X-bar and R Control Charts

An X-bar and R chart pair answers one question about a process measured in small samples: is it behaving the way it normally does, or has something changed? The X-bar chart watches the subgroup average and the R chart watches the spread inside each subgroup. This guide shows how to build them, how the limits are set, and how to read the signals.

To chart your own process data, see the control chart page.

Introduction

Every process varies. Two parts from the same machine are never identical, and two orders from the same warehouse never take exactly the same time. A control chart separates ordinary variation, the kind the process produces on its own, from signals that something specific has changed, such as a worn tool, a new supplier or a changed shift pattern.

Reacting to ordinary variation wastes effort, because adjusting a stable process makes it worse. Ignoring a real shift lets defects pile up. The chart gives a written rule for telling the two apart.

To check whether your own process is stable and can hold its specification, try the free control chart tool.

What is Control Charts (X-bar, R)?

An X-bar chart plots the mean of each subgroup (a small sample taken close together in time) against the order the subgroups were taken. An R chart plots the range of each subgroup, which is its largest value minus its smallest. Each chart has a center line and an upper and lower control limit.

The two are read together. The R chart tells you whether the spread is stable, and the limits on the X-bar chart are only meaningful when it is. A control limit is not a specification limit: control limits describe what the process does, and specification limits describe what the customer needs.

When to Use This Technique

Use X-bar and R charts when you measure a continuous quantity, such as a length, weight, fill volume or cycle time, in small subgroups of roughly 2 to 10 taken at regular intervals, and the subgroup is a set of items made under the same conditions. For one measurement per interval, an individuals chart is the usual choice. For counts of defects or of defective items, use an attribute chart. For catching very small, persistent shifts, CUSUM charts respond faster than a Shewhart chart.

How It Works

Take k subgroups of size n. Compute each subgroup mean and range, then the grand mean X̄ (the average of the subgroup means) and the average range R̄. The limits follow from R̄ and tabulated constants that depend on n:

X-bar chart:  center = grand mean
              UCL, LCL = grand mean ± A2 × R-bar
R chart:      center = R-bar
              UCL = D4 × R-bar,  LCL = D3 × R-bar

For subgroups of size 5, the standard constants are A2 = 0.577, D3 = 0 and D4 = 2.114. The within-subgroup standard deviation is estimated as R̄ / d2, with d2 = 2.326 for n = 5. The X-bar limits sit about three standard errors of the subgroup mean either side of the center line, so a stable, roughly normal process puts about 99.7% of subgroup means inside them.

Step-by-Step Process

1. Choose the measurement and a subgroup size, and keep conditions inside each subgroup as uniform as possible so the range reflects only short-term variation.
2. Collect enough subgroups. Twenty to twenty-five is the usual guidance for setting limits; the example below uses six only to keep the arithmetic readable.
3. Compute each subgroup mean and range, then the grand mean and R̄.
4. Check the R chart first. If a range is outside its limits, find the cause before trusting the X-bar limits.
5. Apply the limits to the X-bar chart and check for signals.
6. When you find and remove a special cause, drop the affected subgroups and recompute the limits. Do not drop points just because they look odd.

Interpreting Results

A point outside a control limit is the basic signal. Run rules add sensitivity to smaller shifts. Commonly used ones are eight points in a row on one side of the center line, six points in a row steadily rising or falling, and two of three points beyond two standard errors on the same side. Each extra rule also raises the false-alarm rate, so choose the rule set before you look at the data and apply it consistently.

No signal means the process looks stable, not that it is good. A stable process can still produce output outside the specification. Capability measures such as Cp and Cpk compare the process spread to the specification, and they only make sense once the process is in control.

Real-World Example

These are synthetic numbers written for this example, not measurements from a real process: a filling line is sampled six times, taking five fill weights in grams each time. The subgroups are:

50.2 49.8 50.1 50.4 49.9
50.0 50.3 49.7 50.1 50.2
49.9 50.5 50.2 49.8 50.0
50.1 50.0 49.6 50.3 50.2
50.4 49.9 50.0 50.2 50.6
49.8 50.1 50.3 49.9 50.0

Each subgroup mean and range is the plain average and the largest minus the smallest of its five values, which gives:

The grand mean is 50.083 g and R̄ is 0.633 g. With n = 5, the X-bar limits are 50.083 ± 0.577 × 0.633, which is 49.718 to 50.449 g. The R chart limits are 0 and 2.114 × 0.633 = 1.339 g. Every subgroup mean and range falls inside its limits, so the line looks stable on this evidence. The estimated within-subgroup standard deviation is 0.633 / 2.326 = 0.272 g. Six subgroups is too few to set limits for real use; it is enough to show the arithmetic.

Best Practices

• Set limits from a period when the process was running normally, and keep them fixed until you have a reason to recompute them.
• Build subgroups from items made close together, so differences between subgroups reflect real change over time.
• Do not draw specification limits on an X-bar chart. Subgroup means vary less than individual items, so the two are not on the same scale.
• Check that the measurement system is itself repeatable before charting. A noisy gauge shows up as a wide R chart. Gauge R&R studies test this.
• Write down what you did when a signal fired and what you found. The record is what makes the chart useful later.

Related Techniques

CUSUM charts accumulate small deviations and detect small sustained shifts sooner. Gauge R&R studies check the measurement system behind the data. To chart your own data, use the free control chart tool.

Conclusion

An X-bar and R pair turns a stream of measurements into a decision rule: leave a stable process alone, and investigate when a signal appears. The limits come from your own process data, not from the specification, and the R chart should always be read first.