Home / Shared Math Details / Control Chart Math / Splits and Phases

Splits and Phases¶

A split divides a control chart into consecutive sections, called phases, and gives each phase its own center line and its own control limits. This page defines how the sections are formed, what each phase estimates for itself, what stays shared across the whole chart, and how the out-of-control rules behave at a boundary.

Quantum XL implements this idea in three separate places, so the same picture on screen (dividers with labels, and limits that step at each divider) is produced by three different calculations. The differences are visible in the minimum section length and in what happens when a break is invalid, so they are worth knowing.

Notation¶

Term	Description
\(n_{\text{pts}}\)	number of plotted positions on the chart
\(S\)	number of sections (phases)
\(b_s\)	first position of section \(s\), with \(b_0 = 0\) and \(b_S = n_{\text{pts}}\)
\(\mathcal{P}_s\)	the set of positions belonging to section \(s\)
\(n_{\text{pts},s}\)	length of section \(s\)
\(L_{\min}\)	shortest section the engine will estimate from
\(g_i\)	value of the split column at position \(i\)
\(\hat{\mu}_s, \hat{\sigma}_s\)	center and sigma estimated inside section \(s\)
\(k\)	sigma multiplier, default \(3\)
\(N_b\)	requested baseline count
\(\mathcal{B}_s\)	baseline window inside section \(s\)
\(\mathcal{F}\)	positions used to fit a transformation
\(\hat{\lambda}\)	fitted transformation parameter

Forming the sections¶

\[ 0 = b_0 < b_1 < b_2 < \dots < b_{S-1} < n_{\text{pts}}, \qquad \mathcal{P}_s = \left[\, b_s,\; b_{s+1} \,\right), \qquad n_{\text{pts},s} = b_{s+1} - b_s \]

\(n_{\text{pts}}\) is the number of plotted positions, \(b_1\) through \(b_{S-1}\) are the break positions (each one is the first position of a new section), and \(b_S = n_{\text{pts}}\) closes the last section. A chart with \(S - 1\) breaks has \(S\) sections, and the sections tile the chart with no gaps and no overlap.

Break positions are entered as 1-based chart point numbers, the numbers shown along the horizontal axis, and are converted to the 0-based positions above. Breaks must be strictly increasing and distinct, and a break at the first position is not a break at all because section 0 always starts there.

When the breaks are read from a column instead of typed:

When the breaks come from a column, a break is recorded at every position whose value differs from the position before it:

\[ b \in \{\, i : 1 \le i < n_{\text{pts}},\; g_i \ne g_{i-1} \,\} \]

\(g_i\) is the column's value at position \(i\). Comparison is on the value as read, and a value returning after an interruption starts a new section rather than rejoining the earlier one, so the sections are runs of equal consecutive values, not groups.

Per-phase estimation¶

Each section is estimated on its own data and nothing is pooled across sections:

\[ \hat{\mu}_s = \hat{\mu}\!\left( \mathcal{P}_s \right), \qquad \hat{\sigma}_s = \hat{\sigma}\!\left( \mathcal{P}_s \right) \]

\[ \text{CL}_i = \hat{\mu}_s, \qquad \text{UCL}_i = \hat{\mu}_s + k\,\hat{\sigma}_{s,\,i}, \qquad \text{LCL}_i = \hat{\mu}_s - k\,\hat{\sigma}_{s,\,i}, \qquad i \in \mathcal{P}_s \]

The same estimator the chart would use on unsplit data is applied to each section separately, so a chart with \(S\) sections carries \(S\) center lines and \(S\) limit sets. Any recursion the chart uses, such as the EWMA weighting or the CUSUM accumulation, is also restarted at \(b_s\) from that section's own center.

Everything downstream of the center and sigma is therefore also per phase: the zone boundaries, the degrees of freedom reported for each sigma, and, on the attribute charts, the overdispersion statistic \(\sigma_Z\). There is no pooled overdispersion value across phases, because phases with different center lines have nothing meaningful to pool.

The three mechanisms¶

Charts	How the sections are computed	What drives it
\(\bar{X}\)-R, \(\bar{X}\)-S, I-MR, I-MR-R/S, EWMA, CUSUM, Levey-Jennings	One statistics pass and one limit computation per section, inside a single call. Each section produces its own statistics block, and the section a point belongs to is carried by its position in the result.	The Split Control Limits group on the chart's own dialog: none, split by column, or manual break points.
p, np, c, u, Laney P', Laney U', G, T	The data is sliced into one copy per phase, each copy is calculated separately, and the per-phase results are stitched back into one series by position. Each phase's distribution parameters are kept per phase (the event probability on a G chart, the Weibull parameters on a T chart), while the single chart-level center and limit values that the summary reports are the first phase's.	The Split Control Limits group on the chart's own dialog: none, column, or manual breaks.
Z-MR	Part runs are found inside the Z-MR calculation itself: a run is a maximal block of consecutive rows with the same part name. The run number of each row is then fed into the same divider drawing the split charts use.	The Part column on the Data tab. Z-MR has no split option and no baseline option.

Z-MR dividers are not split control limits

A Z-MR chart shows dividers and labels that look exactly like split control limits, but they mark part runs, they come from the data rather than from any setting, and they cannot be edited. They are also not limit boundaries: the plotted values are standardized, so the Z panel's limits are fixed and the moving range panel's limits are constants, and neither steps at a divider. The same part appearing again later is a new run, so a part can produce several dividers.

Shared across the chart versus computed per phase¶

Shared, one value for the whole chart	Estimated separately in every phase
The transformation and its fitted \(\hat{\lambda}\)	Center line
The requested baseline count \(N_b\)	Within, overall, and between sigma estimates
The sigma multiplier \(k\)	Control limits and zone boundaries
The rule selection and every rule threshold	Degrees of freedom for each sigma
Any supplied historical mean or sigma, applied to every phase	The baseline window \(\mathcal{B}_s\)
The assumed sample size behind flat attribute limits	The overdispersion statistic \(\sigma_Z\)
Specification limits and the target	Rare-event distribution parameters
	Out-of-control run counters

One baseline count, reused inside each phase¶

\[ N_b^{\text{eff}}(s) = \min\left(N_b,\; M_s\right) \]

One requested count \(N_b\) is reused inside every section \(s\) and capped at that section's own length \(M_s\), so each section estimates from its own first \(N_b^{\text{eff}}(s)\) positions and the window never reaches across a section boundary.

There is one requested count for the chart and no per-phase count. A phase shorter than \(N_b\) falls back to using all of its own positions, while a longer phase in the same chart still uses \(N_b\). Full detail: Baseline and Limit Estimation.

One transformation, fitted on the union of the phase baselines¶

A transformation is fitted once for the whole chart, on the union of the per-section baseline windows, and then applied to every position:

\[ \mathcal{F} = \bigcup_{s=0}^{S-1} \mathcal{B}_s, \qquad \hat{\lambda} = \arg\min_{\lambda} \; g\!\left( \lambda; \mathcal{F} \right), \qquad x_i' = T_{\hat{\lambda}}(x_i) \; \text{ for every } i \]

\(\mathcal{B}_s\) is the baseline window inside section \(s\), \(g\) the chart's dispersion objective, and \(T_{\hat{\lambda}}\) the fitted transformation. There is one \(\hat{\lambda}\) for the chart, never one per section. Each section's center and limits are then estimated in the transformed scale from the transformed values.

This is the interaction readers most often get wrong. Splits do not each get their own lambda, and the fit does not use the whole data set either: it uses exactly the positions that feed the per-phase means and within sigmas. On an individuals chart those per-phase windows are concatenated before the fit, so the dispersion objective sees one join at each boundary.

Out-of-control rules at a boundary¶

The out-of-control rules are evaluated inside each section independently:

\[ \text{run tests on } \mathcal{P}_s \text{ only}, \qquad \text{counters reset at every } b_s \]

A run of points on one side of the center, a trend, an alternating sequence, or an \(N\) of \(M\) zone pattern cannot be assembled from positions on both sides of a break. A run in progress at \(b_s\) is abandoned and counting restarts from the first position of the new section.

The point-by-point test against the limits is unaffected, because each point is compared with its own phase's limits. It is the run-based and zone-based rules whose counters restart.

Minimum section length¶

\[ n_{\text{pts},s} \ge L_{\min} \quad \text{for every } s \]

\(L_{\min}\) is the shortest section the engine will estimate from. It is \(5\) on the variable charts and \(2\) on the attribute and rare-event charts. The last section is checked like every other one.

The two families differ, and so does the failure behavior.

Charts	\(L_{\min}\)	What happens to a break that would create a shorter section
Variable charts	5	The whole analysis is rejected. The dialog checks neither the minimum nor the section arithmetic, so a break can be accepted at entry and then fail when the chart is built.
Attribute and rare-event charts	2	The break is dropped and the chart is built without it. The dialogs also check the two-row rule when the break is entered.

Two further engine rules apply to the variable charts: manually entered limits cannot be combined with splits, since there is no way to supply a separate band per phase, and a section with fewer than five valid values fails the analysis even when the section itself is long enough.

Used by¶

Attribute charts: p, np, c, u, Laney P', and Laney U'.
Rare-event charts: G and T.
Z-MR, for the part-run dividers only.
The variable charts (\(\bar{X}\)-R, \(\bar{X}\)-S, I-MR, I-MR-R/S, EWMA, CUSUM, Levey-Jennings) re-estimate per section in the same way; their math pages are planned.

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.
Chakraborti, S., Human, S. W., and Graham, M. A. (2009). Phase I statistical process control charts: an overview and some results. Quality Engineering, 21(1), 52-62.
Box, G. E. P., and Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society: Series B, 26(2), 211-252.