An np-chart is an attribute control chart that plots the number of defective units in each fixed-size sample. It tells you whether the count of defectives is stable, in statistical control, or whether a special cause has pushed it up or down. It requires a constant sample size, and it is built on the binomial distribution.
It matters because a lot of inspection is pass/fail, not a measurement. You pull 100 units, you count how many are defective, and you want to know whether today's count of 6 is normal noise or a real change. Staring at the raw numbers will fool you, 6 might be perfectly ordinary or a genuine alarm depending on your process average. The np-chart puts statistical limits around the count so the answer is objective. This guide covers what the chart is, when to use it, the formulas, and how it differs from its close relative the p-chart.
What is an np-chart?
An np-chart is one of the attribute control charts in statistical process control used when your data is a count of defective units out of a sample of fixed size. The name comes from its math: n is the sample size and p is the proportion defective, so n × p is the expected number of defectives per sample, exactly what the chart plots. Each point on the chart is the count of defective units in one subgroup, and the control limits tell you the range that count should stay in if nothing but common-cause variation is present.
The key word is defective meaning a whole unit that fails, conforming or not, a binary verdict per unit. That distinguishes the np-chart from charts that count defects, where a single unit can have several flaws. Because each unit is either good or bad, the count of bad units in a fixed sample follows the binomial distribution, and the np-chart's limits come straight from that distribution. It belongs to the same family of control charts as the variables charts, but its statistics are attribute statistics.
When do you use an np-chart?
Use an np-chart when three things are true: your data is attribute (pass/fail) data counting defective units your sample size is constant from subgroup to subgroup, and each unit is judged independently good or bad. A packaging line pulling exactly 200 units every hour and counting how many fail a seal check is the textbook case. So is an electronics line testing a fixed panel of boards each shift and counting failures, or a fill line inspecting a set 50 bottles per lot for cap defects. In each case the question is the same: is today's count of bad units ordinary variation, or has something in the process changed?
The constant sample size is the deciding factor. It is what lets the np-chart plot the raw count instead of a proportion, because with n fixed, the count and the proportion carry the same information and the count is easier to read on the floor. The moment your sample size varies, different batch sizes, whatever came off the line that hour, the raw count is no longer comparable point to point, and you switch to a p-chart. If you are counting defects rather than defective units, or measuring a variable, a different chart applies. The table below places the np-chart among its neighbors.
| Chart | What it plots | Sample size |
|---|---|---|
| np-chart | Number of defective units | Constant |
| p-chart | Proportion (fraction) defective | Constant or varying |
| c-chart | Number of defects (constant area) | Constant |
| u-chart | Defects per unit | Constant or varying |
What are the np-chart formulas?
The formulas come from the binomial distribution and are simple to compute. First find the average proportion defective across all your samples, then the centerline and limits follow:
The average proportion defective is p̄ = (total defectives) ÷ (total units inspected) = Σnp ÷ (k × n), where k is the number of subgroups and n is the constant sample size. The centerline is the average number defective, CL = np̄. The control limits are np̄ ± 3√(np̄(1−p̄)), where the square-root term is the standard deviation of the binomial count. The lower control limit is set to zero whenever the formula produces a negative number, because you cannot have fewer than zero defectives.
A worked example makes the arithmetic concrete. Say you pull a sample of n = 200 units every hour for 25 hours and count 150 defective units in total. The average proportion defective is p̄ = 150 ÷ (25 × 200) = 0.03, so the centerline is np̄ = 200 × 0.03 = 6 defectives per sample. The standard deviation term is √(6 × (1 − 0.03)) = √5.82 ≈ 2.41, so three of them is about 7.2. The upper control limit is 6 + 7.2 = 13.2, and the lower limit would be 6 − 7.2 = −1.2, which is negative, so it is set to 0. On that chart, any hour with 14 or more defectives is an out-of-control signal, and because the lower limit is zero, the chart cannot flag an unusually good hour on the low side, a reminder that a floored lower limit gives up half the chart's sensitivity.
How do you build an np-chart?
The build is a short, repeatable sequence, the discipline is in keeping the sample size fixed and gathering enough subgroups before you set limits:
- Fix the sample size. Choose a constant n, large enough that you typically catch a few defectives per sample, a common guideline is n×p̄ of at least about 5 so the binomial is well behaved.
- Collect the subgroups. Gather at least 20 to 25 subgroups of size n, counting the number of defective units in each.
- Compute the average proportion. Add all the defectives and divide by the total units inspected to get p̄.
- Set the centerline. Multiply to get np̄, the average number defective per sample, that is your centerline.
- Compute the control limits. Apply np̄ ± 3√(np̄(1−p̄)); floor the lower limit at zero if it comes out negative.
- Plot every point. Plot the count of defectives for each subgroup against the centerline and limits, in time order.
- Interpret the signals. Read points outside the limits, and pattern rules, as special causes; investigate them before recalculating limits.
By the numbers. The np-chart is a standard binomial attribute chart, and its centerline and ±3-sigma limits are documented in NIST's engineering statistics handbook, which derives the control limits np̄ ± 3√(np̄(1−p̄)) from the binomial distribution and notes the lower limit is set to zero when the calculation is negative (NIST/SEMATECH e-Handbook, proportions control charts). The convention of gathering at least 20–25 subgroups before fixing limits, and keeping n×p̄ large enough for the binomial approximation to hold, is standard SPC practice covered in the same handbook (NIST/SEMATECH e-Handbook, Process Monitoring).
How is an np-chart different from a p-chart?
They chart the same thing two ways. The p-chart plots the proportion (fraction) defective, defectives divided by sample size, while the np-chart plots the raw count of defectives. When the sample size is constant, the two charts are equivalent: the np-chart is just the p-chart scaled up by n, and they will flag exactly the same points as out of control. The np-chart is preferred in that case because a whole number like “6 defective” is easier for an operator to record and read than a decimal like “0.06.”
The real fork is the sample size. A p-chart can handle a varying sample size, it recomputes the control limits for each subgroup based on that subgroup's n, so the limits step in and out as the sample size changes. The np-chart cannot do that; with a fixed centerline of np̄ it assumes n never moves. So the rule is simple: constant sample size, use the easier-to-read np-chart; varying sample size, use the p-chart. Both count defective units and both rest on the binomial distribution; only the y-axis and the sample-size flexibility differ.
Whichever attribute chart you run, its value depends on someone acting on the signal in time. The interpretation of out-of-control points uses the same logic as the Nelson rules and the broader distinction between common and special cause variation and the choice of chart follows from attribute versus variable inspection. When defect counts and chart signals surface in real time at the line instead of in a report the next morning, an operator can chase the special cause while it is still present, which is the shift the team at CLS made when quality data moved from next-morning paperwork to something visible during the shift. That live feedback is part of what Harmony gives a plant: a control chart only protects you if the out-of-control point is seen while there is still time to act.