated and rolled goods, wiring and cabling, defects per production hour, errors per batch of documents. It sits inside statistical process control as the "how many flaws per unit of opportunity" tool, next to the c-chart for fixed units and the p and np charts for pass-fail defectives. A control chart tells you when the rate shifted; pairing it with a Pareto chart of defect types tells you what to fix first, and a rate that traces to a repeatable cause belongs in your defect tracking records, not a quiet rework bin. Where the defect rate feeds a capability conversation about a measured characteristic, process capability (Cpk) picks up the variables side.

The hard part is rarely the arithmetic, it is capturing an honest count and the sample size at the station, every time. A u-chart drawn from clipboard tallies is only as current as the last time someone keyed them in, and if the sample size is missing, the rate is meaningless. Digitizing the defect count and the unit size at the point of inspection, the way Harmony's live capture and shop-floor visibility tooling does, keeps the chart current and the numbers searchable across shifts, feeding the same quality trends your review meetings already run. See how a connected floor turns raw counts into decisions in the CLS field story. The u-chart's whole promise is fairness: it refuses to let a big batch look worse than a small one just because it offered more room for defects to appear.