Home / Statistical Tools / Control Charts / Individuals / Math Details
Math Details¶
This page gives the exact formulas Quantum XL uses to build an Individuals chart. Each equation lists what it computes and where it appears on the chart. Every formula on this page is shared with other control charts, so each block is followed by a link to the page that owns it.
Notation¶
| Term | Description |
|---|---|
| \(t\), \(i\) | point index, in plotted order. Both letters appear in the shared blocks below and mean the same thing here |
| \(x_t\) | the observation plotted at point \(t\) |
| \(n_{\text{pts}}\) | number of plotted points |
| \(L\) | moving range length, the number of consecutive observations in one window, default \(2\) |
| \(MR_t\) | the moving range ending at point \(t\) |
| \(m\) | number of moving ranges, or of successive differences |
| \(\overline{MR}\), \(\widetilde{MR}\) | the average and the median of the moving ranges |
| \(\bar{x}\) | mean of the observations, the center line |
| \(\hat{\sigma}_{\text{within}}\) | estimated within sigma |
| \(k\) | sigma multiplier (number of standard deviations), default \(3\) |
| \(d_2(n)\), \(d_3(n)\), \(d_4(n)\) | expected value, standard deviation, and median of the range of \(n\) standard normal observations |
| \(c_4'(n)\) | tabulated unbiasing constant for the square root of MSSD |
| \(\mathcal{B}\) | the baseline set of points the estimates are computed from |
| \(\text{CL}\), \(\text{UCL}\), \(\text{LCL}\) | center line, upper and lower control limits |
There is no subgroup size on this chart. Every point carries exactly one observation, which is why nothing below is evaluated at a per-point \(n_i\) and why the limits are straight lines.
Plotted statistic¶
The chart plots the observation itself, so there is nothing to compute for the plotted series. Every observation is plotted, whatever the baseline selection is, because the baseline governs the limits and never what is plotted. A blank row leaves its position empty.
The moving range panel of this dialog is computed even on this variant, where it is not drawn, and its construction is set out on the Moving Range Math Details page.
Center line¶
The center line is the mean of the observations in the baseline set, which is every observation unless First N subgroups: restricts it. A supplied historical mean replaces \(\bar{x}\) after it has been computed, so the baseline still decides which points fed the estimate that was discarded.
Used by: the center line of the chart, the Center row of the summary table, and the center of the band the out-of-control tests measure runs against.
Within sigma¶
The Individuals variant offers three estimators of \(\hat{\sigma}_{\text{within}}\), and one value is estimated once for the whole chart (once per phase when the chart is split).
Square root of MSSD, the method this variant selects for you:
The sum runs over the \(m\) successive differences of consecutive non-missing observations (a missing value breaks the chain). This is the square root of half the mean squared successive difference (MSSD). When Use unbiased constants (within sigma) is checked (the default), the estimate is divided by the tabulated correction factor \(c_4'(m + 1)\).
Average moving range:
Each moving range \(MR_t\) is the maximum minus the minimum of \(L\) consecutive non-missing observations (\(L\) is the moving range length, default \(2\), in which case \(MR_t = |x_t - x_{t-1}|\)), and \(m\) is the number of moving ranges. A missing value breaks the chain of consecutive observations, so no moving range spans a gap.
When Use Nelson moving-range correction is checked, moving ranges larger than \(D_4(L)\,\overline{MR}\), with \(D_4(L) = 1 + 3\,d_3(L)/d_2(L)\), are removed and the average is recomputed. This repeats until no moving range exceeds the recomputed threshold (at most 50 iterations); a moving range removed in one pass stays removed in every later pass. The multiplier in that threshold is always \(3\), whatever sigma multiplier the chart is drawn with. The correction affects only the sigma estimate, not the moving range values themselves, and it is offered only with the average moving range method. If it removes every moving range, no within sigma can be computed.
Median moving range:
\(\widetilde{MR}\) is the median of the moving ranges (the average of the two middle values when their count is even), and \(d_4(L)\) is the unbiasing constant for the median range of \(L\) standard normal observations. The moving ranges are formed exactly as in the average moving range method.
Full details, including the degrees of freedom each estimator carries: Process Sigma Estimation.
Only the first of the three reads the Use unbiased constants (within sigma) checkbox. The two moving range estimators divide by a constant of their own in every case, which is why the checkbox greys out when you select either of them.
The two moving range estimators also need the moving range window, which the square root of MSSD does not:
Each moving range is the largest minus the smallest of the \(L\) values in a window that ends at the current point, where \(L\) is the moving range length (default \(2\), allowed \(2\) to \(100\)). At the default \(L = 2\) the window holds a pair of neighbors, so \(MR_t = \lvert x_t - x_{t-1} \rvert\). The value is stored against the last point of its window, so \(MR_t\) is plotted at position \(t\).
A moving range is produced only when the window is complete. Three consequences follow:
- The first \(L - 1\) points of a series have no complete window and are left unplotted. At the default \(L = 2\) that is the first point only; at \(L = 5\) the first four points are blank.
- A missing value, or a subgroup with no computable statistic, empties its slot and restarts the window, so the \(L - 1\) points that follow the gap are also unplotted. No moving range spans a gap: the sequence is never closed up around one.
- The number of moving ranges is at most the number of available values minus \(L - 1\), and every gap reduces it further.
Excluding a point as an outlier removes it from the sigma estimate, which drops the moving ranges whose window contained it, but the plotted moving range series is rebuilt from the complete data, so an excluded point does not break the plotted chain.
Full details: Moving Ranges.
Control chart constants¶
The range constants behind the two moving range estimators are looked up at the moving range length on this chart, never at a subgroup size.
The range constants describe the distribution of the range \(R\) of \(n\) independent standard normal observations:
So \(R/d_2(n)\) is an unbiased estimate of \(\sigma\) from a subgroup range, \(d_3(n)\) scales the spread of a range, and \(\widetilde{R}/d_4(n)\) is the corresponding estimate from a median range. Quantum XL evaluates these from precomputed full-precision tables for \(n = 2\) to \(100\). Outside that range they are undefined, which is why a subgroup size or moving range length above \(100\) produces no estimate and no limits.
Full details: Control Chart Constants.
The tables run from \(2\) to \(100\), which is exactly the range the Moving range length: box accepts.
Control limits¶
Every limit on a control chart has the same form, a center line plus or minus \(k\) standard deviations of the plotted statistic:
Full details: Control Limits and Zones.
Here the plotted statistic is a single observation, so its standard deviation is \(\hat{\sigma}_{\text{within}}\) itself with no division by a subgroup size, and it is the same at every point:
There is no flooring and no capping. An observation can legitimately be any value, so both limits are drawn wherever the arithmetic puts them.
Sigma zones¶
The zone lines, and the four out-of-control tests that read them, are placed by cutting the drawn band into thirds on each side. The sigma multiplier does not enter the calculation.
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.
Full details, including what the labels "1 sigma" and "2 sigma" really mean at a multiplier other than \(3\): Control Limits and Zones.
The band on this chart is symmetric about the center line, so the zone lines are evenly spaced. At the default \(k = 3\) they sit at one and two estimated standard deviations as their labels say; at any other multiplier they are still thirds of the band you asked for, which changes what tests 5 through 8 are testing. This is the only panel on the chart, so all eight rules run here.
Baseline and split limits¶
Neither of these changes the formulas above; both change which points the sums run over.
Estimate Limits From restricts every estimate to the first \(N\) points and then extends the resulting center line and limits across the whole chart. The number of successive differences or moving ranges that window supplies is smaller than the number of points it holds:
With the default moving range length \(L = 2\), a baseline of \(N_b^{\text{eff}}\) points supplies at most \(N_b^{\text{eff}} - 1\) moving ranges, and fewer when a gap inside the baseline breaks the chain of consecutive values.
On this chart the minimum is 5 points, the window advances past blank rows and excluded outliers rather than spending a slot on them, and the plotted values, the minimum, the maximum, and the counts are always computed from all of the data. Formulas: Baseline and Limit Estimation.
Split Control Limits divides the chart into phases and re-runs the whole estimation inside each one, so each phase gets its own mean, its own within sigma, its own limits, and its own zone lines, and every out-of-control run counter restarts at the boundary. A baseline count is reused inside each phase, capped at that phase's length. One transformation is fitted for the whole chart rather than one per phase. Formulas: Splits and Phases.
Shared Math Details used here¶
This chart uses shared formulas defined once in Shared Math Details.
| Shared concept | Used here for | Reference |
|---|---|---|
| Process sigma estimation | the square root of MSSD, the average moving range, and the median moving range estimates | Process Sigma Estimation |
| Moving ranges | the window the two moving range estimators run over | Moving Ranges |
| Control chart constants | \(d_2\) and \(d_4\) behind the moving range estimators, and \(c_4'\) behind the square root of MSSD | Control Chart Constants |
| Control limits and zones | the general limit form, the zone construction, and the manual and disabled limit types | Control Limits and Zones |
| Baseline and limit estimation | what Estimate Limits From restricts, and how many differences a baseline supplies | Baseline and Limit Estimation |
| Splits and phases | per-phase estimation and the run counters restarting | Splits and Phases |
| Out-of-control rules | flagging out-of-control points | Out-of-Control Rules |
The Individuals chart contributes no formula of its own. Every equation above is one of these shared pieces evaluated for a chart of single observations.
See Also¶
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.
- Duncan, A. J. (1986). Quality Control and Industrial Statistics (5th ed.). Irwin.
- Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product. Van Nostrand.