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Control Limits and Zones

This page defines how Quantum XL places the control limits and sigma zones on a control chart once a chart has supplied its center line and its standard deviation. Every chart provides its own center and sigma; the limit and zone construction here is shared.

Notation

Term Description
\(\text{CL}\) the center line supplied by the chart
\(\sigma_i\) the chart's standard deviation at subgroup \(i\) (may vary by subgroup)
\(k\) the sigma multiplier (number of standard deviations), default \(3\)
\(\text{UCL}, \text{LCL}\) upper and lower control limits

Control limits

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

When the standard deviation varies by subgroup (for example an attribute chart with unequal subgroup sizes), \(\sigma_i\) changes from point to point, so the limits are stepped rather than straight. Attribute charts floor the lower limit at \(0\), since a count or proportion cannot be negative. The p and Laney P' charts also cap the upper limit at \(1\) (a proportion cannot exceed one), and the np chart caps it at the subgroup size \(n_i\); the c, u, and Laney U' charts apply no upper cap.

Sigma zones

The one-sigma width is recovered from the limit spacing rather than recomputed, so the zones follow each point's limits exactly:

\[ \sigma_1 = \frac{\text{UCL} - \text{CL}}{k} \]

The zone boundaries are then placed at one and two of these units on each side of the center line:

Zone Region
Zone C within \(\text{CL} \pm \sigma_1\)
Zone B between \(\text{CL} \pm \sigma_1\) and \(\text{CL} \pm 2\sigma_1\)
Zone A between \(\text{CL} \pm 2\sigma_1\) and the control limit

Limit types

Quantum XL offers three ways to set the limits:

  • Calculated (normal): the formulas above, using the chart's estimated center and sigma.
  • Manual: the user supplies the center, upper, and lower limits directly.
  • None: the center line is drawn but no limits or zones are placed.

Used by

See Also

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.