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¶
The zone lines are not placed the same way on every chart
Read this section before using Tests 5 through 8 or the Westgard rules on a chart drawn with a sigma multiplier other than \(3\).
One option, Show 1-sigma / 2-sigma zone lines, places the zone boundaries on every chart, but the three chart families place them by three different rules. The labels "1 sigma" and "2 sigma" therefore do not mean the same thing on every chart:
| Chart family | How the boundaries are placed | What the "1 sigma" line really is |
|---|---|---|
| Attribute charts (p, np, c, u, Laney P', Laney U') and Z-MR | the upper half band is divided by the sigma multiplier \(k\) | a true one standard deviation, at any \(k\) |
| Variable charts (Xbar-R, Xbar-S, I-MR, I-MR-R/S, Levey-Jennings, EWMA, and their single-panel variants Xbar, R, S, Individuals, Moving Range) | each side of the drawn band is cut into thirds, and \(k\) never enters the calculation | \(k/3\) standard deviations, so one standard deviation only when \(k = 3\) |
| CUSUM | no zones are produced at all, and the checkbox that would show them is hidden | nothing; the zone lines do not exist on this chart |
| Rare-event charts (G, T) | thirds of \(k\) in normal probability space, then pushed through the chart's own inverse distribution | not a fixed distance at all, and not symmetric about the center line |
The zone boundaries are computed once, stored with each plotted point, and read back by the out-of-control tests that count points by zone: Tests 5, 6, 7, and 8, and the Westgard R-4s test on a Levey-Jennings chart. Those tests never recompute a boundary, so what they measure follows the family and the sigma multiplier:
- On a Z-MR chart the tests keep their textbook meaning at any \(k\), and the chart is always drawn at \(k = 3\) in any case.
- On an attribute chart the zone lines are placed at a true one and two sigma for any \(k\), but no zone test is offered there, so nothing reads them.
- On a variable chart drawn at the default \(k = 3\) the tests also keep their textbook meaning, because thirds of a 3-sigma band are 1 sigma and 2 sigma.
- On a variable chart drawn at any other \(k\) they do not. At \(k = 2\) the boundary labeled 1 sigma sits at \(0.667\) standard deviations and the one labeled 2 sigma at \(1.333\), so Test 6 ("4 out of 5 beyond 1 sigma") is really "4 out of 5 beyond \(0.667\) sigma" and Test 5 ("2 out of 3 beyond 2 sigma") is really "beyond \(1.333\) sigma". At \(k = 3.5\) the same two boundaries sit at \(1.167\) and \(2.333\) standard deviations, making those tests harder to trigger. Test 7 ("15 within 1 sigma") widens or narrows the same way.
- 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.
If you change the sigma multiplier away from \(3\) on a variable chart, read the zone lines as thirds of the band you asked for, not as standard deviations.
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¶
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.
Variable charts: thirds of the drawn band¶
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. On the range, standard deviation, and moving range panels the zone lines are built from the unclamped lower limit and then floored at zero, so when the drawn lower limit has been clamped to zero the lower zone lines are not thirds of the band that appears on the chart.
G and T charts: thirds in normal probability space¶
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 lines are not equal distances above and below the center line, and the spacing between them is uneven. The zone rules are not offered on these charts.
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" and the zone tests are measured against the divided manual band | same |
| 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 thirds-of-the-band zones. 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.