A control limit marks a boundary for expected variation on a control chart. The lower control limit (LCL) and upper control limit (UCL) sit below and above a center line representing a process average. A point beyond either limit is a signal to investigate, not proof of a particular cause. Patterns and trends also matter when assessing stability. section 14.1.1 section 14.3.2
A common starting rule places limits three standard deviations below and above the mean. For normally distributed data, approximately 99.73% of observations lie within that range. The width is a decision: two-standard-deviation limits detect smaller changes sooner, while four-standard-deviation limits allow more variation before signaling. Choose the setting for the process and the cost of unnecessary alarms. Sample size can affect calculated limits and their sensitivity. section 14.1.1 section 14.1.5
Calculate limits for the quantity the chart actually tracks:
- A p-chart tracks the proportion of defective items. For each sample, calculate its proportion and σ = √(p̄(1 − p̄) / n); then set LCL = p̄ − 3σ and UCL = p̄ + 3σ. Because n enters the calculation, limits can differ between samples. section 14.2.1
- An np-chart tracks defective-item counts when sample sizes are uniform. A worked version tightens the limits to two standard deviations and flags counts outside them. section 14.2.2
- A c-chart tracks defects per unit. Using the average count c̄, its three-sigma lower limit is max(0, c̄ − 3√c̄): a defect count cannot be negative. section 14.2.3
- For measurements, choose limits around the statistic of interest: sample means on an x-bar chart, within-sample ranges on an r-chart, or sample standard deviations on an s-chart. section 14.3.1 section 14.3.2 section 14.3.3
>>> oj['UCL'] = p_bar + 3 * oj['sigma'] >>> oj['LCL'] = p_bar - 3 * oj['sigma']
The Python assignment computes a separate three-sigma UCL and LCL for each p-chart sample from its estimated standard deviation. section 14.2.1
Read limits alongside the chart
In piston-ring production, several sample means cross the x-bar chart’s limits, including points in a rising sequence near the end. The companion r-chart has all sample ranges within its limits. Investigate the shift in average diameter without treating stable ranges as evidence that every aspect of the process is stable. section 14.3.1 section 14.3.2
Limits also adapt to how observations arrive. An I-MR chart pairs limits for individual measurements with limits for changes between successive measurements; an EWMA chart adjusts its limits for a smoothing constant to help detect subtle shifts. Whichever chart is used, an out-of-limit point calls for investigation, while multiple charts can reveal different aspects of process performance. section 14.3.4 section 14.3.5 section 14.4
