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

Math Details¶

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

Notation¶

Term	Description
\(c_i\)	number of defects in subgroup \(i\)
\(\bar{c}\)	center line (mean defect count)
\(\sigma\)	standard deviation of the defect count
\(k\)	sigma multiplier (number of standard deviations), default \(3\)

Center line¶

\[ \bar{c} = \frac{1}{m}\sum_{i=1}^{m} c_i \]

the mean defect count over the \(m\) baseline subgroups (or a supplied known value).

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

Plotted statistic¶

\[ c_i \]

the raw defect count in subgroup \(i\).

Standard deviation¶

\[ \sigma = \sqrt{\bar{c}} \]

For the Poisson model the variance equals the mean, so the standard deviation is the square root of the center line. It is a single value shared by every point, because the area of opportunity is constant.

Control limits¶

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

The limits are straight (the area of opportunity does not change) and the lower limit is floored at \(0\). See Control Limits and Zones for the shared limit and zone construction.

Overdispersion diagnostic¶

The c 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.