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I-MR chart (individuals and moving range)

The I-MR chart is the right choice when each measurement is a single individual value (n = 1 per subgroup): batch yields, daily pH readings, lot-by-lot test results, or any process where subgrouping is not practical. It plots two charts together: the Individuals (I) chart for the measurements and the Moving Range (MR) chart for the range between consecutive readings.

Constants (for n = 2 successive observations)

d₂ = 1.128 → σ̂ = MR̄ / 1.128
D₄ = 3.267 → UCL_MR = D₄ × MR̄
D₃ = 0 → LCL_MR = 0 (for n = 2)

Worked example

Ten daily batch-yield readings: 23, 25, 22, 27, 24, 26, 23, 25, 26, 24.

X̄ = (23+25+22+27+24+26+23+25+26+24) / 10 = 245 / 10 = 24.5 Moving ranges (|Xᵢ − Xᵢ₋₁|): |25−23|=2, |22−25|=3, |27−22|=5, |24−27|=3, |26−24|=2, |23−26|=3, |25−23|=2, |26−25|=1, |24−26|=2 MR̄ = (2+3+5+3+2+3+2+1+2) / 9 = 23 / 9 = 2.556 I-chart control limits: UCLᵢ = X̄ + 3 × (MR̄ / 1.128) = 24.5 + 3 × (2.556 / 1.128) = 31.30 LCLᵢ = X̄ − 3 × (2.556 / 1.128) = 17.70 MR-chart control limits: UCL_MR = 3.267 × 2.556 = 8.35 LCL_MR = 0

No points fall outside the I-chart limits (31.30, 17.70), and the largest MR (5) is below 8.35. The process appears stable.

Practice problem

Five individual readings: 10, 14, 12, 16, 13. MRs: 4, 2, 4, 3. MR̄ = 13/4 = 3.25. Compute UCLᵢ.

Show answer

X̄ = 65/5 = 13.0. σ̂ = 3.25/1.128 = 2.881. UCL = 13.0 + 3×2.881 = 21.64.

Common exam traps

Use I-MR only for n = 1. If you can form rational subgroups (e.g., five parts per hour), use an X̄-R or X̄-S chart instead—they have more statistical power.
The I chart cannot detect within-subgroup variation. Because n = 1, you cannot distinguish between-subgroup and within-subgroup variation. The MR estimates short-term variability only.
Non-normal data inflate false alarms. The 3σ limits assume normality. Highly skewed distributions (e.g., Poisson-like cycle times) may need a transformation or a different chart type.
MR̄ is computed from n = 2 moving ranges. Each MR is the absolute difference between two consecutive readings, giving n−1 MR values for a dataset of n observations.

Need A₂, D₄, d₂ or c₄ for your subgroup size? See the free control chart constants table (n = 2–25) with an X̄-R / X̄-S limits calculator.

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