Acceptance sampling¶

Acceptance sampling answers a receiving-dock question: a lot of parts has arrived, inspecting every unit is too slow or too expensive, so you inspect a sample and decide whether to accept or reject the whole lot from what you find. The decision rule is a plan. A plan does not measure the true quality of the lot. It trades the cost of inspection against two risks: rejecting a good lot, and accepting a bad one. mfgQC builds those plans, draws the curve that shows exactly how the two risks trade off, and reports the assumptions that the plan rests on but cannot check from the sample.

This page pins every formula and default to the code in mfgqc/sampling.py. The entry points are mfgqc.sampling_plan, mfgqc.find_plan, mfgqc.z14_plan (attributes, ANSI/ASQ Z1.4), and mfgqc.z19_plan (variables, ANSI/ASQ Z1.9).

A note on two cultures of plan. An attributes plan classifies each inspected unit as good or defective and counts the defectives. A variables plan measures a continuous characteristic on each unit and judges the lot from the sample mean and standard deviation. Attributes plans make no distributional assumption about the measurement. Variables plans assume the characteristic is normal, inspect fewer units for the same protection, and break quietly when that normal assumption fails. mfgQC covers both and surfaces the normal assumption loudly on the variables side.

1. Single attribute sampling plans (`sampling_plan`)¶

What it is and when you use it¶

A single attribute plan is a pair of numbers, the sample size $n$ and the acceptance number $c$. The rule is: draw $n$ units at random from the lot, inspect them, count the defectives $d$. Accept the lot if $d \le c$, otherwise reject it. "Single" means one sample and one decision, with no provision for a second sample. You use a single attributes plan for go/no-go inspection: a unit either passes a gauge or it does not, a seal either holds or it does not, a label is either present or absent.

plan = mfgqc.sampling_plan(134, 3)     # n = 134, accept if 3 or fewer defectives

The probability of acceptance¶

Everything about a plan follows from one function: the probability that the plan accepts a lot whose true incoming fraction defective is $p$. Call it $P_a(p)$. Acceptance means $d \le c$, so $P_a(p)$ is the probability that a count of defectives is at most $c$.

mfgQC computes $P_a(p)$ from one of three models (_pa in sampling.py):

model	distribution of the defective count	$P_a(p)$
`binomial` (default)	$X \sim \mathrm{Binomial}(n, p)$	$P(X \le c)$
`hypergeometric`	$X \sim \mathrm{Hypergeometric}(N,\, D{=}\mathrm{round}(Np),\, n)$	$P(X \le c)$
`poisson`	$X \sim \mathrm{Poisson}(np)$	$P(X \le c)$

The binomial is the model for sampling from a lot large enough to treat as effectively infinite (drawing a defective does not appreciably change the fraction defective of what remains). The hypergeometric is the exact model for a finite lot of known size $N$, where drawing without replacement does change the remaining fraction. The Poisson is an approximation to the binomial that holds when $n$ is large and $p$ is small; mfgQC uses it only when you ask for it by name.

How the model is chosen¶

mfgQC does not silently switch models, in keeping with the guardrail design. The rule (_select_model) is:

If you pass model=, that model is used and the reason recorded is "explicitly requested".
Otherwise, if you pass lot_size=N and the sampling fraction $n/N$ exceeds 0.1, the hypergeometric model is selected, because at that fraction the binomial approximation to a finite lot is questionable.
Otherwise the binomial is used (the infinite-lot default).

The choice and its reason are attached to the result as an assumption check, so the report tells you which model produced the numbers and why. If you force the binomial on a finite lot with $n/N > 0.1$, a second check fires recommending the hypergeometric.

The difference is usually small at the sampling fractions plans run at. For $n=134$, $c=3$, $p=0.02$: the binomial gives $P_a = 0.719$; the hypergeometric for a lot of $N=500$ gives $P_a = 0.734$.

The OC curve¶

The operating-characteristic (OC) curve is the plot of $P_a(p)$ against the incoming fraction defective $p$. It is the complete picture of what a plan does. A good lot (small $p$) sits on the high-left of the curve and is almost always accepted; a bad lot (large $p$) sits on the low-right and is almost always rejected; in between, the curve slides from acceptance to rejection. The steeper that slide, the more sharply the plan discriminates good lots from bad. Increasing $n$ steepens the curve; increasing $c$ shifts it to the right.

Call the curve on a plan with plan.oc_curve(), or chart it directly with plan.view() (the OC curve is the canonical view of a plan). The curve evaluates $P_a$ on a grid of $p$ from near zero up to twice the LTPD.

The four risk points¶

From the OC curve mfgQC reads four standard points (_build_plan, via _invert_pa, which inverts the monotone curve by a bracketed root find):

quantity	definition	meaning
AQL	the $p$ at which $P_a = 0.95$	the acceptable quality level: the worst incoming quality that the plan still accepts with high probability (95%).
Indifference quality	the $p$ at which $P_a = 0.50$	the quality at which accept and reject are equally likely.
LTPD / RQL	the $p$ at which $P_a = 0.10$	the lot tolerance percent defective (also rejectable quality level): quality so poor that the plan accepts it only 10% of the time.
producer's risk $\alpha$	$1 - P_a(\text{AQL})$	the chance of rejecting a lot that is actually at the AQL (a good lot).
consumer's risk $\beta$	$P_a(\text{LTPD})$	the chance of accepting a lot that is actually at the LTPD (a bad lot).

By construction $\alpha = 0.05$ and $\beta = 0.10$ on any plan derived this way, because the AQL and LTPD are defined as the $P_a = 0.95$ and $P_a = 0.10$ points. The producer's risk is the supplier's exposure (good work rejected); the consumer's risk is the buyer's exposure (bad work accepted). The AQL is what you promise the supplier; the LTPD is the quality you are protecting yourself against.

Worked example: derive a plan and its OC curve¶

The plan $n = 134$, $c = 3$ is a textbook single attributes plan.

import mfgqc
plan = mfgqc.sampling_plan(134, 3)
print(plan.report())

Real output:

Sampling Plan n=134, c=3 [single]
=================================
n = 134   c = 3   model = binomial

AQL (Pa=0.95)          = 1.03%
Indifference (Pa=0.50) = 2.73%
LTPD/RQL (Pa=0.10)     = 4.92%
Producer risk alpha    = 0.05
Consumer risk beta     = 0.1

Stated assumptions (not data-checkable): random sampling of the lot, lot homogeneity, and binary good/defective classification of each unit.

Assumption checks:
  [PASS] model_choice (Pa model selection): Pa=0; n=134

Recommendations:
  - no lot_size -> binomial (infinite-lot default)

This plan accepts lots at 1.03% defective 95% of the time and lots at 4.92% defective only 10% of the time. Reading $P_a(p)$ along the OC curve makes the shape concrete:

incoming $p$	$P_a(p)$
0.50%	0.9952
1.00%	0.9537
1.03% (AQL)	0.9494
2.00%	0.7192
2.73% (indifference)	0.5010
4.00%	0.2123
4.92% (LTPD)	0.0998
6.00%	0.0371

(For this plan, $P_a(0.01) = 0.9537$ and $P_a(0.05) = 0.0931$.)

Dispositioning a lot¶

Given a plan, plan.inspect(d) applies the accept/reject rule to a sample in which $d$ defectives were found:

plan.inspect(2)     # 2 defectives, c = 3

Lot Disposition: ACCEPT
=======================
decision   = accept
defectives = 2  (acceptance number c = 3)
n          = 134
observed fraction defective = 1.49%
Pa at observed rate         = 0.858

Two defectives is at or below $c = 3$, so the lot is accepted. Five defectives would exceed $c$ and the lot would be rejected.

Finding a plan from the risks you want (`find_plan`)¶

sampling_plan takes $n$ and $c$ and reports the risks. find_plan works the other way: you state the AQL and LTPD (and optionally $\alpha$ and $\beta$, defaulting to 0.05 and 0.10), and it searches for the plan with the smallest $n$ whose OC curve gives $P_a(\text{AQL}) \ge 1 - \alpha$ and $P_a(\text{LTPD}) \le \beta$. Integer $(n, c)$ rarely hits the requested points exactly, so the result records both what you asked for and what the chosen plan actually achieves.

mfgqc.find_plan(aql=0.01, ltpd=0.05)

Sampling Plan n=132, c=3 [find_plan]
====================================
n = 132   c = 3   model = binomial

AQL (Pa=0.95)          = 1.04%
Indifference (Pa=0.50) = 2.77%
LTPD/RQL (Pa=0.10)     = 4.99%
Producer risk alpha    = 0.05
Consumer risk beta     = 0.1

Requested vs achieved:
  AQL  requested 1% -> achieved 1.04%
  LTPD requested 5% -> achieved 4.99%

Assumptions¶

A single attributes plan rests on three assumptions that the sample cannot verify, and mfgQC states them on every plan rather than testing them:

Random sampling. The $n$ units are a random draw from the lot. If inspectors pull from the top of the pallet, the sample is not representative and $P_a$ does not apply.
Lot homogeneity. The lot is a single population, not a mix of a good batch and a bad one.
Binary classification. Each unit is cleanly good or defective.

The one thing mfgQC does check is the model choice itself, described above. The OC-curve probabilities are exact for the chosen model; the honesty limit is that the model is only as right as those three assumptions.

Source¶

The single-sampling plan, the OC curve, the binomial and hypergeometric acceptance probabilities, and the AQL / LTPD / producer's-and-consumer's-risk vocabulary follow Montgomery, Introduction to Statistical Quality Control (see the bibliography).

2. ANSI/ASQ Z1.4, attributes (`z14_plan`)¶

What it is and when you use it¶

Z1.4 is the standard attributes-sampling system used across defense and commercial supply. Instead of designing a plan from scratch, you look one up by lot size and AQL. The standard organizes plans so that, as lots get larger, sample sizes grow and the plans discriminate more finely, and it provides switching rules that tighten inspection after a run of bad lots and relax it after a run of good ones. You use Z1.4 when a contract or a customer specifies inspection "to Z1.4 at AQL such-and-such".

mfgqc.z14_plan(lot_size=1000, aql=1.0)     # aql is a PERCENT here

What mfgQC implements¶

z14_plan implements single sampling, general inspection level II, normal severity. That is the most common configuration and the one most contracts default to. The lookup has two steps:

Lot size to sample-size code letter. The lot size $N$ maps to a letter A through Q (I and O are skipped) from the level-II table _Z14_CODE_LETTERS, and each letter maps to a sample size $n$ in _Z14_SAMPLE_SIZE (A=2, B=3, C=5, ... J=80, ... Q=1250).
Code letter and AQL to the acceptance plan. The master table is encoded as the acceptance-number staircase $[0, 1, 2, 3, 5, 7, 10, 14, 21]$ that the published normal table runs along its diagonals (_z14_cell, anchored so code K at AQL 1.0 gives $A_c = 3$). The cell returns the acceptance number $A_c$ (which mfgQC stores as $c$) and the rejection number $R_e = A_c + 1$.

The tabled AQLs are the standard percent-defective columns: 0.10, 0.15, 0.25, 0.40, 0.65, 1.0, 1.5, 2.5, 4.0, 6.5, 10.0. The AQL argument is a percent (1.0 means 1%), not a fraction, which differs from sampling_plan where the derived AQL is reported as a fraction internally.

When a (letter, AQL) cell falls off the staircase, the published table prints an arrow: down to the first plan below (a larger sample) or up to the first plan above (a smaller sample). _resolve_z14 follows that arrow to the adjacent code letter and uses the resolved row for both the sample size and the $(A_c, R_e)$. The report records when an arrow was followed.

After the plan is fixed, mfgQC computes the same OC-derived risk points as for any plan, using the binomial (the Z1.4 convention for these lot sizes) unless the sampling fraction forces the hypergeometric.

Worked example¶

mfgqc.z14_plan(1000, 1.0)

Sampling Plan n=80, c=2 [Z1.4 normal]
=====================================
n = 80   c = 2   model = binomial
lot size N = 1000   (n/N = 0.08)
code letter = J   Ac = 2   Re = 3

AQL (Pa=0.95)          = 1.03%
Indifference (Pa=0.50) = 3.33%
LTPD/RQL (Pa=0.10)     = 6.52%
Producer risk alpha    = 0.05
Consumer risk beta     = 0.1

Requested vs achieved:
  AQL  requested 1% -> achieved 1.03%

Assumption checks:
  [PASS] model_choice (Pa model selection): Pa=0.08; n=80
  [PASS] z14_lookup (ANSI/ASQ Z1.4 single-sampling normal): ANSI/ASQ=80; n=80

Recommendations:
  - lot_size given but n/N=0.08 <= 0.1 -> binomial
  - lot 1000 -> code J; n=80; AQL 1.0% -> Ac=2, Re=3.

A lot of 1000 falls in the J code letter, giving $n = 80$. At AQL 1.0% the cell is $A_c = 2$, $R_e = 3$: inspect 80 units, accept on 2 or fewer defectives, reject on 3 or more.

What is and is not covered¶

mfgQC encodes only level II, normal severity, single sampling. The other severities and the switching rules that move between them are documented limits, not silent omissions:

severity="tightened" and severity="reduced" raise NotImplementedError. The tightened and reduced master tables are not encoded.
Inspection levels other than II raise NotImplementedError.
Double and multiple sampling are not implemented.

The normal/tightened/reduced switching procedure of Z1.4 is therefore not automated. You get the normal-inspection plan; moving to tightened or reduced inspection after the prescribed run of accepted or rejected lots is a manual step against your copy of the standard.

Source¶

ANSI/ASQ Z1.4, Sampling Procedures and Tables for Inspection by Attributes (the civilian successor to MIL-STD-105). Z1.4 is named here as the source standard; it is not a formatted entry in the bibliography, which carries Montgomery as the general text for the underlying OC-curve mathematics.

3. ANSI/ASQ Z1.9, variables (`z19_plan`)¶

What it is and when you use it¶

A variables plan judges a lot from measurements rather than from a good/defective count. You measure a continuous characteristic (a diameter, a fill weight, a tensile strength) on each of $n$ sampled units, compute the sample mean and standard deviation, and from those estimate how much of the lot lies outside the specification limit. You use Z1.9 when the characteristic is measured anyway and you want the protection of an attributes plan from a smaller sample. The price is an assumption: the estimate of percent nonconforming is only valid if the characteristic is normally distributed.

mfgqc.z19_plan(lot_size=100, aql=1.0, lower=200.0, upper=210.0)   # aql is a PERCENT

What mfgQC implements¶

z19_plan implements the standard-deviation method (sigma unknown), general inspection level II, normal severity. The lookup mirrors Z1.4: the lot size maps to a code letter (_Z19_CODE_LETTERS, anchored to the published cell lot size 100 to code F) and each letter to a sample size (_Z19_SAMPLE_SIZE, F=10, G=15, ... up to P=200). The sample sizes are smaller than Z1.4's at the same lot size, which is the point of a variables plan.

The acceptance rule is built on the quality index $Q$, the distance from the sample mean to a spec limit in sample standard deviations:

\[Q_U = \frac{\text{USL} - \bar{x}}{s}, \qquad Q_L = \frac{\bar{x} - \text{LSL}}{s}.\]

A larger $Q$ means the mean sits farther inside the limit, so less of the lot is estimated to fall outside it. The standard offers two equivalent acceptance forms, and mfgQC reports both:

Form 1, the k-method. Accept the lot if every applicable $Q \ge k$, where $k$ is the acceptability constant for the sample size and AQL. mfgQC derives $k$ from the normal-approximation design that underlies the published table (_z19_k):

$$k = z_{\text{AQL}} - z_\alpha \sqrt{\frac{1}{n} + \frac{z_{\text{AQL}}^2}{2n}},$$

where $z_{\text{AQL}} = \Phi^{-1}(1 - p)$ is the standard-normal quantile at the AQL fraction $p$, $z_\alpha = \Phi^{-1}(1 - \alpha)$ with producer's risk $\alpha = 0.05$, and $\Phi^{-1}$ is the inverse standard-normal CDF. This reproduces the published normal-inspection $k$ to about 0.02 for mid-range $n$. Confirm the smallest and largest code letters against your Z1.9 table copy.

Form 2, the M-method. Estimate the percent nonconforming from $Q$ using normal theory ($\widehat{p} = \Phi(-Q)$, the normal tail beyond the limit), and accept if that estimate is at most $M$. mfgQC sets $M$ as the boundary estimate, the percent nonconforming implied when $Q$ exactly equals $k$ (M = _pct_nonconforming(k)). This is the normal-theory $M$, which is not identical to the published table $M$; the two forms agree on the accept/reject decision at the boundary, which is the property that matters.

z19_plan returns a Z19Plan carrying $n$, the code letter, $k$, $M$, the AQL, and any attached spec limits. It does not by itself judge a lot. Call Z19Plan.inspect(sample, lower=, upper=) to disposition a sample, which computes $\bar{x}$, $s$, the quality indices, the estimated percent nonconforming, and the Form-1 decision.

Worked example: the plan¶

plan = mfgqc.z19_plan(100, 1.0, lower=200.0, upper=210.0)
print(plan.report())

Z1.9 Variables Plan (code F)
============================
lot size 100 -> code letter F   (level II, normal)
n = 10   AQL = 1%
Form 1 (k-method): accept if Q >= k = 1.325
Form 2 (M-method): accept if est. % nonconforming <= M = 9.258%
method: standard deviation (sigma unknown). ASSUMES a normal characteristic.

Assumption checks:
  [PASS] normality (assumed (Z1.9)): assumed (Z1.9); n=10

Recommendations:
  - Z1.9 assumes a normal characteristic; check normality on the sample (inspect() does this) - the % nonconforming estimate is invalid otherwise.

A lot of 100 maps to code letter F and a sample of 10. At AQL 1% the derived acceptability constant is $k = 1.325$.

Worked example: dispositioning a lot¶

inspect applies the plan to a measured sample. With ten readings whose mean sits well inside both limits, the lot is accepted:

plan = mfgqc.z19_plan(100, 1.0)
sample = [206.5, 207.1, 208.4, 206.9, 207.8, 208.0, 207.2, 206.7, 208.9, 207.5]
plan.inspect(sample, lower=200.0, upper=210.0)

Z1.9 Lot Disposition: ACCEPT
============================
n = 10   xbar = 207.5   s = 0.7717   k = 1.325
QL = (xbar-LSL)/s = 9.719   est. % below LSL = 0.000%
QU = (USL-xbar)/s = 3.240   est. % above USL = 0.060%
estimated total % nonconforming = 0.060%
Decision: accept (Form 1: accept if every Q >= k = 1.325).

Assumption checks:
  [PASS] normality (Anderson-Darling): AD=0.119, ...

Both quality indices clear $k$, so the lot is accepted. Push the mean up toward the upper limit and the upper quality index drops below $k$:

sample = [208.2, 209.5, 210.3, 208.8, 209.9, 207.6, 209.1, 210.6, 208.4, 209.6]
plan.inspect(sample, lower=200.0, upper=210.0)

Z1.9 Lot Disposition: REJECT
============================
n = 10   xbar = 209.2   s = 0.9592   k = 1.325
QL = (xbar-LSL)/s = 9.592   est. % below LSL = 0.000%
QU = (USL-xbar)/s = 0.834   est. % above USL = 20.212%
estimated total % nonconforming = 20.212%
Decision: reject (Form 1: accept if every Q >= k = 1.325).

Here $Q_U = 0.834$ is below $k = 1.325$, the normal-theory estimate puts about 20% of the lot above the upper limit, and the lot is rejected.

Assumptions and the normality guardrail¶

The variables plan adds one assumption that the attributes plans do not have: the measured characteristic is normally distributed. The percent-nonconforming estimate is a normal tail area, so if the characteristic is skewed or heavy-tailed, $\Phi(-Q)$ is the wrong number and the accept/reject decision built on it is not trustworthy.

mfgQC does not assume this silently. The plan carries a standing note that Z1.9 assumes normality, and inspect runs an Anderson-Darling normality test on the sample and attaches the result. If the sample fails the test, the recommendation is rewritten to say the estimate and the decision are unreliable. At a Z1.9 sample size (often 10 or fewer), a normality test has low power, so a pass is weak evidence; the report flags the low power rather than overstating the check.

The random-sampling and lot-homogeneity assumptions of the attributes plans apply here too.

Source¶

ANSI/ASQ Z1.9, Sampling Procedures and Tables for Inspection by Variables for Percent Nonconforming (the civilian successor to MIL-STD-414), is the source standard for the code-letter and sample-size tables and the $k$- and $M$-method acceptance forms. Z1.9 is named here inline; it is not a formatted entry in the bibliography. The normal-theory derivation of $k$ and the percent-nonconforming estimate follow the same normal-distribution framework as Montgomery, Introduction to Statistical Quality Control (see the bibliography). As the code states, the derived $k$ approximates the published table to about 0.02 for mid-range sample sizes; confirm the extreme code letters against a copy of the standard.

Result objects and methods¶

sampling_plan, find_plan, z14_plan, and z19_plan return frozen result objects that share the common mfgQC result surface: .report() for the full text, .summary() for a flat scalar dict, .to_dict() for a JSON-serializable payload, and .view() for the chart (the OC curve for attribute plans). The plan objects also carry the analysis methods shown above: oc_curve(), aoq_curve(), and inspect() on the attribute SamplingPlan, and inspect() on Z19Plan. Each method returns a new result object with its own provenance step appended.

For the full signatures and result-object methods, see the API reference. The source standards are listed in the bibliography.

model	distribution of the defective count	\(P_a(p)\)
`binomial` (default)	\(X \sim \mathrm{Binomial}(n, p)\)	\(P(X \le c)\)
`hypergeometric`	\(X \sim \mathrm{Hypergeometric}(N,\, D{=}\mathrm{round}(Np),\, n)\)	\(P(X \le c)\)
`poisson`	\(X \sim \mathrm{Poisson}(np)\)	\(P(X \le c)\)

quantity	definition	meaning
AQL	the \(p\) at which \(P_a = 0.95\)	the acceptable quality level: the worst incoming quality that the plan still accepts with high probability (95%).
Indifference quality	the \(p\) at which \(P_a = 0.50\)	the quality at which accept and reject are equally likely.
LTPD / RQL	the \(p\) at which \(P_a = 0.10\)	the lot tolerance percent defective (also rejectable quality level): quality so poor that the plan accepts it only 10% of the time.
producer's risk \(\alpha\)	\(1 - P_a(\text{AQL})\)	the chance of rejecting a lot that is actually at the AQL (a good lot).
consumer's risk \(\beta\)	\(P_a(\text{LTPD})\)	the chance of accepting a lot that is actually at the LTPD (a bad lot).

Acceptance sampling¶

1. Single attribute sampling plans (sampling_plan)¶

What it is and when you use it¶

The probability of acceptance¶

How the model is chosen¶

The OC curve¶

The four risk points¶

Worked example: derive a plan and its OC curve¶

Dispositioning a lot¶

Finding a plan from the risks you want (find_plan)¶

Assumptions¶

Source¶

2. ANSI/ASQ Z1.4, attributes (z14_plan)¶

What it is and when you use it¶

What mfgQC implements¶

Worked example¶

What is and is not covered¶

Source¶

3. ANSI/ASQ Z1.9, variables (z19_plan)¶

What it is and when you use it¶

What mfgQC implements¶

Worked example: the plan¶

Worked example: dispositioning a lot¶

Assumptions and the normality guardrail¶

Source¶

Result objects and methods¶

1. Single attribute sampling plans (`sampling_plan`)¶

Finding a plan from the risks you want (`find_plan`)¶

2. ANSI/ASQ Z1.4, attributes (`z14_plan`)¶

3. ANSI/ASQ Z1.9, variables (`z19_plan`)¶