Home / Shared Math Details / Control Chart Math / Inverse CDF Control Limits
Inverse CDF Control Limits¶
Most control charts place their limits a fixed number of standard deviations above and below the center line. The two rare event charts do not. The G chart and the T chart build every line (both control limits, the center line, and all four zone boundaries) by converting the sigma multiplier into probabilities on the standard normal distribution and then reading the quantiles of the chart's own event distribution at those probabilities. This page defines that construction, the two distributions it is used with, and what it means for the zones and for the out-of-control rules that read them.
Notation¶
| Term | Description |
|---|---|
| \(x_i\) | the plotted value at point \(i\) (a count between events on the G chart, a time between events on the T chart) |
| \(N\) | the number of non-missing values included in the limit calculation |
| \(\bar{x}\) | the mean of those included values |
| \(k\) | the sigma multiplier (number of standard deviations), default \(3\) |
| \(d\) | the zone step in normal units, \(d = k/3\) |
| \(\Phi\) | the standard normal cumulative distribution function |
| \(q_j\) | the \(j\)-th of the seven probabilities derived from \(k\) |
| \(F\), \(F^{-1}\) | the event distribution's cumulative distribution function and its inverse (quantile function) |
| \(p\), \(\hat{p}\) | the event probability of the geometric model and its estimate (G chart) |
| \(\beta\), \(\alpha\), \(\gamma\) | the Weibull shape, scale, and location parameters (T chart) |
| \(\text{CL}\) | the center line |
Why a Symmetric Band Is Not Used¶
A limit at \(\text{CL} \pm k\sigma\) places the same distance on each side of the center line. That is a reasonable description of a process whose plotted statistic is roughly symmetric, which is what averages, ranges, and proportions with a moderate sample size all are.
Counts between rare events and times between rare events are not symmetric. Both distributions are heavily right-skewed: the most likely gaps are short, and long gaps taper off slowly with no upper bound. Two problems follow from forcing a symmetric band onto them.
- The lower limit lands below zero for any realistic center line, and a gap cannot be negative. Clamping it at zero leaves a chart with no lower limit at all, which discards exactly the signal a rare event chart exists to find: an unusually short gap, meaning the events are happening more often.
- The upper limit is too tight. In a right-skewed distribution the long gaps are ordinary, so a limit placed \(k\) symmetric units above the center flags routine quiet periods as special causes.
Working in probabilities avoids both problems. Instead of asking what value is \(k\) standard deviations away, the charts ask what value has the same tail probability that \(k\) standard deviations would have on a normal distribution, and answer that question with the skewed distribution actually being used. The false alarm rate is what is being held constant, not the distance.
The Probability Ladder¶
The sigma multiplier \(k\) is first turned into seven probabilities on the standard normal distribution. One third of \(k\) sets the zone step:
Each probability corresponds to one line on the chart:
| Probability | Line | Value at \(k = 3\) |
|---|---|---|
| \(q_1 = \Phi(-k)\) | lower control limit | \(0.00135\) |
| \(q_2 = \Phi(-2d)\) | Zone A lower boundary | \(0.02275\) |
| \(q_3 = \Phi(-d)\) | Zone B lower boundary | \(0.15866\) |
| \(q_4 = \Phi(0)\) | center line | \(0.5\) |
| \(q_5 = \Phi(d)\) | Zone B upper boundary | \(0.84134\) |
| \(q_6 = \Phi(2d)\) | Zone A upper boundary | \(0.97725\) |
| \(q_7 = \Phi(k)\) | upper control limit | \(0.99865\) |
Raising \(k\) therefore pushes the probabilities further into the tails rather than stretching a band, and the default \(k = 3\) reproduces the familiar \(0.135\) percent tail probability of a three sigma normal limit.
Inverting the Event Distribution¶
Every line is then the quantile of the chart's own event distribution at its probability:
The center line is \(F^{-1}(0.5)\), the median of the fitted distribution. For a right-skewed distribution the median is below the mean, so the center line sits below the average count or average time between events.
The limits are constant across the chart: every point gets the same seven lines, because the parameters are estimated once from the whole set of included values rather than per point. A row with no value blanks the plotted point, the center line, both limits, and all four zone boundaries at that position, so no line is drawn through the gap.
The G Chart: Geometric Model¶
The G chart plots the count of units between events, where a unit is whatever is being counted between occurrences: opportunities, items produced, days, or patients.
The counts between events are modelled as geometric on the support \(0, 1, 2, \ldots\) (a count of \(0\) means two events in a row with no non-event unit in between):
\(p\) is the probability that any single unit is an event. It is estimated from the values included in the limit calculation:
The mean of this distribution is \((1 - p)/p\), so \(1/(\bar{x} + 1)\) is the method-of-moments estimate of \(p\). The leading factor \((N - 1)/N\) is a finite-sample correction: it lowers \(\hat{p}\) slightly, which lengthens the modelled gap between events and widens the limits. The correction fades as \(N\) grows.
Instead of a historical mean and sigma, the G chart accepts a historical event probability directly. Supplying it replaces the estimate above, and the value must be strictly between \(0\) and \(1\).
The Interpolated Inverse¶
The geometric distribution is discrete, so \(F\) only takes its values at whole numbers and there is generally no count whose cumulative probability is exactly \(q_j\). The quantile is taken from the straight line joining the two nearest points of the CDF. First locate the bracketing integer:
then interpolate between \(F(x_0)\) and \(F(x_0 + 1)\) and floor the result at zero:
The lines are therefore not whole counts, which is intended: a limit of \(0.42\) is a real limit for a count, not a rounding artifact. Interpolating the discrete CDF is not the same as solving \(F(x) = q_j\) for a continuous \(x\). The geometric CDF is concave, so a chord lies below the curve and the interpolated quantile is the larger of the two at small probabilities. With \(p = 0.4\) the interpolated center line is \(0.4167\) against \(0.3569\) from the continuous solution; the two agree closely at the upper control limit.
The floor at zero is doing real work on the lower side. When the target probability \(q_1\) is smaller than \(F(0) = \hat{p}\), the interpolated value is negative, and the lower control limit is set to zero. A lower control limit of zero on a G chart means the process is rare enough that no single short gap can be a signal on its own, which is why the run-based rules matter so much on these charts.
The T Chart: Weibull Model¶
The T chart plots the measured time between events, so the gap is continuous rather than a count.
The times between events are modelled as three-parameter Weibull with shape \(\beta\), scale \(\alpha\), and location \(\gamma\):
The location is fixed at \(\gamma = 0\), because a time between events cannot be negative and no waiting time is excluded by assumption. The center line is the median:
The shape and scale are fitted from the included values with a two-parameter uncensored Weibull fit. Two fitting paths exist, chosen by the data rather than by an option:
- If no included value is exactly zero, the parameters come from a Newton iteration, with plotting positions from the exact median rank formula based on \(0.5^{1/N}\).
- If any included value is exactly zero, the fit falls back to ordinary least squares on the probability plot, using Bernard's approximation \((i - 0.3)/(N + 0.4)\) for the plotting positions. The ranks are computed with the zeros in place and the zeros are then dropped, because the probability plot needs the logarithm of each value.
Fitting requires at least three included values, and a fit that fails to converge or that returns a non-positive parameter is reported as an error rather than charted.
Exponential Mode¶
With the shape fixed at \(\beta = 1\) the Weibull becomes the exponential distribution, which is the constant-rate model: the chance of an event in the next instant does not depend on how long the wait has already been. The scale is then the sample mean of the included values, which is the maximum likelihood estimate:
Exponential mode is the right choice when the events have no memory, that is, when the rate of occurrence is believed constant over the period charted. It is also the more stable choice on a short data set, because one parameter is being estimated instead of two. A fitted shape well away from \(1\) is evidence against the constant-rate assumption: below \(1\) the events cluster, above \(1\) they become more regular.
The T chart accepts historical Weibull parameters in place of the fit. In the general case both the shape and the scale must be supplied, since a half-specified Weibull is rejected; in exponential mode the scale alone is enough because the shape is already fixed at \(1\). A complete set of supplied parameters lets the chart be drawn with fewer than three values, because nothing is being estimated from the data.
Flooring the Lower Limit¶
The lower control limit is clamped at zero. With the location fixed at \(\gamma = 0\) the Weibull quantile function cannot return a negative value, so on the T chart this clamp is a safeguard against numerical noise rather than a routine correction. On the G chart, by contrast, the floor is applied inside the quantile function and does change the answer, as described above.
Zones on These Charts¶
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 zones are therefore placed by probability rather than by distance.
The zone boundaries are the same seven quantiles, so each boundary is placed by its probability rather than by a distance:
| Zone | Region |
|---|---|
| Zone C | between \(F^{-1}(\Phi(-d))\) and \(F^{-1}(\Phi(d))\) |
| Zone B | between \(F^{-1}(\Phi(\pm d))\) and \(F^{-1}(\Phi(\pm 2d))\) |
| Zone A | between \(F^{-1}(\Phi(\pm 2d))\) and the control limit on that side |
Because \(F^{-1}\) is not a straight line, the upper part of a zone is wider than the lower part of the same zone. The zone boundaries are not one and two standard deviations from the center line, and they are not equal distances from it. What is symmetric here is the probability content of each zone, not its width.
The consequence for the out-of-control rules is direct. The rules that count points within a zone or beyond it (two of three beyond Zone A, four of five beyond Zone B, and the two fifteen-point and eight-point zone patterns) assume the zones mean the same thing on both sides of the center line and that being in Zone B is equally unusual above and below. On an inverse CDF chart the probability content still matches side to side, but the geometry does not, and the lower zones can be compressed almost to nothing when the fitted distribution puts most of its mass near zero. For this reason Quantum XL does not offer the four zone-reading rules on the G and T charts. The rules that these charts do offer read the control limits, the center line, or the ordering of the plotted values, none of which depend on zone symmetry.
The zone boundaries are still drawn, so they remain useful for reading the chart. Just do not read them as one and two sigma distances.
These Charts Have No Sigma¶
Neither chart computes a standard deviation at any point in its calculation. The event distribution's parameters carry all of the information that \(\text{CL}\) and \(\sigma\) carry on a Shewhart chart. Several things that look transferable from the other chart families therefore do not apply:
- There is no sigma estimation method to choose, no within or overall sigma, and no unbiasing constants. The Process Sigma Estimation page describes nothing these two charts do.
- The control chart constants (\(d_2\), \(c_4\), \(D_3\), \(D_4\), and the rest) are never evaluated.
- The sigma multiplier \(k\) still has an effect, but it changes tail probabilities rather than a distance. Doubling \(k\) does not double a band width.
- The reported parameter is the event probability \(\hat{p}\) on the G chart, and the fitted shape and scale on the T chart. There is no reported process sigma to compare against a specification.
- Descriptions of limits, zones, and standards-given values written for the center-plus-\(k\)-sigma charts should not be reused on these two charts without checking each statement against this page.
Used by¶
- G Chart, with the interpolated geometric inverse.
- T Chart, with the Weibull quantile function and its exponential special case.
See Also¶
- Control Limits and Zones, the center-plus-\(k\)-sigma construction used by every other chart family
- Standard Normal Distribution
- Out-of-Control tests
- Shared Math Details
References¶
- Benneyan, J. C. (2001). Number-between g-type statistical quality control charts for monitoring adverse events. Health Care Management Science, 4(4), 305-318.
- Nelson, L. S. (1994). A control chart for parts-per-million nonconforming items. Journal of Quality Technology, 26(3), 239-240.
- Santiago, E., and Smith, J. (2013). Control charts based on the exponential distribution: adapting runs rules for the t chart. Quality Engineering, 25(2), 85-96.
- Johnson, N. L., Kemp, A. W., and Kotz, S. (2005). Univariate Discrete Distributions, 3rd ed. Wiley.
- Nelson, W. (1982). Applied Life Data Analysis. Wiley.
- Montgomery, D. C. (2013). Introduction to Statistical Quality Control, 7th ed. Wiley.