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At a similar observed rate, a larger independent sample generally narrows the interval. More units do not help if they duplicate one clustered process condition.
Probability
Defect Rate Confidence Calculator
Calculate an exact Clopper-Pearson interval for an observed defect proportion and compare the unrounded upper limit with a maximum acceptable rate.
EXACT DEFECT-PROPORTION EVIDENCE
Test whether the sample supports a maximum defect-rate claim
The page inverts exact binomial tails, handles zero and all-defect samples explicitly, and makes the quality decision from the unrounded upper limit rather than the displayed percentage.
Target evidence status-
Observed defect rate-
Exact lower limit-
Exact upper limit-
Target margin-
Interval width-
LIVE DECISION RECORD
Exact interval and target decision register
Observed rate, exact limits, target, and margin are presented on one unrounded comparison basis.
| Evidence point | Rate | Distance from target | Rule role | Decision reading |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
Lower=BetaInv(alpha/2; d,n-d+1); Upper=BetaInv(1-alpha/2; d+1,n-d)
| Symbol | Meaning and unit | Current value |
|---|---|---|
| n | Inspected units | 500 |
| d | Defective units | 4 |
| C | Two-sided confidence, percent | 95 |
| T | Maximum acceptable defect rate, percent | 2 |
Waiting for valid inputs.
FIVE-STEP EVIDENCE WORKFLOW
Make the acceptance rule visible
- Confirm inspected units are comparable binary trials from a defined stable process.
- Enter inspected and defective counts without substituting a rounded percentage.
- Select the confidence level required by the decision protocol.
- Enter the maximum acceptable rate before reading the interval.
- Use the upper-limit status, export the exact values, and retain the sampling record.
FIVE CONFIDENCE FUNDAMENTALS
Why the endpoint governs the claim
- Observed proportion
- `d/n`, the sample estimate before uncertainty.
- Exact inversion
- Limits found by inverting binomial tail probabilities.
- Conservative coverage
- Discrete data can make actual coverage exceed the nominal level.
- Upper-limit rule
- Support requires the unrounded upper endpoint to be at or below target.
- Target margin
- Target minus upper limit; positive supports and negative falls short.
DEFAULT SUBSTITUTION
Four defects in 500 units at 95% confidence
pHat=4/500=0.008; alpha=0.05; exact interval=0.2183908% to 2.0355633%
The maximum acceptable rate is 2%. Because the unrounded upper limit 2.0355633% is above 2%, the sample is marked not-supported even though the observed rate is only 0.8%.
THREE DEEPER MODULES
Understand what can tighten or invalidate the interval
Decision strictness
Higher confidence widens the interval and makes support harder. This is a governance choice about evidence strength, not a tuning knob for a desired answer.
Measurement integrity
False negatives lower the observed count and can create unjustified confidence. Validate inspection-system capability separately.
TWO EVIDENCE CASES
Near-target and zero-defect samples
Observed below target, interval above
The default case shows why observed 0.8% alone cannot support a 2% maximum under the chosen exact upper-limit rule.
Zero in one hundred
The observed rate is zero, but the 95% two-sided exact upper limit is positive. The exported result states how much uncertainty remains instead of declaring perfection.
CONFIDENCE GLOSSARY
Six terms in the decision record
- Clopper-Pearson
- Exact binomial interval based on tail inversion.
- Alpha
- One minus confidence level.
- Beta quantile
- Inverse beta-CDF expression used for exact endpoints.
- Coverage
- Long-run interval containment frequency.
- Upper confidence limit
- Highest endpoint used for the target comparison.
- Support status
- Whether upper limit is no greater than target.
LIMITS AND EVIDENCE
Exact arithmetic still needs representative data
- Units are independent binary trials at one stable defect probability.
- The interval is two-sided Clopper-Pearson and can be conservative.
- The status rule compares the exact upper endpoint with the target.
- Inspection misclassification, clustering, drift, and adaptive sampling are excluded.
- The calculator does not replace a contractual or regulatory acceptance plan.
Retain: lot/process definition, randomization, raw unit results, inspection-system study, confidence and target authority, and exported endpoint comparison.
RELIABLE SOURCES
Primary exact-interval references
DEFECT CONFIDENCE FAQ
Questions about exact limits and targets
Why use an exact interval?
Clopper-Pearson limits invert exact binomial tail tests and remain valid at zero or all defects. They are often conservative, which means actual coverage can exceed the nominal level.
Why compare the upper limit with the target?
A low observed rate can still be too uncertain. Requiring the upper confidence limit to stay below the target asks whether the sample supports the target under the stated rule.
Why can 0 defects still have a positive upper limit?
A finite sample can miss a nonzero defect probability. The exact upper limit quantifies rates still compatible with seeing zero defects at the chosen confidence level.
Does not-supported mean the process fails?
No. It means this sample and rule do not support the claim that the rate is at or below target. The process may need more evidence, a lower observed count, or investigation.
Can I use a one-sided limit instead?
A one-sided upper confidence bound is valid for some acceptance rules, but this page implements the specified two-sided exact interval and compares its upper endpoint.
Are repeated units independent?
The interval assumes stable independent Bernoulli classifications. Clustering by cavity, machine, batch, or inspector can make the effective information smaller.
IMPORTANT EVIDENCE NOTE
Not-supported is not the same as process failure
The status evaluates one claim under one confidence rule. It can call for more evidence without establishing that the underlying rate exceeds the target.