Pass/Fail Calculators

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Pass/fail testing (attribute testing) classifies items as either conforming (pass) or nonconforming (fail) without measuring a continuous variable. It is widely used in quality control, diagnostic testing, bioassays, and sterility testing. Statistical methods for pass/fail data include binomial distribution, acceptance sampling plans (ANSI/ASQ Z1.4), probability of detection (POD) analysis, and beta binomial models. Sample size calculations for pass/fail tests require specifying the acceptable false-positive and false-negative rates (α and β) and the defect fraction or detection threshold to be distinguished.

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Binomial Distribution for Pass/Fail Data

If p = probability of failure per item and n = sample size, the number of failures X follows the binomial distribution: P(X = k) = C(n,k) × p^k × (1−p)^(n−k). Probability of passing (0 failures in n items): P(0 failures) = (1−p)^n. Example: with 5% defect rate (p=0.05) and n=20: P(0 defects) = 0.95^20 = 0.358 (36% chance of accepting a 5% defective lot with zero failures in 20).

Acceptance Sampling

Acceptance quality level (AQL) defines the maximum defect fraction considered acceptable. Acceptance sampling plans (ANSI/ASQ Z1.4 for attribute data) specify sample size n and acceptance number c: accept the lot if ≤c failures are found; reject if > c. Operating characteristic (OC) curve shows P(acceptance) vs. true defect rate.

Sample Size for Pass/Fail Tests

To demonstrate with confidence C% that the true defect rate is below a threshold p₀ using zero-failure testing: n = ln(1−C) / ln(1−p₀). Example: 95% confidence that true defect rate < 5%: n = ln(0.05)/ln(0.95) = −2.996/−0.0513 ≈ 58 items.

Diagnostic Test Pass/Fail

For binary diagnostic tests: sensitivity = P(test+ | disease present); specificity = P(test− | disease absent). False negative rate = 1 − sensitivity; false positive rate = 1 − specificity. These define the test's operating characteristics for clinical decision-making.

Glossary

Binomial Distribution
The probability distribution for the number of successes in n independent trials each with probability p; P(X=k) = C(n,k)×p^k×(1−p)^(n−k); foundation of attribute pass/fail testing.
Acceptance Quality Level (AQL)
The maximum defect rate considered acceptable; used in ANSI/ASQ Z1.4 sampling plans to set sample size (n) and acceptance number (c); lots with defect rate ≤ AQL are generally accepted.
Sensitivity / Specificity
Test performance metrics: sensitivity = P(positive | disease present); specificity = P(negative | disease absent); trade-off determines the optimal detection threshold for clinical or quality applications.

Frequently Asked Questions

Pass/fail data (attribute data) follows the binomial distribution: P(X=k) = C(n,k) × p^k × (1−p)^(n−k), where p = probability of failure, n = sample size, k = number of failures. For testing with zero tolerance (c=0): P(acceptance) = (1−p)^n. This formula is used to calculate the required sample size for demonstrating low defect rates. For rare events, the Poisson approximation (λ = np) is used when n is large and p is small.

To demonstrate with confidence (1−α) that the true defect rate is below p₀, using a zero-failure rule: n = ln(α) / ln(1−p₀). For 95% confidence (α = 0.05) and p₀ = 1%: n = ln(0.05)/ln(0.99) = −2.996/−0.01005 ≈ 298 items. For 95% confidence and p₀ = 5%: n ≈ 58 items. This is the standard approach for sterility testing, vaccine lot release, and device reliability validation where zero defects must be demonstrated.

AQL is the maximum defect rate considered acceptable for a lot to pass inspection. ANSI/ASQ Z1.4 standard tables specify: based on lot size and chosen AQL, how many items to sample (n) and how many failures to tolerate (c, acceptance number). If failures ≤ c: accept lot; if > c: reject lot. AQL = 0.65% means lots with ≤0.65% defective are generally accepted; lots above this are generally rejected. Operating characteristic (OC) curves show the probability of accepting lots with various actual defect rates.

Sensitivity = P(positive result | disease truly present) = true positive rate; measures ability to detect disease. Specificity = P(negative result | disease truly absent) = true negative rate; measures ability to correctly clear healthy individuals. False negative rate = 1 − sensitivity (missed cases). False positive rate = 1 − specificity (false alarms). There is usually a trade-off: lowering the detection threshold increases sensitivity but decreases specificity. The optimal threshold depends on the consequences of each error type — for screening, high sensitivity is prioritized; for confirmatory testing, high specificity is required.