Process monitoring

I-MR Control Charts: How to Calculate and Interpret Them

Kitstead Labs · Engineering reference · Synthetic worked example

When each sampling occasion yields one measurement, an I-MR chart can make changes in level or short-range variation visible. This guide builds both charts from an ordered thickness series, identifies the actual limit violations and compares them with specification checks. The example is small so every calculation is inspectable; a real monitoring baseline needs a defensible sampling plan.

Quick answer

An I-MR control chart plots individual measurements and absolute differences between successive values. With moving ranges of length two, estimate within sigma as MR̄/1.128 and Individuals limits as mean ± 3 sigma. Control limits describe process behavior; specification limits express engineering requirements.

What an Individuals and Moving Range chart shows

An I-MR chart pairs each ordered value with the magnitude of its change from the preceding value. It is useful when a meaningful sampling occasion supplies one observation, rather than a rational subgroup. Use subgroup charts when the sampling design actually supports them; do not discard subgroup information simply to use an I-MR template.

The Individuals chart examines location and the Moving Range chart examines local changes. Both require a coherent process sequence. Serial correlation, a change of measurement system or concatenated equipment populations can make the usual limits misleading. Adjacent moving ranges share measurements, so they are not independent observations themselves.

MR(2) formulas and constants

For n ordered values x_i, calculate n−1 absolute differences and average them. There is no moving range for the first value. For moving ranges of length two, d₂ = 1.128, D₃ = 0 and D₄ = 3.267. These conventional rounded constants assume the usual normal-error range model.

MRᵢ = |xᵢ − xᵢ₋₁|; MR̄ = ΣMRᵢ / (n − 1)
I center = x̄; I limits = x̄ ± 3(MR̄ / 1.128)
MR center = MR̄; MR LCL = 0; MR UCL = 3.267 MR̄

All limits have the same units as the measurements. Estimate a baseline from enough representative observations, investigate signals, and document justified phase changes. Do not recalculate limits simply to absorb a concerning new point.

Worked example: ordered synthetic thickness data

This is the same fully synthetic series used in Cpk vs Ppk. We deliberately include a level shift; these ten points illustrate arithmetic, not an approved control-chart baseline.

Synthetic thickness measurements in collection order
Observation	Thickness (µm)	Moving range (µm)
1	10.00	—
2	10.04	0.04
3	10.02	0.02
4	10.06	0.04
5	10.04	0.02
6	10.20	0.16
7	10.24	0.04
8	10.22	0.02
9	10.26	0.04
10	10.24	0.02

x̄ = 101.32/10 = 10.132 µm. MR̄ = 0.40/9 = 0.044444 µm. The within estimate is 0.039401 µm, so the three-sigma distance is 0.118203 µm. Calculations below use unrounded values.

Synthetic I-MR limits in µm
Chart	Lower limit	Center	Upper limit
Individuals	10.013797	10.132000	10.250203
Moving Range	0	0.044444	0.145200

Synthetic data in collection order. Dashed lines are calculated control limits and center line; horizontal axis is observation number. The data table provides the exact plotted values.

Observation 1 (10.00 µm) falls below the Individuals LCL and observation 9 (10.26 µm) exceeds its UCL. The moving range at observation 6 is 0.16 µm, exceeding 0.1452 µm. It marks the jump from 10.04 to 10.20 µm. These are signals for investigation; the example's known synthetic shift does not teach a general root-cause diagnosis.

With independently specified LSL = 9.50 µm and USL = 10.50 µm, none of the ten values is out of specification. Thus this example has two single-point Individuals OOC values, one MR limit signal, and zero OOS values. The MR signal is not an additional Individuals OOC count.

Control limits are not specification limits

OOC means a value breaches a statistical control criterion; OOS means a measurement breaches an engineering specification. A process can produce an in-spec signal, as above, or remain statistically consistent while producing unacceptable values. Keep customer/design limits separate from limits estimated from process behavior.

A signal should prompt a check of sampling, instruments, materials and process context. A chart with no points beyond its limits still does not prove that all assumptions hold. Run patterns may matter, but only interpret rule sets you deliberately chose and validated.

Common mistakes

Sorting measurements by size destroys the temporal information and usually reduces moving ranges.
Using n instead of n−1 when averaging the n−1 ranges changes the limits.
Carrying a moving range across a documented process reset can mix distinct phases. Model phases explicitly and justify any exclusions.
Replacing control limits with LSL/USL changes the question the chart answers.

Review I-MR charts in desktop Excel

The released SPC & Capability Analyzer v0.1 includes Individuals and Moving Range charts and separate specification-limit visibility. Its OOC count evaluates single-point Individuals limit violations only. It does not implement Nelson, Western Electric or other advanced rule sets. Its capability estimates should be interpreted alongside this sequence review.

Sources and further reading

NIST: Individuals control charts — moving ranges and Individuals limits.
NIST: Shewhart X-bar and R control charts — range-chart constants and subgroup context.