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Z-MR How-To¶
This walkthrough builds a complete Z-MR chart from twenty-five measurements of four different parts, none of which is the same size as another. That is what the chart is for: every measurement has its own part's mean subtracted before it is plotted, so parts of any size share one chart.
Two things in this data are worth having seen once. The lines on both panels are fixed numbers that owe nothing to the data. And one part is run twice, which makes two runs but only one mean, and the chart shows the consequence plainly.
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-five turned shafts from a job shop, measured in millimetres. Four part numbers, made in short runs, in the order the shop ran them. Part S-100 was run at the start of the week and again later in the week.
| Part | Diameter |
|---|---|
| S-100 | 12.013 |
| S-100 | 12.004 |
| S-100 | 12.018 |
| S-100 | 11.988 |
| S-100 | 11.997 |
| S-220 | 24.982 |
| S-220 | 24.972 |
| S-220 | 24.983 |
| S-220 | 24.995 |
| S-220 | 24.995 |
| S-340 | 39.970 |
| S-340 | 40.031 |
| S-340 | 40.101 |
| S-340 | 40.038 |
| S-340 | 39.987 |
| S-100 | 12.049 |
| S-100 | 12.041 |
| S-100 | 12.072 |
| S-100 | 12.049 |
| S-100 | 12.052 |
| S-480 | 60.009 |
| S-480 | 60.013 |
| S-480 | 60.013 |
| S-480 | 60.015 |
| S-480 | 59.999 |
The Part column is not a measurement and you do not check it in the measurement list. It goes in the Part Column slot instead.
Two features to keep in mind while reading the finished chart:
- The four part numbers sit at 12, 25, 40 and 60 millimetres. No ordinary chart could plot them together.
- S-100 appears in two blocks, rows 1 to 5 and rows 16 to 20. That is five runs of four parts, and the second S-100 block averages about 0.05 mm higher than the first.
Steps¶
-
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-five measurements in rows 2 through 26.
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Start the chart
From the Excel ribbon: QXL Stat Tools Tab > Control Charts > Short Run > Z-MR (Z Score with Moving Range).
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Give it the cells
In the Data Selection window set Selected Range: to A1:B26. Leave Data in Columns selected and First Row/Column is Header checked. Press Next >.
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Check the measurement column
The Z-MR Chart dialog opens on the Data tab. Check Diameter in the measurement list. Leave Part unchecked: a column can only do one job at a time.
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Choose the part column
Set Part Column to Part.
Finish stays unavailable until both are set
Z-MR is the one variable chart that refuses to run with nothing selected. Finish becomes available only when a measurement column and a part column are both checked, and no message tells you which of the two is missing.
-
Leave the estimation settings alone
On the Limits and Estimation tab, Estimate sigma: is already on Constant, the Moving-range statistic: on Average moving range, and Moving-range length (L): on 2. Those are the defaults this walkthrough uses.
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Finish
Press Finish. The chart is written to a new worksheet named ZMR, with no hyphen and no "Chart" in the tab name, and the Control Chart analysis pane opens on the right.
What you should see¶
Two stacked panels with twenty-five positions each, and vertical dividers with the part names where one run ends and the next begins. There are four dividers, at rows 6, 11, 16 and 21.
| Center line | Lower control limit | Upper control limit | |
|---|---|---|---|
| Z Chart | 0 | -3 | +3 |
| Moving Range Chart | 1.128 | 0 | 3.686 |
Those six numbers are not calculated from your data. They are the same on every Z-MR chart anyone has ever drawn with these settings. The Z panel is fixed at 0 and plus or minus 3 by definition, because the plotted value is already in standard deviations; there is no Number of standard deviations (k): box in this dialog to change them. The Moving Range panel's lines are the moving range constants themselves, 1.128 and 3.686 at a length of 2, because the moving range of standardized values has a known distribution. Nothing in your measurements moves any of them.
What the data does decide is where the points fall. Each part's own mean is subtracted, and one standard deviation is estimated for the whole chart:
| Part | Rows | Mean subtracted |
|---|---|---|
| S-100 | 1 to 5 and 16 to 20 | 12.028 |
| S-220 | 6 to 10 | 24.985 |
| S-340 | 11 to 15 | 40.025 |
| S-480 | 21 to 25 | 60.010 |
The estimated standard deviation is 0.0223 mm. It comes from the moving ranges of the deviations, not of the measurements: subtract each row's part mean first, then take moving ranges of that sequence of deviations. Those twenty-four moving ranges average 0.0251, and 0.0251 divided by the constant 1.128 gives 0.0223. Under the default Constant setting a moving range is allowed to span a part change, because both rows have already had their own part's mean taken off.
One measurement is out of control. Row 13 is an S-340 at 40.101 mm. Its part mean is 40.025, so it sits 0.076 above it, and 0.076 divided by 0.0223 is +3.39, past the upper limit of +3. It is the only marked point on the chart. No other measurement gets past 2.49.
The Moving Range panel is quiet, and it is worth seeing why, because it looks wrong at first. The largest plotted moving range is 3.14, against an upper limit of 3.686. A jump of 3.39 standard deviations breaches the Z panel and a moving range of 3.14 does not breach its own panel: the moving range limit is the wider of the two, so a single stray measurement can be marked above and unremarked below.
Five runs mean five blank positions on the Moving Range panel, at rows 1, 6, 11, 16 and 21. A moving range needs a previous point from the same run, and the first row of each run has none. That is also why the panel has twenty positions with values and not twenty-four.
S-100's two runs sit on opposite sides of the center line. Its first block averages 12.004 and its second averages 12.053, but only one mean is subtracted from both, the mean of all ten S-100 rows, 12.028. So the first run plots entirely below zero, from -0.46 to -1.81, and the second entirely above it, from +0.57 to +1.96. Nothing drifted inside either run. The pooled mean sits between the two levels, and each run is measured against it.
Which test flagged the point
All eight out-of-control tests are on by default and Quantum XL does not report which one marked a given point. On this chart rules 5 through 8 apply to the Z panel only, and the Moving Range panel runs rules 1, 2 and 3 and nothing else, so the alternating test never fires there however the dialog is set. Note also that a part change does not restart the counting tests on the Z panel; only a blank point does. See Out-of-Control tests and the Rules and Display section of the Options page.
Things to try next¶
- Estimate a separate sigma for each part. On the Limits and Estimation tab choose By parts. Each part is then scaled by its own scatter: 0.0176 for S-100, 0.0073 for S-220, 0.0543 for S-340 and 0.0049 for S-480. Watch what happens to row 13: the wild measurement is inside the estimate that scales it, so S-340's sigma more than doubles and row 13 drops from +3.39 to +1.39 and is no longer marked at all. Four of S-100's second-run points are marked instead, by the pattern tests. With four or five measurements per part, a per-part estimate is built from very little.
- Estimate a separate sigma for each run. Choose By runs and S-100's two blocks stop sharing an estimate: they get 0.0137 and 0.0144 while the other three runs keep 0.0073, 0.0543 and 0.0049. The mean subtracted is still S-100's pooled mean, 12.028, because the sigma setting never changes the mean.
- Switch to the median moving range. Choose Median moving range and the estimate drops to 0.0157, because the median ignores the two large moving ranges around row 13. Row 13 rises to +4.81 and row 11 joins it outside the limits. The Moving Range panel's own lines change too, to a center of 0.954 and an upper limit of 3.116, since those constants belong to the method rather than to the data.
- Give S-100 a nominal instead of a mean. In the Part Overrides grid type
12.000into Override Mean for S-100 and leave its Override Sigma blank. Both S-100 runs are then measured against the nominal 12.000 rather than against their pooled mean, so the first run moves up to sit just above zero and the second stays high. This is the setting to reach for when a part has a specified target and the run-to-run average is not it.