Variables Acceptance Sampling

Variables Acceptance Sampling (manufacturing-quality/as9100/variables-acceptance-sampling)

Use when a lot of measured product must be accepted or rejected on the evidence of a sampled variable inspection: the lot size fixes the sample size code letter, the required AQL fixes the sample size n, the acceptability constant k and the maximum allowable percent nonconforming M, and the Q statistic formed from the specification limit, the sample mean and the sample standard deviation decides the verdict against k. This leaf implements a variables acceptance sampling plan model in pure Python, stdlib only: a documented reduced reference table in the style of the public ANSI Z1.9 and MIL-STD-414 k-method resolves the code letter, n, k and M, and an in-leaf normal survival function resolves the estimated percent nonconforming p_hat for the M-method check. It pairs with acceptance-sampling in this pack, which decides lots from counts of nonconforming units instead of measurements, and with statistical-process-control, which watches the ongoing process with charts over time rather than gating individual lots.

Domain quick reference

• Scope: variables acceptance sampling decides each individual lot from the measured values of a single quality characteristic (normal distribution assumed, sigma unknown and estimated by the sample standard deviation s); it is not a chart-based monitoring scheme and not a count-based plan on nonconforming units.
• Code letter lookup: code_letter(lot_size, level) returns the sample size code letter from the general level II rows of the reduced table: 91-150 E, 151-280 F, 281-500 G, 501-1200 H, 1201-3200 J, 3201-10000 K. Lot sizes below 91 or above 10000 have no code letter in the table.
• Plan lookup: plan_lookup(code, aql) returns {n, k, M}. Sample sizes: E 15, F 20, G 25, H 30, J 35, K 40. k by AQL: 0.65 -> 1.75, 1.0 -> 1.62, 1.5 -> 1.47, 2.5 -> 1.28, 4.0 -> 1.09. M pairs with the same AQL order per code: E 4.17/3.61/2.98/2.28/1.66, F 4.05/3.50/2.89/2.21/1.61, G 3.97/3.43/2.83/2.16/1.58, H 3.90/3.37/2.78/2.13/1.55, J 3.85/3.33/2.75/2.10/1.53, K 3.80/3.29/2.72/2.08/1.52. The anchor plan row is code H at AQL 1.0: n = 30, k = 1.62, M = 3.37.
• Q statistics: upper limit Q_u = (USL - xbar) / s and lower limit Q_l = (xbar - LSL) / s, computed by form_q_upper and form_q_lower.
• k-method decision: accept when Q >= k and reject when Q < k, via accept_verdict.
• M-method check: p_hat is the estimated percent nonconforming, 100 * normal_survival(Q) for an upper limit and 100 * normal_cdf(-Q) for a lower limit (equal values through normal symmetry); the M-method accepts when p_hat <= M.
• Reference: the ANSI Z1.9 / MIL-STD-414 style k-method plan structure is named and paraphrased only, never reproduced verbatim; AS9100 clause 8.6 frames the product acceptance context per standards-map.yaml.
• Assumption: the reduced table embeds only the general level II code letter rows and the five AQL rows listed above, so levels I and III, lot sizes outside 91-10000, other code letters and other AQLs raise ValueError by design, the same convention as the attribute acceptance-sampling sibling.

Workflow

1. Fix the lot size (units per lot) and the inspection level, then resolve the sample size code letter with code_letter(lot_size, level).
2. Fix the AQL for the measured characteristic and look up the plan with plan_lookup(code, aql): the sample size n, the acceptability constant k and the maximum allowable percent nonconforming M.
3. Measure a random sample of n units and compute the sample mean xbar and the sample standard deviation s.
4. Form the Q statistic for the applicable single limit with form_q_upper(usl, xbar, s) or form_q_lower(lsl, xbar, s).
5. Decide the k-method verdict with accept_verdict(Q, k): accept when Q >= k.
6. Run the M-method companion: estimated_pct_nonconforming(Q, tail) returns p_hat in percent, and the plan accepts by M when p_hat <= M.
7. Run the whole single-sided flow with variables_sampling_decision(lot_size, aql, usl_or_lsl, xbar, s), which returns {code, n, k, M, Q, p_hat, accept}; report those fields and confirm the deterministic checks with the contract test scripts/test_variables_acceptance_sampling.py.

Worked example

Lot of 800 units at level II, AQL 1.0, measured characteristic with sample mean xbar = 49.97 and sample standard deviation s = 0.12.

• Code letter: code_letter(800, "II") = "H" (band 501-1200).
• Plan: plan_lookup("H", 1.0) = {n: 30, k: 1.62, M: 3.37}.
• Upper limit USL 50.2: Q_u = (50.2 - 49.97) / 0.12 = 1.9167 (module output 1.916667). accept_verdict(1.9167, 1.62) = True, the lot is accepted.
• M-method: p_hat = 100 * normal_survival(1.9167) = 2.764 percent (module output 2.764015), within 0.05 percent of the 2.76 percent anchor and below M = 3.37, so the M-method check also accepts.
• Lower limit LSL 49.4: Q_l = (49.97 - 49.4) / 0.12 = 4.75 (module output 4.750000), accept True; p_hat sits below 0.001 percent.
• Reject case: sample mean 50.1 against USL 50.2 gives Q = 0.8333, below k = 1.62, accept False.
• AQL margin identity: the same stats with a sample mean at Q = 1.50 accept at AQL 1.5 (k 1.47) and reject at the tighter AQL 1.0 (k 1.62) and AQL 0.65 (k 1.75).

Verification

• Confirm code_letter returns H at lot 800, G at 281, H at 501, J at 1201 and K at 10000, and raises ValueError below 91, above 10000, for non-positive lot sizes and for levels I and III with no reduced row.
• Confirm plan_lookup("H", 1.0) returns {n: 30, k: 1.62, M: 3.37} and that the embedded n, k and M rows match the spec table exactly.
• Confirm form_q_upper(50.2, 49.97, 0.12) returns 1.9167 and form_q_lower(49.4, 49.97, 0.12) returns 4.75.
• Confirm accept_verdict accepts at Q >= k including the boundary and rejects below it.
• Confirm estimated_pct_nonconforming(1.9167, "upper") returns 2.76 percent within 0.05 percent and that the upper and lower tail forms agree for any Q.
• Confirm variables_sampling_decision(800, 1.0, 50.2, 49.97, 0.12) returns the anchor dict {code H, n 30, k 1.62, M 3.37, Q 1.9167, p_hat 2.764, accept True} and rejects the xbar 50.1 case.
• Confirm ValueError rejection of non-physical inputs: s <= 0, lot_size <= 0, out-of-band lot sizes, unknown code letters, AQLs outside the table, unknown levels and invalid tails.
• Run the contract test offline: python3 scripts/test_variables_acceptance_sampling.py (33 tests, deterministic).

Related leaves

• manufacturing-quality/as9100/acceptance-sampling: attribute lot plans decided by counts of nonconforming units in the sample; this leaf is the measurement-based counterpart for the same acceptance activity.
• manufacturing-quality/as9100/statistical-process-control: chart-based monitoring of an ongoing process over time; this leaf gates each individual lot from its own sample measurements instead.
• manufacturing-quality/as9100/measurement-systems-analysis: the repeatability and reproducibility context that the Q statistics here treat as negligible measurement error.

Pitfalls

• Treating the bands as the full standard table: the leaf embeds a documented reduced table with only the general level II rows (91-150 E through 3201-10000 K) and five AQL rows, so any other code letter, AQL, level or out-of-band lot size raises ValueError instead of guessing.
• Reading the limit direction from the name: variables_sampling_decision treats the passed limit as an upper specification limit when the limit value exceeds the sample mean and as a lower specification limit otherwise; run a two-sided check or a mean that crossed its limit by forming Q with form_q_upper or form_q_lower and calling accept_verdict directly.
• Assuming the M-method always agrees with the k-method: on this reduced training table the M values are fixed paired constants, so the two checks can disagree at moderate Q; the disposition verdict is Q versus k (spec convention) and p_hat versus M is the secondary estimate check. In the worked anchor they agree, 2.764 <= 3.37.
• Substituting a known sigma or a range-based spread: the model estimates sigma by the sample standard deviation s and assumes a normal characteristic; other spread inputs change Q and can flip the verdict.
• Feeding s = 0, the all-identical sample: the Q statistic is undefined and the module raises ValueError; a zero-spread sample carries no sampling information about the lot position.
• Reading p_hat as a count: p_hat is the estimated percent of the normal population beyond the limit given the sample mean and spread, not the count of nonconforming units found in the sample, which is the attribute sibling's quantity.

Behavior contract (gate 3)

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

python3 scripts/test_variables_acceptance_sampling.py

The test covers the spec validation list: the code letter boundary truth table (spec boundaries 281 -> G, 501 -> H, 1201 -> J plus every band edge), the plan lookup values for every embedded code and AQL (sample sizes 15-40, k set 1.75/1.62/1.47/1.28/1.09, M rows E, H and K), the Q forms and accept verdict at the worked example (Q 1.9167 accept, Q 4.75 lower-limit accept), the reject case (xbar 50.1 against USL 50.2, Q 0.833 reject), p_hat = 2.76 percent within 0.05 percent at Q 1.9167, the AQL margin identity (accept at a looser AQL, reject at a tighter AQL for the same stats), the normal survival and CDF complement identities, upper and lower tail symmetry, the M-method agreement anchor, exact dict keys, determinism, and ValueError rejection of non-physical inputs (s <= 0, lot_size <= 0, out-of-band lots, unknown code letter, unknown AQL, unknown level, invalid tail).

Compliance

• Standards referenced, not reproduced: AS9100 clause 8.6 (product acceptance) frames the context, and the ANSI Z1.9 / MIL-STD-414 style k-method plan structure is named and paraphrased as a small reduced training table, summary data per standards-map.yaml; no verbatim standard tables or text.
• compliance: STANDARDS-REF, gated: false.