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Math Details

This page gives the exact formulas Quantum XL uses to build an np 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 \(i\)
\(\bar{p}\)	pooled proportion defective
\(\sigma_i\)	standard deviation of the count at subgroup \(i\)
\(k\)	sigma multiplier (number of standard deviations), default \(3\)

Center line

\[ \bar{p} = \frac{\sum_i d_i}{\sum_i n_i}, \qquad \text{CL}_i = n_i\,\bar{p} \]

The pooled proportion \(\bar{p}\) is estimated over the baseline subgroups (or supplied as a known value), and the center line at each point is \(n_i\,\bar{p}\). When subgroup sizes vary, the center line moves from point to point.

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

Plotted statistic

\[ d_i \]

the raw count of defective items in subgroup \(i\).

Standard deviation

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

the binomial standard deviation of the count.

Control limits

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

The limits are floored at \(0\) and capped at the subgroup size \(n_i\), since the count lies in \([0, n_i]\). See Control Limits and Zones for the shared limit and zone construction.

If Force Straight Limits is enabled, a single assumed size replaces \(n_i\) in the center line and the standard deviation, producing a level center line and level limits (the pooled \(\bar{p}\) is still estimated from the real subgroup sizes).

Overdispersion diagnostic

The np chart reports the Laney overdispersion factor \(\sigma_Z\) as a diagnostic (not applied to the limits). See Laney Overdispersion (Sigma Z); a value well above \(1\) favors the Laney P' Chart.

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

• np Chart
• 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.