I-MR Control Chart (Individuals and Moving Range)

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Industry Use

How this chart is applied in pharmaceutical manufacturing

The individuals and moving-range (I-MR) pair charts single measurements when rational subgroups are not available — one batch, one assay, one reading at a time. The I chart tracks the process level; the MR chart tracks short-term variability through the absolute difference of consecutive points. Because there is no subgroup averaging, the I chart leans directly on the distribution of individual values — see the normality caution below.

Where this is used:

Batch-level results: potency, yield, assay of one batch per run.
In-process single readings: fill volume, weight checks, environmental data.
Stage 3 continued process verification under the FDA process-validation lifecycle.

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Calculation Explanation

Mathematical basis, limit modes, and the five rules

Moving range and sigma estimate

MRᵢ = |xᵢ − xᵢ₋₁| · MR̄ = Σ MRᵢ / (n − 1) · σ̂ = MR̄ / d₂ , d₂ = 1.128

Control limits

I chart: X̄ ± 3·σ̂ · MR chart: UCL = D₄·MR̄ = 3.267·MR̄ , LCL = 0

Two limit modes. ESTIMATED limits come from the data at hand (above). SPECIFIED limits use a standard mean and sigma from a reference period — in that mode the specified σ̂ drives both charts (I limits at X̄_spec ± 3σ̂_spec and the MR limits via the implied MR̄ = d₂·σ̂_spec), and the software cross-checks the observed mean against the specified one.

Western Electric rules (as implemented, each on the I chart): Rule 1 — one point beyond 3σ. Rule 2 — two of three consecutive points beyond 2σ on the same side. Rule 3 — four of five consecutive points beyond 1σ on the same side. Rule 4 — eight consecutive points on the same side of center. Rule 5 — six consecutive points steadily increasing or decreasing.

Western Electric SQC Handbook (1956) NIST e-Handbook §6.3.2 FDA Process Validation (2011)

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Variable Definitions

All input parameters and their meaning

Symbol	Variable	Units	Description
xᵢ	Individual observation	product units	Single measurement (e.g. potency of one batch).
X̄	Center line (mean)	product units	Mean of the individuals (estimated mode) or the specified standard mean.
MRᵢ / MR̄	Moving range; its average	product units	\|xᵢ − xᵢ₋₁\| between consecutive points; MR̄ estimates short-term variability.
σ̂	Estimated standard deviation	product units	MR̄ / 1.128 (estimated mode) or the specified sigma — which then drives both charts.
d₂ / D₄	Control-chart constants (n = 2)	dimensionless	d₂ = 1.128, D₄ = 3.267 for a moving range of two consecutive points.
Rules 1–5	Western Electric rule toggles	—	Enable per your SOP; more rules = more sensitivity and more false alarms.

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Step-by-Step Tutorial

Worked example: daily potency assay

Scenario: one potency assay per day, ten results (%LC): 99.2, 100.5, 98.8, 101.2, 99.7, 100.1, 101.4, 99.5, 100.8, 99.9.

Compute the center line.

Calculation
X̄ = 1001.1 / 10 = 100.11%
Compute the moving ranges.

Calculation
MR: 1.3, 1.7, 2.4, 1.5, 0.4, 1.3, 1.9, 1.3, 0.9 → MR̄ = 12.7 / 9 = 1.411%
Estimate σ̂ and set the limits.

Calculation
σ̂ = 1.411 / 1.128 = 1.251%
I chart: 100.11 ± 3.753 → LCL 96.36, UCL 103.86
MR chart: UCL = 3.267 × 1.411 = 4.61, LCL = 0
Test the rules: all points and ranges inside limits; no rule fires (longest same-side run 2; longest monotone run 3 — points 5–7 rise 99.7 → 100.1 → 101.4, well short of the 6 Rule 5 needs).

Interpretation
Process is in statistical control — only common-cause variation. Continue charting.

✓ In statistical control — X̄ = 100.11%, limits 96.36–103.86%, MR UCL 4.61, no rule signals

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Interpretation, Assumptions and Limitations

What a signal means — and what the chart presumes

Signals: any rule firing is special-cause variation to investigate per SOP — the zone rules (2, 3) and run rules (4, 5) catch shifts and trends smaller than a 3σ excursion. An MR-chart signal means short-term variability changed; investigate it before interpreting the I chart, whose limits depend on MR̄.
Normality matters here more than on any subgrouped chart: the I chart has no central-limit protection, so skewed data (impurities, particle counts, microbial counts) produce false signals or missed ones. Check the distribution; transform or use an appropriate alternative when it fails.
Independence: autocorrelated data (frequent sampling of a slow process) narrows MR̄ and produces false alarms.
Short-term σ̂: with drifting data MR̄/d₂ understates total variation — which is exactly why Cpk and Ppk are reported together.
Stable reference period: recompute limits only per your SOP, not after every point.

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References

Primary sources for this guide

Western Electric Company, Statistical Quality Control Handbook (1956) — origin of the zone/run rules.
NIST/SEMATECH e-Handbook of Statistical Methods, §6.3.2 (individuals control charts). itl.nist.gov
FDA, Process Validation: General Principles and Practices (2011) — Stage 3 context. fda.gov