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

Math Details¶

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

Notation¶

Term	Description
\(n_i\)	size of subgroup \(i\) (its area of opportunity)
\(c_i\)	number of defects in subgroup \(i\)
\(\bar{u}\)	center line (pooled defects per unit)
\(\sigma_i\)	standard deviation of the rate at subgroup \(i\)
\(k\)	sigma multiplier (number of standard deviations), default \(3\)

Center line¶

\[ \bar{u} = \frac{\sum_i c_i}{\sum_i n_i} \]

the pooled defects per unit over the baseline subgroups (or a supplied known value). It stays flat regardless of subgroup size.

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

Plotted statistic¶

\[ u_i = \frac{c_i}{n_i} \]

the defects per unit in subgroup \(i\).

Standard deviation¶

\[ \sigma_i = \sqrt{\frac{\bar{u}}{n_i}} \]

the Poisson standard deviation of the rate. It depends on the subgroup size, so it changes from point to point when sizes differ.

Control limits¶

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

The lower limit is floored at \(0\); there is no fixed upper cap. Because \(\sigma_i\) varies with subgroup size, the limits step with the size. 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 standard deviation, producing level limits; the center line \(\bar{u}\) is unchanged.

Overdispersion diagnostic¶

The u 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 U' 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

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.