What does each plotted point represent?

The maximum minus minimum within one subgroup.

Can the process mean change while the R chart stays stable?

Yes. A common shift of all subgroup values leaves their range unchanged.

Does a point above the upper limit prove a defect?

No. It signals unusual variability relative to the baseline and calls for investigation.

AI Glossary / Statistics and probability / Intermediate

R chart

Definition

An R chart is a statistical process control chart that tracks the range within successive subgroups of measurements. Each range is the subgroup maximum minus its minimum. The chart helps detect changes in short-term process variability and is often paired with an X-bar chart that monitors subgroup means. Its control limits describe expected statistical behavior under the chart's assumptions; they are not product specifications or guarantees that individual outputs meet requirements.

What it means

The R stands for range. A process can maintain the same average while becoming more variable, and a mean chart alone may miss that distinction. An R chart therefore focuses on spread within carefully defined subgroups.

The subgroup design matters. Measurements grouped together should represent comparable conditions over a suitably short period, so that their within-group variation answers the intended process question. Mixing different products, workloads, or operating regimes can create ranges that reflect the mixture rather than a change in one process. R charts are commonly used with small fixed subgroup sizes; larger groups often favor standard-deviation charts.

How it works

Define a measurement and a consistent rational-subgroup procedure. Collect baseline subgroups, inspect their context, and calculate each range R_i. Average the ranges to obtain R-bar, which becomes the center line for a standard R chart.

For a fixed subgroup size n, conventional control limits are LCL = D3 × R-bar and UCL = D4 × R-bar, using factors appropriate to n. These factors come from the range distribution under the usual normal-process model, with negative lower limits truncated to zero. Do not substitute factors for a different subgroup size.

Plot later subgroup ranges in time order against the established limits. Investigate signals and unusual patterns according to a defined response plan. A signal is evidence to investigate, not a diagnosis of its cause. Consider measurement-system changes, dependence, and distributional assumptions before acting. Assess variability stability before relying on a companion mean chart whose limits use that variability estimate. Recalculate baselines only through a justified, documented process.

Three practical examples

A manufacturing team monitors the range of five comparable component measurements collected at each inspection interval.
An AI operations team considers an R chart for small groups of latency measurements under a controlled, comparable workload, after checking whether chart assumptions are reasonable.
A laboratory tracks within-run spread in repeated measurements to identify potential changes in measurement consistency.

Think of it this way

Imagine checking how far apart the fastest and slowest runners finish in each comparable training group. That gap tells you about within-group spread, not the group's average finishing time. Comparing gaps over time can reveal changing consistency. The analogy depends on comparable groups: combining beginners and experienced runners in one session could widen the gap without indicating a change in either group's underlying consistency.

Worked example

In an original arithmetic example, suppose a suitable established baseline has average subgroup range R-bar = 4 units, with five observations per subgroup. The conventional factors for n = 5 are D3 = 0 and D4 = 2.115.

The center line is 4, the lower limit is 0 × 4 = 0, and the upper limit is 2.115 × 4 = 8.46 units. A new subgroup of 10, 11, 12, 12, and 14 has range 14 - 10 = 4 and plots on the center line. Another subgroup of 9, 10, 11, 12, and 19 has range 19 - 9 = 10, exceeding 8.46 and signaling a need for investigation.

The second subgroup's signal does not establish which observation or mechanism caused the change. It also does not say that a product specification was violated, because no specification limits were supplied. Adding 100 to every observation in the first subgroup changes its mean but leaves its range at 4. That illustrates why an R chart needs a separate location-monitoring companion when shifts in the process mean matter. The baseline value is stipulated; this small example is not a sufficient baseline-estimation study.

Illustrative case study

Fictional case study: Pine Precision records five comparable measurements at regular intervals and uses paired mean and range charts. An R-chart signal appears after a sensor-maintenance event.

The team checks calibration records and measurement procedures before changing the production settings. It preserves the original signal and documents any justified baseline revision. This fictional case illustrates investigation of variability, not a real diagnosis or evidence that every signal following maintenance is caused by the sensor.

Common misconceptions and mistakes

An R chart does not monitor the overall range of every observation ever collected. It monitors a sequence of within-subgroup ranges. Combining subgroup sizes without accounting for their different range distributions can make comparisons misleading.

Control limits are not tolerance limits, confidence intervals for individual measurements, or specification limits. A stable process can still produce unacceptable outputs, and an unstable process may temporarily remain within specifications. Another mistake is treating a lower control limit of zero as a target of zero variation. It simply reflects the conventional lower-limit calculation for certain small subgroup sizes.

Business use

R charts can help teams distinguish ordinary within-process variation from evidence of a change that deserves investigation. This supports more disciplined responses than adjusting a process after every fluctuation.

Define who reviews signals, what contextual data are retained, and how causes are investigated. For AI-related operational measures, examine autocorrelation, skewed latency distributions, workload differences, and model-version changes before applying manufacturing-style limits. A chart is useful only when its subgroup construction and assumptions match the process. Use separate capability or service-level analyses to determine whether the observed performance meets business requirements.

Check your understanding

With R-bar = 4 and D4 = 2.115, does a subgroup range of 9 exceed the upper control limit?

Answer: Yes. The upper limit is 8.46, and 9 is larger. The signal concerns statistical variability, not an unspecified product tolerance.

Related terms

Control chart
Range
S chart
Control limits
Common cause
Special cause
subgroup
X-bar chart
process stability
process capability

Learn next

Learn rational subgrouping and the difference between location and spread. Then compare R and S charts, investigate chart signals, and distinguish statistical control from meeting specifications.

Statistical control

Sources

NIST/SEMATECH explains R-chart construction, assumptions, factors, and the relationship to X-bar and S charts: https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm

The baseline value, subgroups, arithmetic, AI application, and fictional case are original illustrations. The n = 5 constants were checked against NIST's table.

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