Sampling plan tables (such as ISO 2859-1 or ANSI/ASQ Z1.4) may look like a bunch of numbers for reference, but they are based on deep statistical principles and mathematical models. Understanding these principles can turn you from a "tool user" into an "expert who understands the tool's theory".

• The core goal of a sampling plan table is to balance Producer's Risk (α) and Consumer's Risk (β). The underlying math is mainly based on three probability distributions: binomial, Poisson, and hypergeometric.

Core Concept: Operating Characteristic (OC) Curve

To understand sampling plan tables, you must first understand the OC curve. Each sampling plan (defined by sample size n and acceptance number Ac) corresponds to a unique OC curve.

What is an OC curve?

It is a chart with the x-axis as the actual lot defect rate (p) and the y-axis as the probability of the lot being accepted (Pa).

What does it tell us?

It describes the "behavioral characteristics" of a sampling plan. For a given plan (e.g., n=80, Ac=2), the OC curve can answer: *"If the actual defect rate of this lot is 3%, what is the probability it will pass our sampling inspection?"*

Ideal vs. Reality

• The ideal OC curve is a vertical cliff. When the defect rate is below a certain standard, the acceptance probability is 100%; once above the standard, the acceptance probability drops to 0%. This is impossible in reality.
• The real OC curve is a smooth S-shape. This means there is always risk.

The Balance of Two Major Risks

There are two very critical points on the OC curve, which are the two major risks that sampling plans must balance:

The design of sampling plan tables is essentially the design of a series of OC curves, so that for each given AQL, the corresponding producer's risk α is relatively fixed (usually around 5%).

Underlying Mathematical Models (The Calculation Engine)

So how is each point (acceptance probability Pa) on the OC curve calculated? This is where probability distribution models come in.

1. Hypergeometric Distribution

• Application: When samples are drawn from a finite, isolated lot without replacement, this is the most precise model.
• Calculation: Calculate the probability of drawing exactly x defectives in n samples from a lot of N items with D defectives.
• Formula: P(x) = [C(D, x) * C(N-D, n-x)] / C(N, n)
• Practical use: Most accurate, but very complex for large N and D. In practice, the following approximations are often used.

2. Binomial Distribution

• Application: When the lot is very large (theoretically infinite), or sampling from a continuous production stream. It is a good approximation of the hypergeometric when N >> n.
• Calculation: Probability of exactly x defectives in n independent trials (each with defect rate p).
• Formula: P(x) = C(n, x) * p^x * (1-p)^(n-x)
• Acceptance probability (Pa): Sum of probabilities for x=0 to Ac. Pa = Σ P(x) (from x=0 to Ac).

3. Poisson Distribution

• Application: When n is large and p is small, Poisson is an excellent and simpler approximation of the binomial. This is very common in quality management sampling plans.
• Calculation: Probability of x events (defectives) occurring in n samples, with average λ = n*p.
• Formula: P(x) = (e^(-λ) * λ^x) / x!
• Acceptance probability (Pa): Again, sum for x=0 to Ac.
• Core engine: Most ISO 2859-1 calculations are based on Poisson, as it is simple and very close to binomial in typical industrial scenarios.

How a Sampling Plan Table is Made (A Simplified View)

Let's simulate how standards committees think:

1. Determine AQL values: First, define a series of standardized AQL values (e.g., 0.65%, 1.0%, 1.5%, etc.).
2. Set producer's risk α: Decide that when the actual defect rate equals AQL, the acceptance probability Pa should be high, e.g., Pa ≈ 95%. This means producer's risk α ≈ 5%.
3. Calculate for each code letter: For each code letter (e.g., G, H, J, K...), which corresponds to a fixed sample size n.
   • Goal: For this n and given AQL, find an acceptance number Ac so that when p = AQL, Pa is as close as possible to 95%.
4. Start iterative calculation (using Poisson as an example):
   • Example: Code letter "J" (n=80), AQL=1.0%.
   • Calculate λ: λ = n * p = 80 * 0.01 = 0.8.
   • Try Ac=0: Pa = P(0) = (e^-0.8 * 0.8^0) / 0! = 0.449 (44.9%). Too low.
   • Try Ac=1: Pa = P(0) + P(1) = 0.449 + 0.359 = 0.808 (80.8%). Still low.
   • Try Ac=2: Pa = P(0) + P(1) + P(2) = 0.808 + 0.144 = 0.952 (95.2%).
   • Bingo! Ac=2 gives Pa (95.2%) very close to our target (95%).
5. Fill the table: So, in the table, the entry for code letter "J" and AQL 1.0% will be Ac=2, Re=3.
6. Handle arrows: If for some combinations (usually n is too small for a strict AQL), even Ac=0 gives Pa >> 95%, it means the plan is too lenient for the producer. The table will point to a plan with a larger n for better discrimination.

By repeating this for all code letters and AQLs, a complete sampling plan table is created. It is a huge family of OC curves, cleverly condensed into a table for users to apply directly without complex calculations.