Home / Statistical Tools / Control Charts / T Chart / Math Details
Math Details¶
This page gives the exact formulas Quantum XL uses to build a T chart. As on the G Chart, nothing here works the way a Shewhart chart works: there is no standard deviation and no constant. Every line is a quantile of a fitted Weibull distribution, and the sigma multiplier only decides which quantiles.
Notation¶
| Term | Description |
|---|---|
| \(i\) | event index, in plotted order |
| \(x_i\) | the time between event \(i-1\) and event \(i\), the value plotted |
| \(t_i\) | the moment event \(i\) occurred, when the input is date-times |
| \(N\) | number of intervals entering the fit |
| \(\bar{x}\) | mean of those intervals |
| \(\alpha\), \(\hat{\alpha}\) | Weibull scale, and its estimate |
| \(\beta\), \(\hat{\beta}\) | Weibull shape, and its estimate |
| \(\gamma\) | Weibull location, fixed at \(0\) on this chart |
| \(k\) | sigma multiplier, from Number of StDevs:, default \(3\) |
| \(d\) | \(k/3\), the zone step in normal-probability space |
| \(\Phi\) | cumulative distribution function of the standard normal distribution |
| \(q_j\) | the \(j\)th of seven probabilities the lines are drawn at |
| \(F\), \(F^{-1}\) | cumulative distribution function of the fitted Weibull distribution, and its inverse |
| \(u\) | the Time Unit factor, in days |
The plotted statistic¶
When the input column holds date-times, each interval is the difference between consecutive stamps divided by the unit factor:
for Days, Hours, Minutes and Seconds respectively. Nothing is truncated: the fraction is kept, which is the whole difference between this chart and a G chart built from the same dates. The first stamp establishes the starting point and produces no plotted value, so \(N\) stamps give \(N - 1\) intervals, and the stamps must be non-decreasing.
The fitted distribution¶
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:
Full details: Inverse CDF Control Limits.
The location \(\gamma\) is fixed at \(0\) and there is no control that changes it, so the distribution starts at zero: there is no minimum waiting time built into the model.
How the two parameters are estimated¶
With Fit exponential distribution only ticked, the shape is forced and the scale is the sample mean:
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:
Every line on the chart is then a fixed multiple of the average interval, which makes the whole chart checkable by hand. At the default \(k = 3\):
With the box cleared, which is the default, both parameters are fitted from the intervals. The fit is an iterative maximum-likelihood solve on the ranked intervals; when any interval is exactly zero the engine switches to a least-squares fit on the ranks instead, because the likelihood method cannot use a zero. A shape estimate above 1 means the event becomes more likely the longer the wait has been, and below 1 means less likely. That is information a constant-rate model cannot express, and it is why the general fit is the default.
With both Historical Weibull boxes filled, neither parameter is estimated and the data affects nothing but the plotted points. Supplying exactly one in general-Weibull mode stops the analysis. Under the exponential box, \(\hat{\beta}\) is 1 and a supplied \(\alpha\) is used as given.
From the sigma multiplier to seven probabilities¶
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 | \(0.99865\) |
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.
Full details: Inverse CDF Control Limits.
The lines that result¶
The lower limit is clamped at zero, though the clamp rarely bites: the Weibull quantile at \(q_1 = 0.00135\) is already positive. What it is, at the default multiplier and shape 1, is \(0.00135\,\bar{x}\), a fraction of a percent of the average interval. Two consequences:
- The band is extremely asymmetric. In the exponential case the upper limit is about 6.6 times the average interval while the lower limit is about a 740th of it. The center line at \(0.693\,\bar{x}\) sits far nearer the bottom of the band than the top.
- In practice only the upper limit can flag a point. An interval would have to be almost exactly zero to fall below the lower limit, so Rule 1 on this chart reports a wait longer than the model expects, which means the events became rarer.
Sigma zones¶
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 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.
Full details: Control Limits and Zones and Inverse CDF Control Limits.
Out-of-control tests¶
The T chart runs Rule 1 by default and offers rules 2, 3 and 4, all off when the dialog opens. Rule 9 is not offered here, because it tests for counts of exactly zero and belongs to the G chart. Rules 5 through 8 are not offered and are disabled in the engine.
Point \(i\) is flagged when the plotted value falls outside the limits drawn at that point:
Both comparisons are strict, so a point sitting exactly on a control limit is not flagged. The test uses the limits as drawn, which means it follows stepped limits, manual limits, and transformed data with no special handling. The rule has no threshold of its own: its width is the chart's sigma multiplier \(k\), chosen with the control limits.
The engine also supports an independent multiple \(k^{*}\) for this rule. When one is supplied, one sigma of the plotted statistic is recovered separately for each side from the drawn limits,
and the test is applied at \(k^{*}\) of those units instead:
Recovering each side on its own means an asymmetric band gives an asymmetric per-side sigma. The only chart that uses an independent multiple is Levey-Jennings, whose 1-2s warning is this same test at \(k^{*} = 2\). On every other chart Rule 1 tests the drawn limits.
Point \(i\) is flagged when the \(N\) points ending at \(i\) all lie strictly on one side of the center line:
A point exactly on the center line belongs to neither side and ends both runs. Default \(N = 9\).
Point \(i\) is flagged when every step in the \(N\) steps ending at \(i\) moves the same way:
\(N\) counts steps between points, not points, so the pattern spans \(N+1\) points. With the default \(N = 6\) the seventh point of a monotone run is the first one flagged. Both comparisons are strict: two equal consecutive values end both runs.
Write the direction of the step into point \(j\) as
Point \(i\) is flagged when the \(N\) steps \(s_{i-N+1}, \ldots, s_i\) are all non-zero and each one has the opposite sign to the step before it. A tie between two consecutive points gives \(s_j = 0\) and ends the alternation. The pattern spans \(N+1\) points, so with the default \(N = 14\) the fifteenth point of a zigzag is the first one flagged.
A blank row keeps its position on the chart: it is plotted as a gap, and the points after it do not shift left. Each rule handles that gap in its own way.
| Rule | Effect of a blank point |
|---|---|
| 1 | the point is skipped and is never flagged |
| 2 | both same-side counts reset, so the gap breaks the run |
| 3 | both counts reset when either the current or the previous point is blank |
| 4 | the alternation count resets when any of the three points forming the two steps is blank |
| 5 and 6 | every window containing the blank point is discarded, so the gap blocks all \(M\) windows that overlap it |
| 7, 8, and 9 | the run count resets, so the gap breaks the run |
| R-4s | the pair containing the blank point is skipped |
A gap therefore breaks a run rather than bridging it.
Two other situations remove a point's limits, and they are not the same as a gap:
- A subgroup with no usable observation has no plotted value either, so it is a gap and the table above applies.
- An incomplete set of manual limits is rejected for the whole panel, and the panel is replaced by an error message instead of a chart, so no rule is evaluated or reported for it.
Turning the control limit type off is different again, and is covered separately below.
Full details: Out-of-Control Rules.
Baseline and split limits¶
Neither changes the formulas above; both change which intervals the fit runs over.
Baseline Estimation fits from the first \(N\) events and extends the resulting lines across the whole chart. The minimum is 3 events, or 4 when the column holds date-times. Formulas: Baseline and Limit Estimation.
Split Control Limits divides the chart into phases and re-fits inside each one, so each phase carries its own shape, scale, median and limits. Formulas: Splits and Phases.
What the chart refuses¶
| Input | Result |
|---|---|
| Fewer than 3 intervals, or fewer than 4 date-times | The analysis stops: not enough data, unless both parameters were supplied outright |
| Any negative value | The analysis stops |
| Date-times out of chronological order | The analysis stops, naming both rows and both dates |
| One historical parameter without the other, in general-Weibull mode | The analysis stops: invalid override |
| A historical parameter at or below zero | The analysis stops |
| Intervals the Weibull fit cannot converge on | The analysis stops: estimation failed |
| A missing value in the middle of the column | That position plots nothing, and no center line, limit or zone line is drawn through it either |
Shared Math Details used here¶
| Shared concept | Used here for | Reference |
|---|---|---|
| Inverse CDF control limits | the probability ladder, the Weibull and exponential models, and the zone shape | Inverse CDF Control Limits |
| Control limits and zones | the rare-event zone construction | Control Limits and Zones |
| Out-of-control rules | rules 1 to 4 and the blank-point behaviour | Out-of-Control Rules |
| Baseline and limit estimation | what Baseline Estimation restricts | Baseline and Limit Estimation |
| Splits and phases | per-phase re-fitting | Splits and Phases |
What this chart contributes of its own is the second parameter. The G chart has one number to estimate and no choice about how; this one has two, and the check box that fixes the shape at 1 is the difference between a chart you can verify with a calculator and a chart that can tell you whether the waits are getting longer.
See Also¶
References¶
- Montgomery, D. C. (2013). Introduction to Statistical Quality Control (7th ed.). Wiley.
- Nelson, L. S. (1994). "A Control Chart for Parts-Per-Million Nonconforming Items." Journal of Quality Technology, 26(3), 239-240.
- Wheeler, D. J. (2011). "Working with Rare Events." Quality Digest.