Home / Statistical Tools / Control Charts / Laney P' Chart / Math Details

Math Details¶

This page gives the exact formulas Quantum XL uses to build a Laney P' chart. It is a p chart whose standard deviation is scaled by the overdispersion factor σ_Z. 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)
\(\sigma_i\)	ordinary binomial standard deviation at subgroup \(i\)
\(\sigma_Z\)	overdispersion factor
\(k\)	sigma multiplier (number of standard deviations), default \(3\)

Center line and plotted statistic¶

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

The center line is the pooled proportion (or a supplied known value), and each point is the subgroup proportion, exactly as on the p Chart.

Ordinary standard deviation¶

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

the binomial standard deviation before the overdispersion correction.

Overdispersion-adjusted control limits¶

The binomial standard deviation is multiplied by the overdispersion factor \(\sigma_Z\) (see Laney Overdispersion (Sigma Z), estimated from the standardized values \(z_i = (p_i - \bar{p})/\sigma_i\)):

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

When \(\sigma_Z = 1\) this reduces to the ordinary p chart. See Control Limits and Zones for the shared limit and zone construction.

Shared Math Details used here¶

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

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

References¶

Laney, D. B. (2002). Improved control charts for attributes. Quality Engineering, 14(4), 531-537.
Montgomery, D. C. (2013). Introduction to Statistical Quality Control, 7th ed. Wiley.