Control charts
Monitor observations in time order for signals inconsistent with a stable process.
Concepts and reasoning
Monitoring a process over time
A control chart compares time-ordered observations with variation expected from a stable process model. Its purpose is to identify changes worth investigating, not to label every fluctuation as a problem. Reacting to ordinary variation as though it were a special cause can make a process less stable.
The center line represents the estimated process level. Control limits represent expected statistical variation under the chart model. They are different from specification limits, which express acceptable product performance. An in-control process may fail specifications, and an out-of-control signal may occur while every item remains within specification.
Match the chart to the data
For continuous measurements, an individuals and moving-range chart, I–MR, is useful when one observation is collected at a time. X-bar/S charts use rational subgroups and monitor subgroup means and SDs. The subgroup structure must reflect the process.
A p chart monitors the fraction of items that are defective, allowing sample sizes to vary. An np chart monitors the number of defective items with a constant sample size. A c chart monitors defect counts with constant opportunity; a u chart monitors defects per unit when opportunity varies. A defective item is a binary classification, while one item may contain several defects.
Chart selection therefore starts with the data-generating process. NIST's control-chart guide organizes charts by continuous and attribute data and explains the associated assumptions.
Mathematical foundations
A proportion chart
If sample i contains nᵢ independent items with a common defective probability p, its observed proportion has variance p(1 − p)/nᵢ. Estimate the baseline proportion using the total defective count divided by the total inspected:
Approximate three-sigma limits for sample i are
Different sample sizes produce different limits. A simple unweighted average of sample proportions can estimate the baseline differently when sample sizes vary. For rare events or small samples, the normal approximation behind these limits can be poor.
For independent normally distributed observations with known mean and SD, the probability of exceeding ±3 SD in one observation is about 0.0027. Repeated monitoring creates repeated opportunities for false signals. Estimated parameters, dependence, discreteness, and added run rules change the operating behavior.
Worked example
Worked example: why sample size changes the limits
Suppose a stable baseline has a defective fraction of 0.10.
| Sample size | Standard error | Lower limit | Upper limit |
|---|---|---|---|
| 25 | 0.060 | 0.000 | 0.280 |
| 100 | 0.030 | 0.010 | 0.190 |
| 400 | 0.015 | 0.055 | 0.145 |
For n = 100, the SE is √(0.1 × 0.9/100) = 0.03, so limits are 0.10 ± 0.09. Observing 20 defective items gives p = 0.20, just above the upper limit of 0.19, and warrants investigation.
For n = 25, five defective items also gives 0.20, but it is within the wider limits. The same observed fraction carries less statistical evidence when based on fewer items. This is why a variable-sample-size p chart should not use one fixed pair of limits without justification.
Suppose an engineering requirement allows at most 5% defective. The baseline process at 10% fails that requirement even if every point lies inside the control limits. The chart evaluates consistency with the baseline, not acceptability of the baseline.
If the baseline has already improved after a documented intervention, continually recomputing limits from mixed old and new observations can obscure the change. Establish a defensible new baseline once the changed process is sufficiently understood.
Interpreting the results
A signal is a reason to investigate
A point outside control limits is unusual relative to the chart's estimated stable-process model. It suggests investigation of a special cause, but does not identify that cause or prove a specific failure. Conversely, points inside limits do not establish stability; persistent patterns may contain information that a single-point rule misses.
The center line summarizes the reference process, while control limits reflect its sampling variation. Neither is a product specification or a confidence interval for an individual item's acceptability. A process can be statistically stable yet fail requirements.
The plotted statistic depends on chart type: an individual measurement, subgroup mean, fraction defective, number defective, defect count, or defect rate. A change in a denominator can change limits even when the underlying rate is constant. Interpret the statistic and its opportunity basis together.
Assumptions and limitations
Match the chart to the data-generating process
Continuous measurements and attribute counts need different charts. A defective item is classified once, while a unit can contain several defects; confusing these leads to the wrong variance model. Proportion charts use binomial reasoning, and defect-count/rate charts use Poisson reasoning under suitable conditions.
Observations should retain their operational time order. Rational subgroups should capture comparable short-term conditions. Serial correlation, varying opportunity, overdispersion, or a changing measurement process can invalidate standard three-sigma limits.
Phase I limits estimated from the same data are exploratory and can be influenced by unusual observations. Future monitoring needs a justified stable baseline and rules for updating it. Recalculating limits after every undesirable point can conceal deterioration rather than control the process.
Diagnostics and sensitivity analysis
Investigate signals using process records
Link unusual points to timestamps, machines, batches, staffing, materials, and measurement changes. Check data-entry errors and denominator changes before altering operations. Preserve the original observations and the reason for any excluded special cause.
Inspect both location and variation: an acceptable mean chart can coexist with increasing subgroup spread or moving ranges. Review runs and patterns using prespecified rules when appropriate, while recognizing that adding rules changes false-alarm behavior.
Compare the selected chart's variance assumptions with the data. A p or u chart with varying denominators should have correspondingly varying limits. If repeated correlated measurements or excess count variation dominate, consider a model that represents them. NIST's control-chart reference organizes charts by measurement type.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- NIST handbook: control charts — Chart selection and statistical process monitoring.