Operating Characteristic Curve

The Operating Characteristic curve (OC curve) is the function Pa(p) that maps a lot's true defect rate p to the probability of acceptance under a given acceptance-sampling plan (n, c). It is the foundational tool for plan design: at p=0 every plan gives Pa=1 (a perfect lot is always accepted); at p=1 every plan gives Pa=0 (a fully defective lot is always rejected); between them the curve takes an S-shape. The plan's numerical anchors are: **AQL = p_α** such that Pa(AQL) = 1−α (typically 95% acceptance), and **LQL** (or LTPD) **= p_β** such that Pa(LQL) = β (typically 10% acceptance). For single sampling the curve is computed from the binomial distribution (or hypergeometric for small lots); for double, multiple and sequential sampling the curve is derived from the multi-stage probability tree. OC curves let practitioners compare plans on protection profile, not just on (n, c), and to compute downstream quality metrics: AOQ (Average Outgoing Quality), AOQL (Average Outgoing Quality Limit) and ASN (Average Sample Number). Foundational reference Montgomery (2019) *Introduction to Statistical Quality Control*; Schilling and Neubauer (2017) cover OC-curve construction across plan families.