Home / Statistical Tools / Control Charts / Laney P' Chart / Laney P' Chart How-To
Laney P' Chart How-To¶
This walkthrough builds the same twenty subgroups twice, once as a p chart and once as a Laney P' chart, because the difference between them is the whole point of this chart. On this data the p chart marks seven of the twenty points out of control. The Laney P' chart marks none. Nothing about the process changed between the two runs.
The data is on this page rather than in a file to download. Press Copy for Excel, then paste it into a blank worksheet.
The data¶
Twenty production lots of a stamped part. Around two thousand pieces were gauged from each lot, and Rejects is how many failed.
| Rejects | Gauged |
|---|---|
| 95 | 2000 |
| 120 | 2200 |
| 83 | 1900 |
| 84 | 2100 |
| 95 | 2050 |
| 43 | 1950 |
| 90 | 2150 |
| 39 | 2000 |
| 106 | 1900 |
| 110 | 2100 |
| 67 | 2000 |
| 76 | 2200 |
| 98 | 1950 |
| 69 | 2050 |
| 80 | 2100 |
| 44 | 1900 |
| 64 | 2150 |
| 14 | 2000 |
| 86 | 2050 |
| 96 | 1950 |
Nothing here looks wrong at a glance. The reject rate runs from under 1 percent to about 5.5 percent, and the samples are large.
Steps¶
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Put the data in Excel
Press Copy for Excel above the table. In Excel, open a blank worksheet, click cell A1, and press Ctrl+V. Headers land in row 1 and the twenty lots in rows 2 through 21.
-
Build it as a p chart first
From the Excel ribbon: QXL Stat Tools Tab > Control Charts > Attribute Charts > P, NP, and Laney's P'. Set Selected Range: to A1:B21 and press Next >. Leave Variant: on P, check Rejects in Defectives Column, set Sample Size to Column and choose Gauged. Press Finish.
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Read the p chart, and then read its diagnostic
Seven of the twenty points are marked. Then look at the summary table: sigma_Z is 3.00.
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Build it again as a Laney P' chart
Start the analysis again, exactly as before, but set Variant: to Laney P'. Press Finish.
What you should see¶
The p chart, on a worksheet named P Chart:
| Value | |
|---|---|
| p-bar | 0.0383 |
| sigma_Z | 3.00 |
| UCL at 2000 gauged | 0.0512 |
| LCL at 2000 gauged | 0.0254 |
| Points marked | 7 of 20: lots 2, 6, 8, 9, 10, 16 and 18 |
The Laney P' chart, on a worksheet named Laney P Chart:
| Value | |
|---|---|
| p-bar | 0.0383 |
| sigma_Z | 3.00 |
| UCL at 2000 gauged | 0.0770 |
| LCL at 2000 gauged | 0 |
| Points marked | none |
The center line did not move. Both charts sit at 0.0383, because \(\sigma_Z\) affects the spread and not the average.
The band is exactly \(\sigma_Z\) times wider. The p chart reaches 0.0129 above the center line at a sample of 2000; the Laney P' chart reaches 0.0387, which is 3.00 times as far. That is the entire adjustment: every subgroup's standard deviation is multiplied by \(\sigma_Z\) before the limits are drawn. The lower limit, 0.0254 on the p chart, is pushed below zero by the same factor and clamped there.
Seven marked points became none. Not because anything was suppressed, but because the p chart's limits were describing a process tidier than the one that produced this data.
What sigma_Z actually measured¶
A p chart's limits come from the binomial model, which assumes every piece in every lot had the same chance of being rejected. \(\sigma_Z\) tests that assumption against the data. It standardizes each subgroup, then measures how much those standardized values move from one subgroup to the next, and compares that with the movement the model predicts:
- Near 1: the lots behave as the binomial model says. A p chart is describing the process correctly.
- Substantially above 1, which is 3.00 here: the lots scatter three times more than binomial variation alone allows. Something varies from lot to lot beyond the piece-to-piece chance of a reject, so the p chart's limits are about three times too tight, and points fall outside them routinely.
- Substantially below 1: the reverse, and the limits are too wide to catch real signals.
With a sample of two thousand pieces this matters more, not less. Binomial limits tighten as the sample grows, so a large sample plus any real lot-to-lot variation gives a chart that flags almost everything. That is the situation this chart exists for.
The summary table gives you a legend, not a verdict
Under the numbers, both charts print an Overdispersion Interpretation block with three rows: near 1.0, significantly above 1.0, and significantly below 1.0, each with a meaning and a recommendation. All three rows are always printed. Quantum XL does not tell you which row your \(\sigma_Z\) falls into, and it does not define "significantly".
Only one test is running
An attribute chart opens with Rule 1 only. On the p chart above, all seven marked points are outside the limits; no pattern test was involved. Rules 2, 3 and 4 are available and unchecked, and rules 5 through 8 are not offered on this dialog at all.
Things to try next¶
- Check the p chart's own diagnostic before trusting its limits. Every p chart and np chart reports \(\sigma_Z\) whether or not you plan to use a Laney chart. The number costs nothing and it is the fastest check on whether the binomial model fits.
- Compare the two lower limits. The p chart has a real lower limit at 0.0254, which is why lots 6, 8, 16 and 18 were marked: they were flagged for being unusually good. Widening the band pushed that limit below zero and clamped it at 0, so the Laney chart has nothing below to breach. On a chart of this kind, a clamped lower limit means low rejects can no longer be signalled at all.
- Try the np variant on the same data. It plots counts rather than proportions and reports the same \(\sigma_Z\), so it inherits the same problem and the same fix.
- Split it. If the lot-to-lot variation came from a known change partway through, Split Control Limits re-estimates \(\bar{p}\) and \(\sigma_Z\) inside each phase, and a Laney chart with two well-separated phases may return a \(\sigma_Z\) near 1 in both.
- Note what Force Straight Limits does here: nothing. The engine never substitutes an assumed sample size on the Laney P' variant, so the limits keep stepping with the real sample sizes however that box is set.