Gage Linearity Bias Study

Gage Linearity and Bias Study (manufacturing-quality/as9100/gage-linearity-bias-study)

Use when the task is judging a gage's bias and linearity at the conceptual level: reference-level masters are measured repeatedly, the bias (observed minus reference) is recorded at each level, and the study asks whether the gage reads true on average (bias significance test) and whether the bias grows across the measurement range (linearity regression of bias on the reference value). This leaf implements the per-level bias, the mean bias, the least-squares linearity statistics, the t test against the two-sided 95 percent critical at the study degrees of freedom, and the percent-of-reference acceptability band, in pure Python, stdlib only. It pairs with manufacturing-quality/as9100/measurement-systems-analysis for the repeatability and reproducibility variance decomposition (which does not compute bias or linearity studies), and with manufacturing-quality/as9100/calibration-control for instrument calibration state and traceability.

Domain quick reference

• Conventions: references are the certified master values in mm, one strictly increasing unique value per level; one bias per level with bias = observed - reference (signed mm, positive means the gage reads high); n = number of reference levels (3 or more).
• Per-level bias and percent of reference: bias_pct_i = 100 * bias_i / reference_i. A level is acceptable when |bias_pct| <= 10 percent of reference (ACCEPTANCE_PCT_BAND).
• Mean bias: bias_bar = sum(bias_i) / n. The spread terms: sst = sum (bias_i - bias_bar)^2 and s = sqrt(sst / (n - 1)).
• Bias significance: t = bias_bar / (s / sqrt(n)) against the two-sided 95 percent t critical at df = n - 1 (12.706 at 1 df down to 2.086 at 20 df, 1.96 above 20 df); significant when |t| >= t_crit.
• Linearity regression: least squares fit bias = intercept + slope * reference, with slope = Sxy / Sxx and intercept = bias_bar - slope * xbar over the level values; sse = sum (bias_i - predicted_i)^2 and r2 = 1 - sse / sst measure how well a straight line carries the bias trend across the range.
• Overall verdict: ACCEPT when the mean bias is not statistically significant AND every level is inside the percent band, else REVIEW. A significant mean bias alone forces REVIEW even when all levels are inside the band.
• AS9100 frames monitoring and measuring resources as controlled and fit for purpose; the bias and linearity study is one common measurement-system evidence practice in that context, summarized here without clause text.

Workflow

1. Collect the study inputs: reference masters (mm) and the measured bias at each level (mm), as equal-length lists with 3 or more strictly increasing positive references.
2. Get the per-level table with per_level_bias(references, biases): reference, bias and bias_pct_of_reference per level.
3. Read the overall mean bias with mean_bias(biases).
4. Fit the linearity evidence with linearity_regression(references, biases): slope, intercept, sse, r_squared, n, xbar, bias_bar.
5. Test the mean bias with bias_significance(biases): t_stat, t_crit, df and the significant flag.
6. Run the whole study with gage_bias_linearity_study(references, biases) for the combined verdict: worst_bias_pct, worst_reference, per_level_acceptable and overall ACCEPT or REVIEW.
7. Confirm the deterministic checks with the contract test scripts/test_gage_linearity_bias_study.py.

Worked example

Reference study: masters [2, 4, 6, 8, 10] mm with measured biases [0.07, 0.10, 0.16, 0.18, 0.24] mm. Module outputs:

• Per-level bias percent of reference: 3.50, 2.50, 2.67, 2.25, 2.40 percent (real module values 3.5, 2.5, 2.6667, 2.25, 2.4), all inside the 10 percent band.
• Mean bias = 0.150 mm; sst = 0.0180; s = 0.06708 mm.
• Linearity regression: slope = 0.0210 mm/mm, intercept = 0.0240 mm, sse = 0.00036, r2 = 0.980 (bias grows about 0.021 mm per mm of reference).
• Bias significance: t = 0.150 / (0.06708 / sqrt(5)) = 5.000; df = 4; t_crit = 2.776; |5.000| >= 2.776, so significant = True.
• Worst percent bias = 3.50 percent at the 2 mm level; all levels are inside the 10 percent band, yet the study verdict is REVIEW because the mean bias is statistically significant.

Verification

• Confirm per_level_bias on the anchor returns the five rows with bias_pct_of_reference 3.50, 2.50, 2.67, 2.25, 2.40 within 1e-2, and that mean_bias returns 0.150.
• Confirm linearity_regression returns slope 0.0210 within 1e-4, intercept 0.0240 within 1e-3, sse 0.00036 within 1e-5 and r2 0.980 within 1e-3, that the regression residuals sum to zero within 1e-9, and that r2 equals 1 - sse/sst exactly.
• Confirm bias_significance returns t = bias_bar / (s / sqrt(n)) exactly per the formula (5.000 on the anchor), t_crit 2.776 at 4 df, and significant True; a small-bias fixture with all biases 0.01 returns significant False.
• Confirm gage_bias_linearity_study returns REVIEW on the anchor (significant bias despite the band) and ACCEPT on the all-biases-0.01 fixture.
• Confirm ValueError rejection of mismatched arrays, fewer than 3 levels, references that are not strictly increasing, any reference at or below zero, and empty bias lists.
• Deterministic, offline: run python3 scripts/test_gage_linearity_bias_study.py (29 tests, exit 0).

Related leaves

• manufacturing-quality/as9100/measurement-systems-analysis: the repeatability and reproducibility variance decomposition with percent contributions and distinct categories; it does not compute bias or linearity studies.
• manufacturing-quality/as9100/attribute-agreement-analysis: agreement analysis for attribute go/no-go judgments instead of variable-data bias studies.
• manufacturing-quality/as9100/calibration-control: calibration state, recall and traceability of the instrument behind the study.
• cross-cutting/numerics/least-squares-regression: the generic fitting machinery without the gage verdict framing.

Pitfalls

• Reading a significant mean bias as an automatic failure: the t test compares the mean bias against the spread of the level biases, so the anchor's 0.150 mm mean bias is significant (t = 5.000 against 2.776) even though every level sits inside the 10 percent band; the verdict REVIEW, not ACCEPT, is the correct joint reading of both gates.
• Reading the percent band as the only gate: a mean bias can be tiny but statistically significant when the level biases are tightly grouped, so the significance test and the band must both pass before ACCEPT.
• Dividing by zero on uniform bias: identical biases across levels give s = 0 and an undefined t; the study reports t_stat 0.0 and significant False (zero-dispersion convention) and lets the percent band decide.
• Forgetting the bias sign: bias is observed minus reference, so a positive bias means the gage reads high and the regression slope carries the sign of the linearity drift.
• Using the wrong degrees of freedom: df = n - 1 with n the number of reference levels, not the number of readings per level; above 20 df the t critical is 1.96, not an extrapolated table value.
• Feeding non-positive or unordered masters: the percent of reference blows up near zero and an unordered level list breaks the fit; references must be strictly increasing positive values (ValueError).
• Routing variance-decomposition questions here: the repeatability and reproducibility study with its percent-of-total verdicts belongs to measurement-systems-analysis; this leaf owns only the reference-level bias and linearity study.
• Routing generic fit questions here: a bare least-squares regression without gage verdict framing belongs to cross-cutting/numerics/least-squares-regression.

Behavior contract (gate 3)

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

python3 scripts/test_gage_linearity_bias_study.py

The test covers the anchor worked example (masters [2, 4, 6, 8, 10] mm, biases [0.07, 0.10, 0.16, 0.18, 0.24] mm: per-level percents within 1e-2, slope 0.0210, intercept 0.0240, sse 0.00036, r2 0.980, t 5.000 with t_crit 2.776 at 4 df, verdict REVIEW), the residual-sum-zero and r2 = 1 - sse/sst and t-formula identities, the small-bias fixture (all biases 0.01: not significant, ACCEPT), the zero-dispersion convention, the large-df normal critical 1.96, the t critical table spot checks, the ACCEPT/REVIEW verdict logic including the band-breach case, and ValueError rejection of mismatched arrays, fewer than 3 levels, non-increasing or non-positive references and empty biases.

Compliance

• Standards referenced, not reproduced: AS9100 is cited as reference context only (measurement system analysis in the AS9100/AS9102 environment); no standard text is reproduced and the study equations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.