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Math Details

This page gives the exact formulas Quantum XL uses to build a p chart. Each equation lists what it computes and where it appears on the chart.

Notation

Term	Description
\(n_i\)	size of subgroup \(i\)
\(d_i\)	number of defective items in subgroup \(i\)
\(\bar{p}\)	center line (pooled proportion defective)
\(\sigma_i\)	standard deviation of the proportion at subgroup \(i\)
\(k\)	sigma multiplier (number of standard deviations), default \(3\)
\(\sigma_Z\)	Laney overdispersion factor (diagnostic only on this chart)

Center line

\[ \bar{p} = \frac{\sum_i d_i}{\sum_i n_i} \]

The center line is the pooled proportion over the baseline subgroups. A known value may be supplied instead, in which case that value is used for \(\bar{p}\).

Used by: the center line of the chart and the p̄ row of the summary.

Plotted statistic

\[ p_i = \frac{d_i}{n_i} \]

the proportion defective in subgroup \(i\).

Standard deviation

\[ \sigma_i = \sqrt{\frac{\bar{p}\,(1 - \bar{p})}{n_i}} \]

This is the binomial standard deviation. It depends on the subgroup size \(n_i\), so it changes from point to point when subgroup sizes differ.

Control limits

\[ \text{UCL}_i = \min\!\left(1,\ \bar{p} + k\,\sigma_i\right), \qquad \text{LCL}_i = \max\!\left(0,\ \bar{p} - k\,\sigma_i\right) \]

The limits are placed \(k\) standard deviations from the center line, floored at \(0\) and capped at \(1\) because a proportion lies in \([0, 1]\). Because \(\sigma_i\) varies with subgroup size, the limits step with the sample size.

Sigma zones

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:

\[ \sigma_1 = \frac{\text{UCL} - \text{CL}}{k} \]

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.

Full details, including the manual and disabled limit types: Control Limits and Zones.

Overdispersion diagnostic

The p chart also reports the Laney overdispersion factor \(\sigma_Z\) as a diagnostic (it is not applied to the limits here). A value near \(1\) supports the binomial model; a larger value indicates overdispersion, and the Laney P' Chart is preferred. With the plotted proportion \(p_i\) as \(v_i\) and \(\bar{p}\) as \(\text{CL}\):

Each subgroup is standardized by its own ordinary sigma:

\[ z_i = \frac{v_i - \text{CL}}{\sigma_i} \]

where \(v_i\) is the plotted statistic at subgroup \(i\), \(\text{CL}\) is the center line, and \(\sigma_i\) is the ordinary binomial or Poisson standard deviation. If \(\sigma_i \le 0\) the value is set to \(z_i = 0\).

The factor is the average moving range of the \(z_i\) (a window of two, so \(MR_i = \lvert z_i - z_{i-1}\rvert\)) divided by \(d_2\):

\[ \sigma_Z = \frac{\overline{MR}_z}{d_2(2)}, \qquad d_2(2) \approx 1.128379 \]

The engine uses the full-precision table value rather than a rounded one, so a hand check from the printed digits will differ in the sixth significant figure.

Any moving-range window that spans a missing subgroup is dropped rather than bridged, and \(\overline{MR}_z\) is the plain mean of the surviving ranges. When \(\sigma_Z\) cannot be computed it is taken to be \(1\) (no correction).

Full details: Laney Overdispersion (Sigma Z).

Shared Math Details used here

This chart uses shared formulas defined once in Shared Math Details.

Shared concept	Used here for	Reference
Control limits and zones	the UCL/LCL and zone construction	Control Limits and Zones
Laney overdispersion (σ_Z)	the overdispersion diagnostic	Laney Overdispersion (Sigma Z)
Out-of-control tests	flagging out-of-control points	Out-of-Control tests

See Also

• p Chart
• np Chart Math Details
• Laney P' Chart Math Details

References

1. Montgomery, D. C. (2013). Introduction to Statistical Quality Control, 7th ed. Wiley.
2. Wheeler, D. J., and Chambers, D. S. (1992). Understanding Statistical Process Control, 2nd ed. SPC Press.