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Laney U' Chart How-To

This walkthrough builds the same twenty subgroups twice, once as a u chart and once as a Laney U' chart, because the difference between them is the whole point of this chart. On this data the u chart marks six of the twenty points out of control. The Laney U' chart marks none. Nothing about the process changed between the two runs.

If you have already read the Laney P' Chart How-To, this is the same argument for defect counts instead of defective items.

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 rolls of coated film. Each roll was inspected over a different number of square metres, and Defects is the number of coating faults found.

Defects Sq Metres
31 40
17 45
1 38
33 42
18 44
10 39
39 43
23 41
38 40
30 46
29 42
20 38
24 44
42 41
22 45
3 39
6 43
14 40
16 42
19 44

The rolls are all about the same size, and the fault counts run from 1 to 42. That range is the thing to keep in mind.

Steps

  1. 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 rolls in rows 2 through 21.

  2. Build it as a u chart first

    From the Excel ribbon: QXL Stat Tools Tab > Control Charts > Attribute Charts > C, U, and Laney's U'. Set Selected Range: to A1:B21 and press Next >. Leave Variant: on U, check Defects in Defects Column, set Inspection Size to Column and choose Sq Metres. Press Finish.

  3. Read the u chart, and then read its diagnostic

    Six of the twenty points are marked. Then look at the summary table: sigma_Z is 2.34.

  4. Build it again as a Laney U' chart

    Start the analysis again, exactly as before, but set Variant: to Laney U'. Press Finish.

What you should see

The u chart, on a worksheet named U Chart:

Value
u-bar 0.5203
sigma_Z 2.34
UCL at 40 square metres 0.8625
LCL at 40 square metres 0.1782
Points marked 6 of 20: rolls 3, 7, 9, 14, 16 and 17

The Laney U' chart, on a worksheet named Laney U Chart:

Value
u-bar 0.5203
sigma_Z 2.34
UCL at 40 square metres 1.3206
LCL at 40 square metres 0
Points marked none

The center line did not move. Both charts sit at 0.5203 faults per square metre: \(\sigma_Z\) scales the spread, not the level.

The band is exactly \(\sigma_Z\) times wider. The u chart reaches 0.3422 above the center line at 40 square metres; the Laney U' chart reaches 0.8003, which is 2.34 times as far. Every subgroup's standard deviation is multiplied by \(\sigma_Z\) before the limits are drawn, so the band keeps its stepped shape and simply moves out. The lower limit, 0.1782 on the u chart, is pushed below zero by the same factor and clamped there.

Six marked points became none, and note which ones they were: rolls 3, 16 and 17 were flagged for being unusually clean, not dirty. A u chart with a real lower limit can signal in both directions, and three of its six signals here were on the good side.

What sigma_Z measured

A u chart's limits come from the Poisson model, which assumes faults arrive independently at a constant rate per square metre. \(\sigma_Z\) tests that against the data: it standardizes each subgroup, 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 rolls behave as Poisson says, and a u chart describes them correctly.
  • Substantially above 1, which is 2.34 here: the rolls scatter more than twice as much as Poisson variation allows. Something varies from roll to roll beyond the chance arrival of faults, so the u chart's limits are more than twice too tight.
  • Substantially below 1: the reverse, and the limits are too wide to catch real signals.

For counts, overdispersion usually means the rate itself moves between subgroups: different batches of coating, different operators, different ambient conditions. The Poisson model has no room for that, and a Laney chart makes room by measuring how much of it there is.

The summary table gives you a legend, not a verdict

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 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 u chart above, all six 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.

Things to try next

  • Check the u chart's diagnostic before trusting its limits. Every c chart and u chart reports \(\sigma_Z\) whether or not you plan to use a Laney chart. It costs nothing and it is the fastest check on whether the Poisson model fits.
  • Notice what the widening cost you. The u chart had a real lower limit at 0.1782 and the Laney chart does not: the correction pushed it below zero and it was clamped. That is a genuine trade. A Laney chart stops crying wolf, and in exchange it can no longer flag an unusually clean roll at all.
  • Try the c variant on the same data to see the mistake it prevents. Set Variant: to C and the Inspection Size group greys out, so the chart plots raw fault counts as though every roll were the same size. The rolls here are close in size, so it will look plausible, which is exactly why it is worth seeing that the chart will not warn you.
  • Split it. If the roll-to-roll variation came from a known change partway through, Split Control Limits re-estimates \(\bar{u}\) and \(\sigma_Z\) inside each phase, and two well-separated phases may each return a \(\sigma_Z\) near 1.
  • Note that Force Straight Limits still works here. Unlike the Laney P' chart, where an assumed n does nothing, this variant does substitute one into the limits, so you can flatten a Laney U' band as well as widen it.

See Also