Evidence strength
Two samples can have the same observed rate but different widths because more independent trials provide tighter sampling information.
Probability
Binomial Event Confidence Calculator
Calculate a Wilson confidence interval for an observed binomial proportion and classify a target probability as supported, contradicted, or still inside the interval.
OBSERVED-PROPORTION UNCERTAINTY
Put an interval around the observed event rate before judging a target
The Wilson score interval adjusts the center and half-width using sample size and a selected z value, then compares the entire interval - not only the point estimate - with the target.
Observed success rate -
Wilson lower bound -
Wilson upper bound -
Wilson half-width -
Observed failures -
Target interpretation -
LIVE DECISION RECORD
Wilson interval construction ledger
Observed counts, z adjustment, bounds, and target status remain visible as separate calculation layers.
| Layer | Success / confidence | Failure / target | Sample / denominator | Rate or bound | Interpretation |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
p_hat=x/n; center=(p_hat+z^2/2n)/(1+z^2/n); half=z*sqrt(p_hat(1-p_hat)/n+z^2/4n^2)/(1+z^2/n)
| Symbol | Meaning and unit | Current value |
|---|---|---|
| trials | Observed trial count - Completed binary observations in the sample. | 80 |
| successes | Observed success count - Successes cannot exceed observed trials. | 54 |
| confidenceLevel | Confidence level (%) - Supported values: 80, 90, 95, 98, or 99. | 95 |
| targetProbabilityPct | Decision target (%) - Reference probability compared with the entire interval. | 60 |
Waiting for valid inputs.
WHO THIS MODEL SERVES
A scoped decision aid, not a universal forecast
Primary audience: Experiment, quality, product, survey, and operations teams interpreting one observed binary proportion.
Decision boundary: Use for a binomial proportion under representative independent sampling; it quantifies sampling uncertainty, not bias, dependence, or future drift.
HOW TO INTERPRET THE SAMPLE
Five steps from counts to a target conclusion
- Define the sampled population, trial, and success rule before entering counts.
- Enter completed trials and observed successes; never substitute expected counts.
- Select a confidence level appropriate to the decision protocol.
- Compare the target with both bounds, not just the observed rate.
- Retain the interval, counts, exclusions, and sampling dates in the exported record.
INTERVAL FUNDAMENTALS
Five distinctions behind Wilson bounds
- Point estimate
- x/n is the observed sample proportion.
- Coverage
- Long-run frequency with which the interval procedure contains the true p.
- Wilson center
- An adjusted center that is not always identical to x/n.
- Half-width
- Distance from Wilson center to each untruncated bound.
- Target classification
- Supported only if the lower bound meets target; contradicted if the upper bound is below it.
FORMULA AND DEFAULT SUBSTITUTION
Construct the bounds from counts and z
Wilson interval = adjusted center +/- adjusted half-width
Defaults give p_hat=54/80=67.5%. At 95% confidence z=1.959964. The denominator is 1+z^2/80; the model computes adjusted center and half-width without intermediate rounding before comparing a 60% target.
DEEPER CONFIDENCE ANALYSIS
Three reasons an interval changes the decision
Boundary behavior
At zero or all successes, Wilson avoids the implausible claim that the true rate is known exactly.
Target ambiguity
When the target lies inside the interval, the data do not cleanly support or contradict it at the chosen coverage level.
WORKED CONFIDENCE CASES
Two samples with different interpretations
Pilot conversion test
Fifty-four successes in 80 trials yield 67.5%. Whether a 60% target is supported depends on the lower Wilson bound, not on the point estimate alone.
Zero-success safety signal
Zero observed events does not prove zero underlying probability. Wilson returns a positive upper bound that shrinks only as credible independent exposure grows.
INTERVAL TERMINOLOGY
Six terms used in the result
- Observed proportion
- Success count divided by trial count.
- Confidence level
- Nominal long-run coverage of the procedure.
- Critical value
- Standard-normal z associated with the selected level.
- Lower bound
- Smallest p retained by the Wilson interval.
- Upper bound
- Largest p retained by the Wilson interval.
- Margin of error
- Wilson half-width around its adjusted center.
EVIDENCE RETENTION
Counts need a sampling provenance
Keep trial inclusion rules, timestamps, population frame, success classification, missing observations, deduplication, confidence level chosen in advance, and target authority. Store the raw numerator and denominator, not only a rounded percentage.
LIMITS AND EXCLUSIONS
What a Wilson interval cannot repair
- Trials should be representative and sufficiently independent.
- Coverage describes repeated sampling, not a Bayesian probability for p.
- Selection bias, misclassification, and nonresponse are not included.
- Multiple comparisons and sequential peeking require adjusted procedures.
- Future regime change is outside the interval.
RELIABLE SOURCES
Primary references for binomial proportion intervals
PROPORTION-INTERVAL FAQ
Questions about coverage and target claims
Why use Wilson instead of observed rate +/- a simple error?
Wilson behaves better for modest samples and rates near zero or one and keeps bounds within 0-100%.
Is this the prospective SLA confidence page?
No. This page starts from observed successes and trials to estimate a plausible interval for p.
Does 95% confidence mean a 95% chance p is inside this interval?
In frequentist terms, the procedure captures the fixed true p in 95% of repeated comparable samples; it is not a posterior probability statement.
What if successes are zero or equal trials?
Wilson still returns a bounded one-sided-looking interval rather than a zero-width false certainty.
Why restrict confidence levels?
The model uses audited standard-normal critical values for the listed levels.
Can dependence be fixed by increasing n?
No. Correlated observations can make the nominal interval too narrow regardless of raw sample size.
IMPORTANT INFERENCE NOTE
Precision is not representativeness
A narrow interval from biased or dependent data can be misleading. Validate sampling design and classification before using bounds in a release, quality, or policy decision.