Calculator Inputs
Example Data Table
| Sample Size | Acceptance Number | Defective Fraction | Interpretation |
|---|---|---|---|
| 50 | 0 | 1.0% | Strict plan. Any observed defect rejects the lot. |
| 50 | 1 | 1.0% | One observed defect may still allow acceptance. |
| 125 | 2 | 2.0% | Larger inspection effort with a limited defect allowance. |
| 200 | 4 | 3.0% | Plan balance depends on the product risk. |
Formula Used
The exact binomial model calculates acceptance probability by adding the probabilities of every acceptable defect count.
Pa is acceptance probability. n is sample size. c is acceptance number. p is defective fraction as a decimal.
For the Poisson approximation, λ = n × p and Pa ≈ Σk=0c e−λ × λk ÷ k!.
How to Use This Calculator
- Choose whether you want acceptance probability or defective fraction.
- Enter the random sample size used by the inspection plan.
- Enter the maximum defects allowed before rejection.
- Enter either the known defective fraction or target acceptance probability.
- Select exact binomial for the main calculation or Poisson for a low-defect approximation.
- Set the decimal precision, then select Calculate.
- Review the result, nearby quality levels, and exported result data.
Acceptance Sampling and Defect Risk
Acceptance sampling checks a small portion of a lot. It supports quick decisions when full inspection is impractical. The plan defines a sample size and an acceptance number. Inspectors count defective units inside the sample. The lot is accepted when defects do not exceed the stated limit. The calculator connects that decision rule with the likely defective fraction.
Probability of acceptance measures plan leniency. A high value means many samples will pass. A low value means the plan will reject more lots. The correct level depends on product risk, customer requirements, cost, and process capability. Sampling plans should match the consequences of a missed defect.
The defective fraction is the expected share of faulty units. It may be written as a decimal or percentage. For example, 0.02 equals two defective units per hundred. This calculator can estimate that fraction from a target acceptance probability. It can also estimate acceptance probability from a known fraction. That makes plan comparison easier before inspection begins.
The default binomial model assumes each inspected unit has the same defect chance. It also assumes one selection does not meaningfully change another selection. This is often suitable for large lots or random samples. The model sums the probabilities of zero defects through the acceptance number. The resulting total is the chance that the plan accepts the lot.
The Poisson approximation is useful for low defective fractions and larger samples. It replaces the binomial calculation with an average defect count. Still, it is an approximation. Use the exact binomial option when accuracy is important, when defect rates are not very small, or when decisions affect contracts, safety, or regulatory work.
Sample size has a strong effect. Larger samples reveal problems more often. A lower acceptance number makes plans stricter. For example, accepting zero defects creates a demanding plan. Allowing several defects raises the probability of acceptance at the same quality level. Changing only one setting can create an unintended decision rule.
An estimated defective fraction is not a guarantee about every unit. It describes the quality level that is consistent with the selected probability. Production variation can still create different sample results. Random selection matters. Biased sampling can make an excellent plan unreliable. Record the lot, sample method, inspection criteria, and measurement conditions with every decision.
Enter a reasonable sample size and acceptance number. Select whether you know the defect rate or the required acceptance probability. Choose a model, then calculate. Compare nearby quality levels using the result table. Discuss the plan with quality, operations, and supplier teams before applying it routinely.
Do not treat sampling as a substitute for process control. A weak process can produce costly variation between inspections. Track recurring failure modes. Improve incoming material, equipment settings, training, and measurement systems. Sampling is most valuable when it supports a wider quality program. It helps teams balance inspection effort, customer protection, and practical production flow.
Frequently Asked Questions
What is probability of acceptance?
It is the chance that a sampling plan accepts a lot at a stated defective fraction. It depends on the sample size, acceptance number, and calculation model.
What is a defective fraction?
It is the estimated proportion of defective units in a lot or process. A defective fraction of 0.02 equals 2% or two units per hundred.
What does the acceptance number mean?
It is the highest number of observed defects that still allows lot acceptance. A lower acceptance number makes the inspection plan stricter.
When should I use the exact binomial model?
Use it for the main result when each sampled item has the same defect chance. It is especially useful when defects are not extremely rare.
When is the Poisson approximation useful?
Use it for quick estimates when defective fractions are low and samples are relatively large. Verify important decisions with the exact binomial model.
Why does a larger sample change acceptance probability?
A larger sample gives more chances to observe defects. For the same quality level and acceptance number, it usually lowers the chance of acceptance.
Can this calculator choose my sample plan?
It evaluates a selected plan. Plan design should also consider risk levels, lot size, inspection costs, contractual requirements, and applicable quality standards.
Does a high acceptance probability prove the lot is good?
No. It only describes expected sampling behavior at a stated defect level. Random variation can still produce an accepted sample from a weak lot.
Why must the sample be random?
Random selection helps the sample represent the lot. Biased selection can hide defects or exaggerate them, making the probability result misleading.
Can I enter zero as the defective fraction?
Yes. The model returns a 100% acceptance probability because a sample from a defect-free process contains no defects under the stated assumptions.
What should I do after calculating?
Compare the result with your quality goals. Document the sample plan, review nearby quality levels, and use process controls to reduce recurring defects.
Use sound sampling plans for consistently better quality decisions.