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u Chart How-To¶
This walkthrough builds a complete u chart from twenty inspection lots of different sizes. The varying size is the point: it is what makes the control limits step while the center line stays flat, and it is the difference between this chart and a c Chart.
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 batches of upholstered panels. Each batch contained a different number of panels, and every panel was inspected for surface blemishes. Blemishes is the total found in the batch, and Panels is how many panels were in it. One panel can carry several blemishes, which is why this is a defect chart and not a defective-item chart.
| Blemishes | Panels |
|---|---|
| 8 | 12 |
| 4 | 15 |
| 5 | 10 |
| 6 | 18 |
| 6 | 14 |
| 4 | 11 |
| 12 | 16 |
| 7 | 13 |
| 9 | 12 |
| 7 | 17 |
| 13 | 15 |
| 18 | 10 |
| 10 | 14 |
| 11 | 19 |
| 1 | 12 |
| 6 | 16 |
| 7 | 13 |
| 5 | 11 |
| 4 | 15 |
| 9 | 14 |
Batch 12 is the one to watch: 18 blemishes across only 10 panels.
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 batches in rows 2 through 21.
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Start the chart
From the Excel ribbon: QXL Stat Tools Tab > Control Charts > Attribute Charts > C, U, and Laney's U'.
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Give it the cells
In the Data Selection window set Selected Range: to A1:B21. Leave Data in Columns selected and First Row/Column is Header checked. Press Next >.
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Check the variant and the two columns
The dialog opens on the Data tab with Variant: already on U. Check Blemishes in Defects Column.
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Point the inspection size at the Panels column
In the Inspection Size group, select Column and choose Panels.
This is the step that is easy to miss
Inspection Size opens on Constant. Leave it there and every batch is charted as though it had the same area of opportunity, which turns a u chart into a c chart of the raw counts with no error message. If your sizes are in a column, you must switch to Column yourself.
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Finish
Press Finish. The chart is written to a new worksheet named U Chart and the Control Chart analysis pane opens on the right.
What you should see¶
One panel with twenty points, its Y axis titled Defect Rate.
| Value | |
|---|---|
| u-bar | 0.5487 |
| sigma_Z | 1.40 |
| LCL | 0 |
The center line is 0.5487 blemishes per panel, and it is pooled. It is 152 blemishes across 277 panels, not the average of the twenty batch rates. The chart draws the pooled figure.
The center line is flat and the limits step. The upper limit is at 1.2515 for the batches of 10 panels and at 1.0586 for the batch of 19. More inspection gives a tighter limit, because a rate measured over more panels is a more precise estimate. Ten distinct sizes give ten distinct upper limits.
That combination, a flat center line with a stepped band, is the u chart's signature. An np Chart steps both, and a c Chart steps neither.
The lower limit is zero for most subgroups but not all of them, and that is worth reading carefully. Three standard deviations below 0.5487 is negative at the smaller inspection sizes, so the limit is clamped to zero there. At the three largest sizes it lifts off zero: 0.0097 at 17 panels, 0.0249 at 18, and 0.0389 at 19. A batch can therefore be marked for being unusually clean, but only if it was one of the large ones. Batch 15, a single blemish across 12 panels, plots at 0.0833 against a lower limit of zero, so it is not a signal; the same rate in a 19-panel batch would have been.
Batch 12 is out of control. It plots at 18/10 = 1.8000 against its own upper limit of 1.2515, and is drawn in a different color. It is the only marked point, and note that its limit is one of the loosest on the chart: it had the fewest panels. It cleared a loose limit.
Where the numbers came from. For any subgroup the standard deviation of the plotted rate is
so it shrinks as the inspection size grows, and the limits are \(\bar{u}\) plus and minus three of those. Only \(n_i\) changes from point to point. There is no upper cap: unlike a proportion, a defect rate has no ceiling, because one panel can carry any number of blemishes. The formulas are on the Math Details page.
And there is a diagnostic you did not ask for. The summary table reports sigma_Z of 1.40 with a three-row Overdispersion Interpretation legend under it. That figure compares the scatter actually present with the scatter the Poisson model predicts. It is reported on every u chart and applied on none: applying it is what a Laney U' Chart does.
Only one test is running
An attribute chart opens with Rule 1 only. Rules 2, 3 and 4 are present and unchecked, and rules 5 through 8 are not offered at all on this dialog. So a u chart with no marked points means nothing broke that single test. See the Rules & Display section of the Options page.
Things to try next¶
- Flatten the limits with Force Straight Limits. Tick it on the Limits & Splits tab and type an Assumed n: of 14, near the average batch. The band becomes a pair of straight lines and the center line does not move, because \(\bar{u}\) never involved a single subgroup's size. Compare with an np chart, where the same control also flattens the center line.
- Switch the variant to C and see what you lose. Set Variant: to C and finish again. The Inspection Size group greys out entirely, and the chart plots the raw blemish counts against straight limits, as if every batch had been the same size. Batch 12's 18 blemishes and batch 14's 11 are then treated as comparable, though one came from 10 panels and the other from 19. See the c Chart.
- Switch to Laney U'. With \(\sigma_Z\) at 1.40 the Laney limits come out 40 percent wider. On this data batch 12 stays marked; on a chart where the diagnostic is larger the difference decides everything, and the Laney U' Chart How-To is built on data where a u chart marks six points and a Laney U' chart marks none.
- Turn on the zone lines. On an attribute chart a zone step is one full standard deviation of the plotted statistic, not a third of the band, so at the default multiplier of 3 the lines land at one and two sigma and the band edge at three.
- Turn on the pattern tests. Tick rules 2, 3 and 4 and rebuild. Nothing new is marked on this data.