Home / Lean Six Sigma / SPC & process capability
When the result is pass/fail or a defect count, capability is measured by the rate: how much comes out defective, how sure you are, and whether you can claim the process meets the customer target. For each sample (lot, shift, day) you need the number of defectives or defects and how much was inspected; you get the rate with its uncertainty range, the Z equivalent to Cpk, the P or U chart to see whether the process is stable and how much more to inspect to prove the target.
Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.
Key: % defective (or DPU) = average rate of all samples together; lower and upper CI = range where the true process rate should lie; PPM = defectives per million; process Z = the same rate expressed as a distance in standard deviations (higher is better; Z = 3 equals 0.135%).
Proportion defective of each sample, center line and 3σ limits (stepped when the size varies). Red points fail a test; the number above is the test.
How to read: horizontal axis = samples in order; vertical axis = defective rate (or defects per unit) of each sample. Middle line = average rate; dashed = control limits (how much the rate varies by chance alone). Red numbered points = special-cause signals. Good: all points inside, with no long runs on one side.
Estimate recomputed at each sample, with confidence bounds. When the line levels off, there are enough samples.
How to read: the solid line is the rate computed with the samples accumulated up to each point; the dashed lines are the confidence band, which narrows with more data. When the line goes flat, the estimate no longer depends on the next samples. A line still rising or falling at the end = collect more data.
Points scattered at random around the mean show that the rate does not depend on sample size; the funnel marks the 3σ limits.
How to read: when the size varies: horizontal axis = size of each sample, vertical = rate; the dashed funnel shows that small samples vary more. Points with no pattern around the mean are good; points rising or falling with size mean the rate is not constant. When the size is fixed, the chart compares observed and expected counts: points near the diagonal = adequate model.
Distribution of the rate per sample. In a capable process almost all bars are left of the target.
How to read: each bar counts how many samples had a rate in that bin. Line p̄ (or ū) = mean; green line = target. Good: almost all bars left of the target. Isolated bars far to the right are bad samples worth investigating.
Confidence bounds of the rate as the total inspected grows, at the current rate. Where the upper bound crosses the target, the sample starts to prove the process meets it.
How to read: horizontal axis = total inspected (log scale, each mark multiplies by 10); lines = upper and lower bounds of the rate, which close in with more inspection. The "today" point is where you are; the "prove" diamond is where the upper bound crosses the green target. If the average rate is above the target, no amount of inspection proves the target.
Estimated rate applied to annual production, with the confidence-interval range.
Rate, limits and z of each sample; z = distance from the center line in standard deviations.
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It measures how much a process produces out of specification when the result is not a continuous measurement but pass/fail (binomial) or a defect count (Poisson). Instead of Cpk you use % defective, PPM, process Z or defects per unit, always with a confidence interval.
Binomial when each part is classed as good or defective and you know how many were inspected. Poisson when you count defects and one unit can have several: bubbles per panel, flaws per meter of weld, errors per invoice. If a part with 3 defects counts as 1 defective, it is binomial.
It is the distance, in standard deviations of the normal curve, that would leave the same proportion defective in the tail: 2.3% equals Z = 2; 0.135%, Z = 3. It lets you compare an attribute process with the Cpk of a variables process (Cpk ≈ Z/3 for a centered, one-sided process).
Because the estimate comes from a sample. If the upper confidence bound exceeds the target, the true rate may still be above it. The "How much to inspect" chart shows how many parts, at the current rate, would make the proof possible.
The reference assistant asks for at least 25 samples, each with n·p̄ of at least 0.5 (in the connector example, at 2.3% defective, at least 22 parts per sample). With very low rates, combine periods or use the G chart for rare events.
It compares the variation of rates between samples with what the binomial or Poisson predicts. Above 130%, samples vary more than the model says (common with samples of thousands of parts) and the P or U chart gives false alarms: use the Laney P′ or U′ chart.
The formulas are those published for the binomial and Poisson capability reports: exact Clopper-Pearson interval via the F distribution and exact Poisson via chi-square. With 511 defectives in 5,342 items, the page gives 9.57% (CI 8.79% to 10.39%), 95,657 PPM and Z 1.31, the same figures as the published unanswered-calls example.
Important notice: Indus Metrics tools produce estimates and pre-sizing results from the data you enter and from published engineering methods. They do not replace design, calculations or reports signed by a licensed professional engineer (PE), nor verification against the applicable codes, standards and regulations, such as OSHA rules and the ASME codes. Material and fuel properties vary: check manufacturer data or your own process analyses before deciding. By using this site you agree to the Terms of Use.
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