Home / Statistical Tools / Control Charts / p Chart / Math Details

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 \(3\)

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. See Control Limits and Zones for the shared limit and zone construction.

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. See 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

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.