Xbar-R Chart

An Xbar-R chart monitors a variable process characteristic using rational subgroups. The Xbar chart monitors subgroup means, and the R chart monitors subgroup ranges.^[1]

Prerequisites

Prerequisites: control chart basics and rational subgrouping.

Process Context

Use Xbar-R when each plotted point summarizes a small subgroup, commonly 2 to 10 observations, collected close enough in time that within-subgroup variation represents short-term common-cause variation.

Definition

For subgroup $i$ with size $n$ , the subgroup mean is ${\bar{x}}_{i}$ and the subgroup range is $R_{i} = x_{i, max} - x_{i, min}$ . The chart pair is interpreted together: first check the R chart for stable short-term variation, then interpret the Xbar chart.

Assumptions / Requirements

Variable measurement data.
Same subgroup size for the constants shown below.
Independent subgroups in time.
Approximate normality is most important for small-sample probability interpretation.
Rational subgrouping is defensible.

Notation

Symbol	Meaning
$n$	Subgroup size
${\bar{x}}_{i}$	Mean of subgroup $i$
$R_{i}$	Range of subgroup $i$
$\bar{\bar{x}}$	Average of subgroup means
$\bar{R}$	Average of subgroup ranges
$A_{2}, D_{3}, D_{4}$	Control chart constants based on $n$

Control Limits / Formula

For equal subgroup size:

\begin{aligned} U C L_{\bar{X}} & = \bar{\bar{x}} + A_{2} \bar{R}, & C L_{\bar{X}} & = \bar{\bar{x}}, & L C L_{\bar{X}} & = \bar{\bar{x}} - A_{2} \bar{R}, \\ U C L_{R} & = D_{4} \bar{R}, & C L_{R} & = \bar{R}, & L C L_{R} & = D_{3} \bar{R} . \end{aligned}

For $n = 5$ , common constants are $A_{2} = 0.577$ , $D_{3} = 0$ , and $D_{4} = 2.115$ .^[1:1]

Interpretation Rules

Investigate any point outside control limits.
Investigate nonrandom patterns, especially runs, trends, or cycles.
If the R chart is unstable, never interpret Xbar shifts like stable mean behavior.
Do not compare points to specification limits on the control chart.

Worked Example

Four subgroups of size $n = 5$ have means $10.02, 9.98, 10.05, 10.00$ and ranges $0.12, 0.15, 0.10, 0.13$ . Then $\bar{\bar{x}} = 10.0125$ and $\bar{R} = 0.125$ .

U C L_{\bar{X}} = 10.0125 + 0.577 (0.125) = 10.0846

L C L_{\bar{X}} = 10.0125 - 0.577 (0.125) = 9.9404

U C L_{R} = 2.115 (0.125) = 0.2644, L C L_{R} = 0.

All listed subgroup means and ranges are inside these limits, so this small example shows no control-limit signal.

Common Mistakes

Using Xbar-R for one observation at a time.
Ignoring the R chart and interpreting only the Xbar chart.
Forming subgroups from mixed machines, shifts, or materials when the goal is short-term variation.
Using specification limits as the Xbar chart limits.

Connections

Related note	Use
Control charts	Chart selection
Xbar-S chart	Alternative using subgroup standard deviations
I-MR / X-MR chart	Individual observations
Control Limits and Specification Limits	Avoid limit confusion
Process capability	After stable control

References

NIST/SEMATECH, e-Handbook of Statistical Methods, "Shewhart X-bar and R and S Control Charts", https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm ↩︎ ↩︎