Individuals And Moving Range Chart

Individuals and Moving Range Chart (manufacturing-quality/as9100/individuals-and-moving-range-chart)

Use when the process yields one measurement per lot or subgroup, so subgroup charts cannot be built: destructive testing consumes the sample, a single unit ships per batch, or one bond or coating coupon is pulled per lot. The I-MR chart turns the time-ordered individual measurements into moving ranges between successive values, draws both charts from the average moving range with the E2 and 3.267 factors, and returns the lot-to-lot stability verdict. It pairs with manufacturing-quality/as9100/statistical-process-control, which owns the subgroup charts (X-bar and R with the A2/D3/D4 constants, Cp/Cpk) that apply when subgroup data exist.

Domain quick reference

• Moving range at position i: MR_i = |x_i - x_{i-1}| for i >= 1; N individuals give N-1 moving ranges.
• Average moving range: mr_bar = mean(MR_i).
• Individuals (X) chart central line and limits: centerline = mean of the individuals; UCL = mean + E2 * mr_bar; LCL = mean - E2 * mr_bar, with the n = 1 constant E2 = 2.66.
• Process sigma estimate from the moving ranges: sigma_hat = mr_bar / d2, with d2 = 1.128 (the n = 2 constant for two-point moving ranges).
• Moving range chart limit: UCL = 3.267 * mr_bar; the MR lower limit is 0 because moving ranges cannot be negative.
• Verdict: in-control when no individual and no moving range falls outside its limits, else out-of-control.
• A point exactly on a limit is inside (inclusive interval).
• AS9100 clause 8.5.1 frames production process control; the I-MR chart is the common evidence of stability for a single-measurement special process, summarized here without clause text.

Workflow

1. Collect the time-ordered individual measurements, one per lot, in production sequence (moving_ranges and the limits all assume order).
2. Compute the moving ranges between successive values with moving_ranges(values).
3. Get the central line and X chart limits with individuals_limits(values); read mean, mr_bar, sigma_hat, UCL, LCL from the returned dict.
4. Get the MR chart limit with moving_range_limits(values); read mr_bar and UCL (LCL is 0).
5. Flag the out-of-limit points with flag_points(values, ucl, lcl); the returned indices point into the values list. Flag the moving ranges with flag_points(mr_values, mr_ucl, 0.0), where MR index i spans original measurements i and i + 1.
6. Combine both flag lists with stability_verdict(individual_flags, mr_flags) for the lot-to-lot verdict, or run imr_summary(values) once to get mean, mr_bar, sigma_hat, x_ucl, x_lcl, mr_ucl, flagged_individuals, flagged_moving_ranges and verdict in one dict.
7. Reject inputs first: fewer than 2 values raise ValueError (a chart needs at least one moving range); flag_points rejects an empty list.

Worked example

Twelve bond-lot pull-off force measurements, one destructive coupon per lot: values [42.1, 41.6, 43.0, 42.4, 41.2, 44.1, 43.5, 42.8, 41.9, 43.2, 44.0, 42.6]. Real module outputs:

• mean = 42.700; mr_bar = 1.118; sigma_hat = 0.991.
• X chart: UCL = 45.674; LCL = 39.726 (mean plus and minus 2.66 * 1.118).
• MR chart: UCL = 3.653 (3.267 * 1.118); LCL = 0.
• Flagging: no individual outside [39.726, 45.674] and no moving range above 3.653 (the largest, 2.90, stays inside), so the verdict is in-control: lot-to-lot stability is demonstrated with no subgroup data.
• Appending a 50.0 coupon flags individual index 12 and moving range index 11, and the verdict flips to out-of-control.

Verification

• Confirm the worked series returns mean 42.700, mr_bar 1.118, sigma_hat 0.991, UCL 45.674, LCL 39.726, MR UCL 3.653 (each within 1e-3) and verdict in-control.
• Confirm [1, 5, 2] gives moving ranges [4, 3] and a constant series gives all-zero moving ranges, zero sigma_hat and X limits collapsed to the constant mean.
• Confirm a two-value series gives mr_bar equal to the absolute difference and UCL_X = mean + 2.66 * |difference|.
• Confirm adding a large outlier widens the X limits (the moving range grows) and flags that position.
• Confirm ValueError rejection: fewer than 2 values for moving_ranges/individuals_limits/moving_range_limits, an empty list for flag_points.
• Confirm determinism: identical inputs give identical dicts.
• Run the contract test offline: python3 scripts/test_individuals_and_moving_range_chart.py (35 tests, deterministic).

Related leaves

• manufacturing-quality/as9100/statistical-process-control: X-bar and R subgroup charts with the A2/D3/D4 constants, subgroup d2 sigma estimate and Cp/Cpk; use it when subgroups exist, not for n = 1.
• manufacturing-quality/as9100/cusum-ewma-monitoring: CUSUM and EWMA statistics for detecting small sustained mean shifts on individual values.
• manufacturing-quality/as9100/attribute-control-charts: p, np, c and u charts for defect and nonconformity rates when the response is counted, not measured.

Pitfalls

• Feeding values out of production order: moving ranges pair successive measurements, so the limits, flags and verdict all assume time-ordered one-per-lot sequence - a reordered series produces a different chart.
• Using the I-MR chart where subgroup data exist: X-bar and R with the A2/D3/D4 constants belong to the statistical-process-control sibling; I-MR is the n = 1 fallback (destructive testing, single unit per batch), not a replacement for subgroup charting.
• Treating a moving-range flag as an individual flag: MR index i spans original measurements i and i + 1, so flagging must be read against the MR list, not the individuals list.
• Ignoring the outlier-limits interaction: injecting a 50.0 coupon both flags the point and inflates mr_bar, widening the X limits - an early extreme value can mask a later excursion of the same size.
• Misreading a point exactly on a limit: limits are inclusive, so a value equal to UCL or LCL is inside the chart, not a flag.
• Charting with fewer than two measurements: a single lot yields no moving range, and fewer than 2 values (or an empty flag list) raises ValueError rather than producing a chart.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_individuals_and_moving_range_chart.py

The test covers the worked-example limits (mean 42.700, mr_bar 1.118, sigma_hat 0.991, X UCL 45.674 / LCL 39.726, MR UCL 3.653 within 1e-3), moving range arithmetic and the N-1 count, the constant-series collapse and two-value identities, boundary points on the limits, outlier flagging at the injected index with the out-of-control verdict, the exact convenience dict keys, determinism, and ValueError rejection of fewer than 2 values and of empty flag input.

Compliance

• Standards referenced, not reproduced: AS9100 (SAE) is cited as the production process control frame; the I-MR equations above are standard engineering methodology (the n = 1 control chart constants E2 = 2.66 and MR factor 3.267), summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.