I-MR / X-MR Chart

An I-MR chart monitors individual measurements and their moving ranges. It is the correct Shewhart chart when each time point has one observation rather than a rational subgroup.^[1]

Prerequisites

Prerequisites: control chart basics and ordered observations.

Process Context

Use I-MR for low-volume production, destructive tests, long cycle times, administrative processes, or any setting where subgrouping is not meaningful.

Definition

The individuals chart plots $x_{i}$ . The moving range chart plots the absolute difference between consecutive observations:

M R_{i} = | x_{i} - x_{i - 1} | .

Assumptions / Requirements

One observation per time point.
Consecutive observations are meaningful for estimating short-term variation.
The process is not strongly autocorrelated.
Measurement resolution is adequate.
Phase I data are screened for special causes before final limits are used.

Notation

Symbol	Meaning
$x_{i}$	Individual observation at time $i$
$\bar{x}$	Average of individual observations
$M R_{i}$	Moving range between $x_{i}$ and $x_{i - 1}$
$\overset{―}{M R}$	Average moving range
$d_{2}$	Range constant, $1.128$ for moving ranges of length 2

Control Limits / Formula

For the individuals chart:

U C L_{X} = \bar{x} + 3 \frac{\overset{―}{M R}}{1.128} = \bar{x} + 2.66 \overset{―}{M R}

C L_{X} = \bar{x}

L C L_{X} = \bar{x} - 2.66 \overset{―}{M R} .

For the moving range chart with range length 2:

U C L_{M R} = 3.267 \overset{―}{M R}, C L_{M R} = \overset{―}{M R}, L C L_{M R} = 0.

Interpretation Rules

Check the moving range chart for unstable short-term variation.
Investigate individual points outside limits and nonrandom patterns.
If autocorrelation is visible, use a time-series or residual approach instead of naive I-MR limits.
Never use specification limits like chart limits.

Worked Example

Five individual measurements are $10.1, 9.9, 10.3, 10.2, 10.0$ . Then $\bar{x} = 10.10$ , moving ranges are $0.2, 0.4, 0.1, 0.2$ , and $\overset{―}{M R} = 0.225$ .

U C L_{X} = 10.10 + 2.66 (0.225) = 10.6985

L C L_{X} = 10.10 - 2.66 (0.225) = 9.5015

U C L_{M R} = 3.267 (0.225) = 0.7351 .

No listed point or moving range exceeds these limits.

Common Mistakes

Calling an individuals chart an Xbar chart.
Creating artificial subgroups after the fact.
Ignoring autocorrelation from automated or continuous processes.
Using a moving range when consecutive observations are not comparable.

Connections

Related note	Use
Control charts	Chart selection
Xbar-R chart	True subgroup chart
Autocorrelated data	When I-MR assumptions fail
Control Limits and Specification Limits	Avoid limit confusion

References

NIST/SEMATECH, e-Handbook of Statistical Methods, "Individuals Control Charts", https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc322.htm ↩︎