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Laney U' Chart¶
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QXL Stat Tools Tab > Control Charts > Attribute Charts > C, U, and Laney's U'
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Laney U' Chart How-To builds the same twenty subgroups twice, once as a u chart and once as a Laney U' chart, from sample data you copy off the page. The u chart marks six points out of control and the Laney U' chart marks none.
The Laney U' chart, read "U-prime", plots exactly what a u Chart plots and draws its limits differently. It multiplies the Poisson standard deviation by an estimate of how much the subgroups really scatter, so a process whose defect rate genuinely moves between subgroups gets limits that describe it rather than limits that flag most of it.
| Panel title on the worksheet | Laney U' Chart, one panel only |
| What it plots | defects divided by inspection size, the same as a u chart |
| Y axis | Defect Rate |
| X axis | Subgroup |
| Distribution behind the lines | Poisson, with the spread rescaled by \(\sigma_Z\) |
One ribbon entry opens three charts. The button reads C, U, and Laney's U', and the Variant: radio row at the top of the dialog selects which one you get. The worksheet it writes is named "Laney U Chart", without the apostrophe, because Excel strips characters it does not allow in a sheet name.
When to use it¶
- You would draw a u Chart, but its \(\sigma_Z\) diagnostic is well above 1, so the counts scatter more than Poisson variation alone explains and the u chart's limits are too tight.
- Large inspection sizes make this more likely, not less. Poisson limits tighten as the area of opportunity grows, so a large sample plus any real subgroup-to-subgroup movement in the rate gives a u chart that flags almost everything.
- The counterpart for defective items rather than defect counts is the Laney P' Chart.
How it works¶
The plotted statistic and the center line are the u chart's, unchanged: \(\bar{u}\) pooled across the baseline subgroups. Only the spread differs.
The factor is \(\sigma_Z\). Each subgroup is standardized, turned into how many Poisson standard deviations it sits from the center line. If the Poisson model were right, consecutive standardized values would move about as much as consecutive values from a standard normal distribution. \(\sigma_Z\) measures how much they actually move, as an average moving range over the usual constant, and the ratio is the correction:
- \(\sigma_Z\) near 1: the model fits, and a Laney U' chart is barely distinguishable from a u chart.
- \(\sigma_Z\) above 1: overdispersion. The Laney limits are wider than the u chart's by exactly that factor.
- \(\sigma_Z\) below 1: underdispersion. The Laney limits are narrower.
Every subgroup's standard deviation is multiplied by the one \(\sigma_Z\) estimated for the chart, then the limits are placed three of those either side of the center line by default, floored at zero. The center line never moves. Because the correction is a single multiplier, the band keeps its shape: it still steps with the inspection size, just further out.
An inspection size is still needed per subgroup, and it can come from a column or from the Constant box: a constant simply makes every subgroup's standard deviation the same, which straightens the band without changing the correction.
\(\sigma_Z\) is reported on the plain c and u charts too, where it is a diagnostic only. On this chart it is the arithmetic.
The exact formulas are on the Math Details page, and \(\sigma_Z\) is defined in Laney Overdispersion.
Output¶
The analysis writes one worksheet named Laney U Chart. Under GroupBy there is one worksheet per group, each named Laney U Chart - {GroupName}.
Worksheets are divided by GroupBy group, not by chart. Every chart belonging to a group is stacked down that one worksheet, one below the other, so selecting three measurement columns without GroupBy produces three charts on a single sheet rather than three sheets. The User Input block is written once at the top of each worksheet; everything after it repeats for each chart on the sheet.
The sheet carries a User Input block, the Laney U' Chart panel, a summary table listing u-bar, UCL, LCL, Total subgroups, OOC points and Outliers marked, and an Overdispersion Interpretation block headed by the computed \(\sigma_Z\) and then a three-row legend for reading it, which is a guide rather than a verdict on yours. Every row of that table is read at the LAST plotted point, so on a chart with varying inspection sizes read the steps off the chart. A split chart adds a Per-Split Breakdown block with a \(\sigma_Z\) per phase, and the pooled \(\sigma_Z\) above the legend is then not printed at all. There is no capability block and no histogram.
When the analysis finishes, the Control Chart task pane opens on the right. See Task Pane.