# Interpret a small success rate with its model and uncertainty

Constructed teaching cases with full context. No actual people, accounts, payments, research participants or forecasts.

Define distinct eligible units and one binary outcome before calculating a rate. Under a declared independent common-probability binomial model, a nominal 95% Wilson score interval describes sampling uncertainty. It does not fix voluntary selection, repeated events or missing outcomes, and it is not a95% probability statement about a fixed parameter.

## Choose a unit that can succeed or fail once

A well-defined task can classify each eligible unit once as success or failure. Reactions plus comments over impressions cannot normally do this: people can contribute repeatedly, events overlap and impressions are displays. Choose the actual outcome and eligible population rather than borrowing a denominator because it is visible.

The simple model assumes independent binary trials with a common success probability. Repeated tasks from one person, clustered teams and materially different conditions can violate it. A checkbox records your declaration, not evidence that those assumptions or a probability sample exist.

## Read the exact interval method and level

The local review uses the two-sided Wilson score interval without continuity correction, with z=1.959963984540054 for nominal 95% confidence. For k successes among n known outcomes, p=k/n; the center is (p+z²/(2n))/(1+z²/n), with half-width z×sqrt(p(1−p)/n+z²/(4n²))/(1+z²/n). The result stays within 0 and 1.

NIST distinguishes Wilson from adjusted Wald and exact-binomial procedures. Its published 8-of 30 Wilson example gives 95% limits approximately 0.141827 and 0.444480; that independent table checks our implementation. This is a method check, not our own participant result. Different confidence levels or methods need explicit labels, not whichever interval looks narrowest.

## Review known and missing outcomes separately

Enter whole counts for successes, failures and eligible units whose outcome is unknown; enter an actual zero explicitly. The tool rejects malformed values and totals above 1,000,000 without truncation. It computes nothing until distinct eligible binary units are declared, and computes Wilson only after the model declaration with at least one known outcome.

Missing-outcome bounds use the whole declared eligible cohort; Wilson uses known outcomes only. The two outputs answer different questions and must not be combined into one certified interval. The example button loads 8 successes,2 failures and 4 unknowns with declarations unchecked. No account, identity or study data is requested.

## Resolve three constructed uncertainty statements

I1 — Constructed rates:8/10 and 80/100 both equal 80%. Under the stated model, Wilson limits are about 49–94% and 71–87%. Decision: report bases and methods alongside the common point estimate. Outcome: the larger supplied sample has a narrower conditional interval, not proof of a more representative or better product.

I2 — Constructed invalid unit:45 comment events over 300 impressions are labeled 45 successful people out of 300. Decision: hold the binomial interval and correct the event/display units. Outcome: no person-level confidence interval is manufactured from overlapping repeated events.

I3 — Constructed zero successes:0 successes among 10 measured trials does not establish that success is impossible. The nominal Wilson upper limit is about 27.75%. Decision: report the observed 0/10 and conditional interval. If the denominator itself is 0, there is no rate or interval, rather than a measured 0%.

## State what nominal confidence does not promise

Confidence describes a procedure’s repeated-sampling behavior under its model; Wilson’s nominal 95% level is not an exact small-sample coverage guarantee. Do not say there is a95% probability that this particular fixed true rate lies inside the computed interval, or that your next ten readers will produce the same rate.

AAPOR requires explicit model and assumption treatment for precision claims from nonprobability samples. An opt-in collection does not gain a population margin of error merely by entering its count here. If selection, dependence or missingness is unresolved, retain a descriptive result or an explicitly conditional illustration instead of certifying general audience preference.

## Save a bounded numerical review

The text download retains the three parsed counts, known/eligible bases, separate bounds, conditional method and declarations. Any input change clears the previous result until reviewed again. Nothing is automatically saved or sent. Keep the source unit, selection and outcome definition with the record privately; the calculator cannot establish them.

Current primary method pages were read on 1 November 2026. All I-cases and loaded counts are constructed. The NIST numeric reference is attributed. These examples do not establish actual participant or business outcomes; real reporting still needs its actual source and appropriate design.

## Primary sources

- [NIST: Wilson and exact-binomial interval distinctions](https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm)
- [NIST Dataplot: published Wilson reference calculation](https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/propconf.htm)
- [AAPOR: precision and model disclosure](https://aapor.org/standards-and-ethics/disclosure-standards/)