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Laney Overdispersion (Sigma Z)

This page defines the Laney overdispersion factor \(\sigma_Z\), which corrects an attribute chart's control limits when the data are more (or less) variable than the binomial or Poisson model assumes. Quantum XL reports \(\sigma_Z\) as a diagnostic on every attribute chart and applies it to the limits on the Laney P' and Laney U' charts.

Notation

Term	Description
\(v_i\)	the plotted statistic at subgroup \(i\) (proportion for p, defects-per-unit for u)
\(\text{CL}\)	the center line (\(\bar{p}\) or \(\bar{u}\))
\(\sigma_i\)	the ordinary binomial or Poisson standard deviation at subgroup \(i\)
\(z_i\)	the standardized subgroup value
\(\overline{MR}_z\)	the average moving range of the \(z_i\)
\(d_2\)	the range-to-sigma constant for a moving-range window of two, \(d_2 = 1.1284\)
\(\sigma_Z\)	the overdispersion factor

Standardized values

Each subgroup is standardized by its own ordinary sigma:

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

If \(\sigma_i \le 0\) the value is set to \(z_i = 0\).

Overdispersion factor

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 \(d_2\):

\[ \sigma_Z = \frac{\overline{MR}_z}{d_2}, \qquad d_2 = 1.1284 \]

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).

Applying the factor

On the Laney charts the ordinary control limit is used with its sigma scaled by \(\sigma_Z\):

\[ \text{UCL} = \text{CL} + k\,\sigma_i\,\sigma_Z, \qquad \text{LCL} = \text{CL} - k\,\sigma_i\,\sigma_Z \]

Diagnostic use

On the ordinary p, np, c, and u charts, \(\sigma_Z\) is computed and reported but not applied to the limits. A value near \(1\) indicates the binomial or Poisson model fits and the ordinary chart is appropriate; a value meaningfully above \(1\) indicates overdispersion, and the Laney P' or Laney U' chart is preferred.

Used by

• Laney P' Chart and Laney U' Chart apply \(\sigma_Z\) to the limits.
• p, np, c, and u report \(\sigma_Z\) as a diagnostic.

See Also

• Control Limits and Zones
• Shared Math Details

References

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