Home / Shared Math Details / Control Chart Math / Control Limits and Zones
Control Limits and Zones¶
This page defines how Quantum XL places the control limits and sigma zones on a control chart once a chart has supplied its center line and its standard deviation. Every chart provides its own center and sigma; the limit and zone construction here is shared.
Notation¶
| Term | Description |
|---|---|
| \(\text{CL}\) | the center line supplied by the chart |
| \(\sigma_i\) | the chart's standard deviation at subgroup \(i\) (may vary by subgroup) |
| \(k\) | the sigma multiplier (number of standard deviations), default \(3\) |
| \(\text{UCL}, \text{LCL}\) | upper and lower control limits |
| \(\sigma_1\) | the one-sigma zone width recovered from the limit spacing |
| \(A_i^{+}, A_i^{-}\) | the upper and lower outer zone lines at point \(i\) (the boundary between Zone A and Zone B) |
| \(B_i^{+}, B_i^{-}\) | the upper and lower inner zone lines at point \(i\) (the boundary between Zone B and Zone C) |
The \(i\) subscript on the zone lines marks a per-point value, so the lines follow stepped limits rather than running straight.
Control limits¶
When the standard deviation varies by subgroup (for example an attribute chart with unequal subgroup sizes), \(\sigma_i\) changes from point to point, so the limits are stepped rather than straight. Attribute charts floor the lower limit at \(0\), since a count or proportion cannot be negative. The p and Laney P' charts also cap the upper limit at \(1\) (a proportion cannot exceed one), and the np chart caps it at the subgroup size \(n_i\); the c, u, and Laney U' charts apply no upper cap.
Sigma zones¶
Most charts draw no zone lines, and none do at a sigma multiplier other than 3
Read this section before using Tests 5 through 8 or the Westgard zone rules. A test that reads a zone runs only where the zone lines exist.
Two conditions decide whether a chart carries zone lines at all, and both must hold.
- The panel plots a measurement against a band that is symmetric about the center line. That is the Xbar panel, the Individuals panel, and the Z panel of a Z-MR chart, and nothing else. A Range, Standard Deviation, or Moving Range panel is a spread, whose distribution is not symmetric; an attribute chart's limits are not symmetric about the center line; a G or T chart inverts a skewed distribution; and the EWMA, CUSUM, T-squared, and Generalized Variance panels plot statistics with no sigma zones of their own.
- The control limits are at three standard deviations. Set Number of Standard Deviations (k) to any other value and no zone lines are drawn on any panel, because the several zone constructions below agree on the same numbers only at \(k = 3\). The chart dialog greys its Show zone lines checkbox and says so. Z-MR is fixed at \(k = 3\) and so is never affected.
Where both hold, thirds of the drawn band are one and two estimated standard deviations exactly, and the "1 sigma" and "2 sigma" labels mean what they say.
| Chart family | Zone lines drawn? |
|---|---|
| Xbar panel of Xbar-R and Xbar-S, Individuals panel of I-MR and I-MR-R/S, the single-panel Xbar and Individuals charts, Levey-Jennings | yes, at \(k = 3\) only |
| Z panel of a Z-MR chart | yes, always; Z-MR's limits are fixed at \(k = 3\) |
| Range, Standard Deviation, and Moving Range panels, including the single-panel R, S, and Moving Range charts | no |
| EWMA, CUSUM, T-squared and Generalized Variance | no |
| Attribute charts (p, np, c, u, Laney P', Laney U') | no |
| Rare-event charts (G, T) | no |
The constructions in the sections below are how the values are placed where they are placed. The engine still computes zone values for families that draw no lines; they are simply not plotted and nothing reads them.
What this means for the tests. The out-of-control tests that count points by zone are Tests 5, 6, 7, and 8, and the Westgard R-4s, 2-2s, and 4-1s tests on a Levey-Jennings chart. A test that reads a zone runs only where the zone lines exist, so it is switched off on every panel that has none and at every sigma multiplier other than \(3\), whatever the dialog's checkboxes say. A rule that did not run is left out of the Rules in effect block on the worksheet, so the sheet records what actually happened rather than what was ticked.
Test 1 is unaffected on every family, because it reads the control limits themselves. Tests 2, 3, and 4 are unaffected, because they read only the center line and the ordering of the points.
Which tests read a zone, and what each one counts, is on Out-of-Control Rules.
Attribute charts and Z-MR: the half band divided by k (drawn on Z-MR only)¶
On the attribute charts and on Z-MR the one-sigma width is recovered from the limit spacing rather than recomputed, so the zones follow each point's limits exactly:
where \(\text{CL}\) is the center line and \(k\) the sigma multiplier. The zone boundaries are then placed at one and two of these units on each side of the center line:
| Zone | Region |
|---|---|
| Zone C | within \(\text{CL} \pm \sigma_1\) |
| Zone B | between \(\text{CL} \pm \sigma_1\) and \(\text{CL} \pm 2\sigma_1\) |
| Zone A | between \(\text{CL} \pm 2\sigma_1\) and the control limit |
Because \(\sigma_1\) comes from the upper half band and the lower boundaries mirror that same distance, a lower boundary can fall below a lower control limit that has been floored at \(0\), and can itself be negative. Z-MR always uses a multiplier of \(3\), so its zone lines are always at one and two standard deviations. If the center line or the upper limit is missing, or the multiplier is not positive, no zone boundaries are produced.
These boundaries are drawn on the Z panel of a Z-MR chart and nowhere else. On an attribute chart the values are computed and then left off the chart: the band is not symmetric about the center line, so a zone read from the upper half does not describe the lower half, and no zone test is offered on an attribute chart in any case. The attribute chart dialogs carry no zone line checkbox.
Variable charts: thirds of the drawn band (drawn on the location panel, at k = 3 only)¶
On the variable charts the band that is actually drawn is cut into thirds on each side, and the sigma multiplier \(k\) does not enter the calculation:
Each side is measured from its own limit, so an asymmetric band gives asymmetric zone lines, and a manual band is divided the same way as a calculated one.
Only the location panel draws them, and only at \(k = 3\). The Xbar panel, the Individuals panel, and the Levey-Jennings panel carry these lines; the range, standard deviation, and moving range panels compute the same values and draw none, and no panel draws them when the control limits are at a width other than three standard deviations. At \(k = 3\) thirds of the drawn band are one and two estimated standard deviations exactly, which is what makes the "1 sigma" and "2 sigma" labels true.
G and T charts: thirds in normal probability space (not drawn)¶
On the G and T charts, limits and zones are not placed at a fixed distance from the center line at all. Thirds of \(k\) are taken in standard normal probability space,
seven probabilities are formed with the standard normal CDF \(\Phi\), and each is pushed through the chart's own inverse CDF \(F^{-1}\) (an interpolated geometric distribution for the G chart, a three-parameter Weibull for the T chart):
The distribution being inverted is asymmetric, so these values are not equal distances above and below the center line, and the spacing between them is uneven. That is why the G and T charts draw no zone lines: the four values are computed, the dialog offers no zone line checkbox, and nothing is plotted. The zone rules are not offered on these charts either. The limits themselves, \(\text{UCL}_i\) and \(\text{LCL}_i\), are drawn as normal.
The distributions these two charts invert, and the probabilities they are evaluated at, are on Inverse CDF Limits.
Limit types¶
Quantum XL offers three ways to set the limits:
- Shewhart (the default): the formulas above, using the chart's estimated center and sigma.
- Manual: the center line, upper limit, and lower limit are supplied instead of estimated. The variable charts take one set per panel; the attribute charts take one set for the whole chart. EWMA and CUSUM do not offer manual limits at all, and a manual band cannot be combined with split control limits, with a transformation, or with historical mean and sigma values.
- None: no limits and no zone lines are drawn. The center line is still estimated and still drawn, honoring the baseline selection and any historical center value.
What the out-of-control tests do under each type differs:
| Limit type | Attribute and rare-event charts | Variable charts |
|---|---|---|
| Shewhart | all enabled tests run | all enabled tests run |
| Manual | all enabled tests run against the manual band, so "beyond \(k\) sigma" means "outside the manual band" | the same, and a zone test still needs the sigma multiplier left at 3, since that is what decides whether the zone lines exist at all |
| None | the tests that need only the center line still run and can still flag points (the same-side, trend, alternating, and consecutive-zeros tests, that is Tests 2, 3, 4, and the G chart's zeros test); only the limit-based and zone-based tests go quiet | nothing runs; every point is reported as in control, because the entire limit-derived result is cleared |
Under None the zone lines cannot be produced on any family, since there is no limit to divide.
Manual limits¶
On a variable chart the manual values are read one panel at a time, and a panel needs all three values, in order:
- the upper control limit,
- the center line,
- the lower control limit,
with the three strictly increasing from the lower limit through the center to the upper limit. Three outcomes are possible per panel:
| What was entered on that panel's row | Result |
|---|---|
| all three values | the manual band replaces the computed band on that panel, at every point |
| none of the three | that panel falls back to its computed limits, even though Manual is selected |
| one or two of the three, or three values that are not strictly increasing | that panel is not drawn: it is replaced by an error message where the chart would be |
So a partly filled grid is legal only if it is filled a whole row at a time. Filling the location row and leaving the dispersion row blank gives a manual location panel next to a computed dispersion panel, which is the intended way to set a band on one panel only. Filling two of the three boxes in a row loses that panel, and the variable chart dialog does not warn you before the chart is built.
The attribute charts take a single set of three values for the whole chart, validate them in the dialog, and show a message describing what is wrong. There a blank center line is allowed and means no center line is drawn, which also removes the zone lines and the tests that read the center.
Used by¶
- Attribute charts: p, np, c, u, Laney P', and Laney U'.
- Variable charts: Xbar-R, Xbar-S, I-MR, I-MR-R/S, Levey-Jennings, and EWMA, for the limit construction, and the first five of those for the thirds-of-the-band zones on their location panel. EWMA takes the limit construction only: it draws no zone lines and its dialog offers no zone line checkbox. The center line and sigma each of them supplies are on Process Sigma Estimation.
- CUSUM uses neither construction on this page. Each of its two panels carries a one-sided decision interval, \(+h\,\sigma_i\) on the upper sum and \(-h\,\sigma_i\) on the lower, the sigma multiplier is ignored, and no zone lines are produced. See CUSUM Math Details.
- Z-MR, for the divided-half-band zones, and the G and T charts, for the limit types and the probability-space zones.
See Also¶
- Out-of-Control Rules
- Out-of-Control tests
- Process Sigma Estimation
- Varying Subgroup Size Limits
- Inverse CDF Limits
- Shared Math Details
References¶
- Montgomery, D. C. (2013). Introduction to Statistical Quality Control, 7th ed. Wiley.
- Wheeler, D. J., and Chambers, D. S. (1992). Understanding Statistical Process Control, 2nd ed. SPC Press.