Zero observed failures does not prove zero risk
With a finite test, zero failures still produces a positive Wilson upper bound. The sample size determines how much residual uncertainty remains.
Reliability
Component Failure Confidence Calculator
Calculate a Wilson failure-rate confidence interval and compare its upper bound with a design limit.
STATISTICAL CONFIDENCE
Compare a defensible interval - not a raw rate - with the design limit
For qualification teams documenting uncertainty from finite pass/fail tests under a predeclared acceptance rule.
Upper Wilson bound-
Observed rate-
Lower bound-
Design margin-
Conservative future failures-
CURRENT DECISION RECORD
Failure-rate confidence ledger
Every row is regenerated from the active inputs and carried into Copy, TXT, and the page-specific PDF payload.
| Measure | Input A | Input B | Result |
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CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
Wilson CI = [center +/- zsqrt(p-hat(1-p-hat)/n+z^2/(4n^2))/(1+z^2/n)]
Waiting for valid inputs.
HOW TO USE
Five steps from test counts to a design claim
- Enter tested and failed counts.
- Choose a two-sided confidence level.
- Enter the design failure-rate limit.
- Use the upper Wilson bound for conservative comparison.
- Archive test definition, censoring rules, and exclusions.
CONFIDENCE FUNDAMENTALS
Five concepts behind the interval
- Observed failure rate
- Failures divided by tested units; it describes the sample and is not by itself a bound on the underlying population rate.
- Wilson score interval
- A binomial proportion interval that adjusts its center and width to behave sensibly near zero, one, and modest sample sizes.
- Confidence level
- The long-run coverage rate of the interval procedure over repeated equivalent studies, not the probability that this fixed interval is true.
- Upper confidence bound
- The conservative endpoint compared with a predeclared maximum failure-rate requirement.
- Design margin
- The design limit minus the upper bound; negative margin means the evidence does not demonstrate the entered limit at the selected confidence.
DEEP STATISTICAL ANALYSIS
Interpret zero failures, sample size, and acceptance rules
More tests improve evidence, not formatting
Additional independent equivalent trials usually narrow the interval. Showing extra decimal places does not create information or compensate for a small sample.
Acceptance rules must precede the result
Choose confidence, failure definition, exclusions, and design limit before inspecting outcomes. Post-hoc thresholds turn the page into a moving target rather than a qualification record.
WORKED DECISION CASES
A qualification case and a zero-failure boundary
Routine connector qualification
Four failures among 400 tested connectors gives an observed rate of 1%. The decision uses the 95% Wilson upper bound against the 2% requirement, not the observed rate alone.
Zero-failure pilot boundary
A pilot with zero failures among 50 units still has a positive upper bound. It may be encouraging operationally yet insufficient to demonstrate a stringent production failure-rate target.
EVIDENCE RECORD
Preserve the test behind the interval
Retain the sampling frame, lot identifiers, exposure duration, environment, failure definition, exclusions, censoring treatment, tested and failed count reconciliation, confidence and limit selected before analysis, protocol revision, deviations, and reviewer approval.
MODEL LIMITS
Conditions behind the Wilson result
- Trials are independent, equivalent, and Bernoulli with one common failure probability.
- Every tested unit has a comparable exposure and a consistently applied failure definition.
- The calculation does not handle censoring, duration adjustment, lot clustering, repeated measures, or process drift.
- The conservative future count is an upper-bound planning projection, not a prediction interval for a particular lot.
CONFIDENCE GLOSSARY
Six distinct statistical terms
- Binomial
- Fixed trials with two outcomes and common probability.
- Coverage
- Frequency intervals contain the true parameter over repetitions.
- z critical
- Normal quantile for the selected confidence.
- Half-width
- Distance from Wilson center to an unbounded endpoint.
- Design limit
- Maximum acceptable failure percentage.
- Conservative count
- Future units multiplied by upper bound.
FREQUENTLY ASKED QUESTIONS
Questions specific to finite failure evidence
Why not compare the observed rate directly with the limit?
The observed rate ignores finite-sample uncertainty. The upper interval endpoint provides a conservative comparison under the binomial model.
Can the confidence level be 100%?
No finite two-sided normal-score interval has a finite 100% critical value. Use a predeclared confidence below 100% and report the remaining uncertainty.
Does an interval below the limit prove every future lot will pass?
No. It summarizes one population parameter under sampling assumptions; future lots can differ because of process drift, environment, suppliers, or dependence.
Can repaired units be counted as new independent trials?
Only when the protocol defines equivalent independent exposures. Reusing units can introduce unit history and within-unit dependence that the simple binomial model omits.
What if units are censored before equal test exposure?
Use life-data or survival methods that retain time and censoring information. Converting unequal exposures into simple pass/fail counts can bias the claim.
Why use Wilson instead of the basic Wald interval?
Wilson avoids several poor boundary behaviors of p-hat +/- zsqrt(p-hat(1-p-hat)/n), especially with small samples or rates near zero.
RELIABLE SOURCES
Binomial and reliability-test references
IMPORTANT STATISTICAL NOTE
An interval cannot repair a biased test
The Wilson calculation assumes the counts represent equivalent independent trials from the population of interest. A narrow interval from unrepresentative, clustered, censored, or inconsistently judged data is not a defensible reliability claim.