Why pair it with an X-bar chart?

Spread and location can change separately.

Can it use one observation per subgroup?

A sample standard deviation requires more than one; individual-measurement charts address another setup.

Are the constants universal?

No; they depend on subgroup size and the chosen construction.

AI Glossary / Statistics and probability / Intermediate

S chart

Definition

An S chart is a statistical process control chart that plots the sample standard deviation within successive subgroups. It monitors whether short-term process variability remains consistent over time. Its center line and control limits describe expected subgroup variation under a baseline model; they are not product specification limits or guarantees that the process meets customer requirements.

What it means

Two production periods can have the same average output but very different spread. An S chart focuses on that spread, helping detect an increase or decrease in within-subgroup variability that a chart of means alone may miss.
The subgroup is central to the interpretation. Measurements collected under similar short-term conditions should form a rational subgroup so that the variation within it reflects the intended process component. Arbitrarily mixing distant periods or different machines can obscure the pattern being monitored.

How it works

Define the measurement, sampling interval, and subgroup construction. Collect numeric observations on a consistent scale using a suitable measurement system. Measurement-system changes can create chart signals even when the underlying process has not changed.
For each subgroup of size n, calculate the sample standard deviation using squared deviations from that subgroup's mean and denominator n−1. Plot these standard deviations in time order. An S chart is not simply a histogram of all measurements pooled together.
Use an adequate, reviewed baseline period to estimate ordinary process behavior. In the common equal-subgroup-size approach, the center line is the average subgroup standard deviation, s-bar. Baseline points with documented special causes require investigation rather than automatic deletion.
For conventional three-sigma-style limits under normality, use UCL = B4 times s-bar and LCL = B3 times s-bar. The constants depend on subgroup size. B3 is truncated at zero because standard deviation cannot be negative.
These constants incorporate c4, the factor relating the expected sample standard deviation to the population standard deviation under the normal model. B4 = 1 + 3 sqrt(1−c4²)/c4, and B3 is the larger of zero and 1 − 3 sqrt(1−c4²)/c4.
Keep the distinction between a three-sigma construction and exact probability limits. The sampling distribution of a standard deviation is asymmetric, particularly for small subgroups. Do not assume that every S-chart configuration has exactly the same false-alarm probability as a normal individual observation within three standard deviations.
Investigate signals using process knowledge and prespecified rules. A point beyond a limit is evidence of inconsistency with the baseline model, not a diagnosis of a particular cause. Unusually low variation can also reflect rounding, duplicated readings, or a changed measurement procedure.
Use a compatible chart of subgroup means when monitoring process location as well. If subgroup sizes vary, use a method that accounts for those sizes rather than applying one set of constants indiscriminately. Autocorrelation, non-normality, and changing operating conditions may require a different chart or model.

Three practical examples

Manufacturing: a team monitors within-batch variability in component dimensions.
Service operations: analysts track variation in comparable short-interval handling times after examining distributional assumptions.
AI-enabled inspection: engineers monitor the spread of a stable measurement process while separately tracking changes in the AI system or reference instrument.

Think of it this way

Think of checking how tightly a group of arrows clusters each round, separately from where the cluster lands. The S chart tracks the tightness. A companion mean chart tracks the location. A tightly clustered group can still be consistently off target, so the two questions should not be confused.

Worked example

Suppose a fictional, reviewed baseline uses subgroups of five measurements and has average subgroup standard deviation s-bar = 2 units. Under the stated normal-model construction, c4 for n = 5 is approximately 0.939986.
Then B4 = 1 + 3 sqrt(1−0.939986²)/0.939986, approximately 2.089. The corresponding B3 expression is negative and is therefore set to zero.
The center line is 2 units, the upper limit is approximately 2.089×2 = 4.178 units, and the lower limit is zero. A new subgroup with standard deviation 4.5 units exceeds the upper limit and triggers investigation under a single-point rule.
This does not say which measurement is defective or whether specifications are exceeded. It says the subgroup's spread is unusually high relative to this stipulated baseline and chart construction. Actual baseline adequacy and assumptions still require assessment.

Illustrative case study

Fictional case: Larch Packaging monitors fill-volume variation in short subgroups from one filling line. An S-chart signal appears after a maintenance change while the average fill volume remains near its usual level.
The team checks the measurement system and operating records, then investigates a valve-setting inconsistency. It documents the cause and corrective action before deciding whether a new stable baseline is justified. It does not continually widen the limits to absorb unexplained signals.

Common misconceptions and mistakes

Control limits are not specification limits. A stable process can still be incapable of meeting requirements.
An S chart monitors standard deviations, not subgroup averages or counts of defects.
No signal does not prove that nothing has changed; detection depends on sample size, shift magnitude, and chart design.
A zero lower limit is a consequence of the construction, not proof that zero variability is operationally plausible.
Recalculating limits after every inconvenient point can hide meaningful changes.

Business use

Use S charts to support disciplined investigation of changing variability. Their value depends on consistent measurements, sensible subgrouping, and a response plan. For AI-supported operations, record model, sensor, and preprocessing changes so a signal can be linked to the relevant process history rather than attributed vaguely to the algorithm.

Check your understanding

If all subgroup means stay constant while spreads increase, which chart is directly aimed at that change?

Answer: The S chart.

Does a point below the upper limit prove conformance to specifications?

Answer: No; control and specification questions differ.

Related terms

Statistical process control
Sample Standard Deviation
R chart
Control limits
subgroup
X-bar chart
control limit
process capability

Learn next

Construct rational subgroups for a process you understand. Compare a chart of subgroup means with a chart of subgroup standard deviations, explaining which changes each can reveal.

Rational subgroups

Sources

NIST/SEMATECH e-Handbook, Shewhart X-bar and R and S Control Charts: https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm The baseline values and case are original fictional illustrations. The c4-based arithmetic was independently checked.

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