Learn · updated 2026-10-22
Laney p′ chart: when your p chart says everything is out of control
A p chart assumes the only variation between samples is binomial chance. With thousands of items per sample that chance variation is tiny, so the limits squeeze in around the average and ordinary month-to-month differences look like special causes. The Laney p′ chart measures how much your samples really vary and widens the limits by that much.
The short answer. Turn each sample into a z-score on the p chart, measure the spread of those z-scores with moving ranges (σz = MR̄ / 1.128), and multiply the p chart's sigma by σz.
σpᵢ = √(p̄(1 − p̄)/nᵢ)zᵢ = (pᵢ − p̄) / σpᵢσz = MR̄ of the z's / 1.128UCL = p̄ + 3 · σpᵢ · σzLCL = p̄ − 3 · σpᵢ · σz
If σz is close to 1, the data are binomial after all and the Laney chart is the same as the p chart.
Worked example: 16 months, 9,000 to 22,000 per month
SPC for Excel's Laney p′ article uses this data from The Health Care Data Guide: the share of health-plan members managed by phone each month.
| Month | Members (n) | By phone | % |
|---|---|---|---|
| Jan-07 | 8,755 | 3,852 | 44.00% |
| Feb-07 | 9,800 | 4,100 | 41.84% |
| Mar-07 | 17,000 | 7,083 | 41.66% |
| Apr-07 | 16,400 | 7,339 | 44.75% |
| May-07 | 19,500 | 9,406 | 48.24% |
| Jun-07 | 19,800 | 9,310 | 47.02% |
| Jul-07 | 21,200 | 7,250 | 34.20% |
| Aug-07 | 22,300 | 10,400 | 46.64% |
| Sep-07 | 21,600 | 9,250 | 42.82% |
| Oct-07 | 20,500 | 9,950 | 48.54% |
| Nov-07 | 18,700 | 9,846 | 52.65% |
| Dec-07 | 18,900 | 9,854 | 52.14% |
| Jan-08 | 14,300 | 8,034 | 56.18% |
| Feb-08 | 14,800 | 8,162 | 55.15% |
| Mar-08 | 14,500 | 8,122 | 56.01% |
| Apr-08 | 14,600 | 8,200 | 56.16% |
- p̄ = 130,158 / 272,655 = 0.4774.
- Ordinary p chart: with n in the tens of thousands, σp is only about 0.53% even in the smallest month (Jan-07), so its limits are 46.1% to 49.3%. 13 of 16 months fall outside their limits. A chart that flags almost everything tells you nothing.
- Laney p′: the z-scores vary far more than 1: σz = 9.23. So the limits are 9.2 times wider. Jan-07: 32.96% to 62.52%. Jul-07: 38.24% to 57.24%.
- Only Jul-07 (34.20%) is outside, below its lower limit. That is the month worth investigating.
When to use it
- Your samples are large (hundreds to thousands or more per point): pick accuracy per day, claims per month, calls handled per week, defective units per shift on a high-volume line.
- The p chart flags many points that don't feel special, and nobody looks at it any more.
- Check σz: well above 1 means overdispersion (use Laney); below 1 is underdispersion, which usually means the samples aren't independent.
The same idea for defects per unit is the Laney u′ chart, built from the u chart.
A caution
Laney's correction treats the extra month-to-month variation as part of the normal process. That is the right default for big administrative or logistics data, but if the variation comes from a real cause you could remove (one shift, one supplier), stratify the data first: chart each source separately.
Sources
- SPC for Excel, “Laney p′ Control Chart” (example data and limits)
- D. B. Laney (2002), “Improved Control Charts for Attributes”, Quality Engineering 14(4), 531–537
- L. P. Provost and S. K. Murray, The Health Care Data Guide (source of the example data)
- NIST/SEMATECH e-Handbook, §6.3.3.2 Proportions Control Charts