I-MR Control Chart Calculator — Individuals & Moving Range SPC

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Industry Use

Individuals and Moving Range Control in Pharma

The Individuals and Moving Range (I-MR) chart is used when individual measurements are collected without the ability to form rational subgroups. This is extremely common in pharmaceutical manufacturing: one tablet sampled per time point, one batch per day, or equipment producing only single items. The I chart monitors location (process mean); the MR chart monitors spread (process variation).

I-MR charts are required by FDA Process Validation Guidance (2011) for Stage 3 continued process verification. They detect special-cause variation using Western Electric rules: out-of-control points, runs, trends, and cycling patterns. The moving range is computed from consecutive pairs of observations, making I-MR highly sensitive to process changes.

Typical applications: potency assay (one batch = one test), tablet weight sampling (one tablet per shift), fill volume checks, assay of reference standards, and dissolution testing. I-MR data is also used to estimate σ̂ for Cpk calculations.

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Calculation Explanation

Mathematical Basis and Control Limits

The I-MR chart consists of two companion charts with linked control limits based on moving range variation.

I Chart — Individuals (Process Mean)

X̄ = (Σ xᵢ) / n

UCL_I = X̄ + 3 × σ̂

LCL_I = X̄ − 3 × σ̂

MR Chart — Moving Range

MRᵢ = |xᵢ − xᵢ₋₁| (consecutive differences)

MR̄ = (Σ MRᵢ) / (n − 1)

σ̂ = MR̄ / d₂ where d₂ = 1.128

UCL_MR = D₄ × MR̄ where D₄ = 3.267

LCL_MR = 0

Control limits: The ±3σ̂ limits correspond to the standard 99.73% statistical control limit. Points beyond UCL or LCL indicate special-cause variation requiring investigation.

Western Electric Rules (AIAG) detect sustained shifts:

1 point beyond 3σ
9 consecutive points on same side of center line
6 consecutive points increasing or decreasing (trend)
2 of 3 points beyond 2σ on same side

FDA Process Validation (2011) AIAG SPC Reference Manual Montgomery Control Charts

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Variable Definitions

Symbols, Units, and Descriptions

Symbol	Name	Units	Description
xᵢ	Individual Observation	product units	Single measurement (e.g., potency of one batch)
X̄	Center Line (Mean)	product units	Average of all individual observations
MRᵢ	Moving Range	product units	Absolute difference between consecutive observations: \|xᵢ − xᵢ₋₁\|
MR̄	Average Moving Range	product units	Mean of all moving range values. Used to estimate process variation.
σ̂	Estimated Standard Deviation	product units	σ̂ = MR̄ / 1.128. Reflects short-term process variation.
UCL_I	Upper Control Limit (I Chart)	product units	UCL = X̄ + 3σ̂. Points above trigger out-of-control signal.
LCL_I	Lower Control Limit (I Chart)	product units	LCL = X̄ − 3σ̂. Points below trigger out-of-control signal.
UCL_MR	Upper Control Limit (MR Chart)	product units	UCL = 3.267 × MR̄. Detects sudden increases in variation.
d₂	Control Chart Constant	dimensionless	For moving range (n=2): d₂ = 1.128
D₄	Control Chart Constant	dimensionless	For moving range (n=2): D₄ = 3.267
n	Sample Size	count	Number of individual observations collected

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Step-by-Step Tutorial

Worked Example — Potency Assay I-MR Chart

Scenario: A potency assay is performed on one batch per day. Ten consecutive daily assay results (%LC) are: 99.2, 100.5, 98.8, 101.2, 99.7, 100.1, 101.4, 99.5, 100.8, 99.9. Construct an I-MR chart and assess control.
Data Collection:

Potency Assay Results (n=10 batches)
Day 1: 99.2%LC
Day 2: 100.5%LC
Day 3: 98.8%LC
Day 4: 101.2%LC
Day 5: 99.7%LC
Day 6: 100.1%LC
Day 7: 101.4%LC
Day 8: 99.5%LC
Day 9: 100.8%LC
Day 10: 99.9%LC
Step 1: Calculate the mean (center line for I chart).
X̄ = (99.2 + 100.5 + 98.8 + 101.2 + 99.7 + 100.1 + 101.4 + 99.5 + 100.8 + 99.9) / 10
X̄ = 1001.1 / 10 = 100.11%
Step 2: Calculate moving ranges.
MR₁ = |100.5 − 99.2| = 1.3
MR₂ = |98.8 − 100.5| = 1.7
MR₃ = |101.2 − 98.8| = 2.4
MR₄ = |99.7 − 101.2| = 1.5
MR₅ = |100.1 − 99.7| = 0.4
MR₆ = |101.4 − 100.1| = 1.3
MR₇ = |99.5 − 101.4| = 1.9
MR₈ = |100.8 − 99.5| = 1.3
MR₉ = |99.9 − 100.8| = 0.9
Sum of MR = 12.7
Step 3: Calculate average moving range and estimate σ̂.
MR̄ = 12.7 / (10 − 1) = 12.7 / 9 = 1.411%

σ̂ = MR̄ / d₂ = 1.411 / 1.128 = 1.251%
Step 4: Calculate I chart control limits.
UCL_I = X̄ + 3 × σ̂ = 100.11 + 3 × 1.251 = 100.11 + 3.753 = 103.86%

LCL_I = X̄ − 3 × σ̂ = 100.11 − 3 × 1.251 = 100.11 − 3.753 = 96.36%

Center line = 100.11%
Step 5: Calculate MR chart control limits.
UCL_MR = D₄ × MR̄ = 3.267 × 1.411 = 4.61%

LCL_MR = 0

Center line = 1.411%
Step 6: Check for out-of-control signals.
I Chart Check: All 10 observations fall between 96.36% and 103.86%. No points beyond control limits. ✓

MR Chart Check: All 9 moving ranges ≤ 4.61%. No out-of-control points. ✓

Western Electric Rules: No 9-point runs above/below center; no 6-point trends detected visually. ✓
Interpretation and Action:
Process is IN STATISTICAL CONTROL.

Potency assay shows only common-cause variation with no evidence of special causes. Process is stable and suitable for continued monitoring. Recommend continuing daily I-MR charting to maintain SPC surveillance during Stage 3 process verification.

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