About the Wilson Score Interval
The Wilson Score Interval Calculator estimates a confidence interval for a proportion, such as a survey response rate or a test pass rate, using a method built specifically to hold up when sample sizes are small or the observed proportion sits near 0% or 100%. It takes the number of successes out of a sample and a chosen confidence level and returns a range of plausible values for the true underlying proportion. This answers the same real-world question the standard proportion confidence interval answers, but computed with a formula that behaves more reliably at the extremes.
How It Works
Enter how many successes occurred, the total sample size, and a confidence level of 90, 95, 98, or 99 percent, which the calculator maps to a corresponding z-score. The math shifts and rescales the raw sample proportion rather than simply adding and subtracting a margin of error from it, which is what keeps the resulting interval from running below 0% or above 100%. The output is the Wilson interval expressed as a percentage range, shown alongside the plain sample proportion for comparison.
Formula & Methodology
To compute it by hand, first find the sample proportion by dividing successes by sample size, and square your chosen z-score. Compute the denominator as 1 plus that squared z-score divided by sample size. The center of the interval is the sample proportion plus the squared z-score over twice the sample size, all divided by that denominator, which is why it sits slightly closer to 0.5 than the raw sample proportion does. The margin comes from taking the square root of the sample proportion times one minus itself divided by sample size, plus the squared z-score divided by four times sample size squared, multiplying by z, and dividing by the same denominator. Adding and subtracting the margin from the center gives the final interval.
Examples
Small sample, high success rate
Out of 10 trials, 9 succeed, giving a sample proportion of 90%. At the 95% confidence level with a z-score of 1.96, the Wilson interval works out to roughly 59.6% to 98.2%, a wide but bounded range that reflects how little a sample of just 10 can pin down the true rate.
Zero observed successes
With 0 successes out of 20 trials, the sample proportion is 0%, yet at 95% confidence the Wilson interval still runs from about 0% up to roughly 16.1%. This shows the method's key advantage: it does not collapse to a single point just because no successes were observed.
Advantages
- Stays mathematically bounded between 0% and 100% even for small samples or proportions at the extremes, where the standard normal-approximation interval can misbehave.
- Lets you compare intervals across four common confidence levels without needing a separate z-table lookup.
- Reports the raw sample proportion next to the adjusted interval so you can see exactly how much the Wilson correction shifted and widened the estimate.
Common Mistakes
- Using this method interchangeably with the simple normal-approximation interval and assuming they always give nearly the same answer, when the gap is largest exactly in the small-sample, extreme-proportion cases this calculator is built for.
- Entering a raw percentage instead of a count for successes, which produces a nonsensical result once it is divided by the sample size.
- Forgetting that a wide interval from a small sample is not a flaw in the method but an honest reflection of how much uncertainty remains with limited data.
Edge Cases to Watch For
- Successes must fall between 0 and the sample size, and the sample size must be greater than zero, or the calculator returns an error instead of an interval.
- With very small samples or proportions at either extreme, this method still produces a bounded, non-degenerate interval, unlike the simpler normal-approximation interval which can shrink to zero width or extend past 0% and 100% in those same situations.
- As the sample size grows large, the correction terms shrink toward zero and the Wilson interval converges toward the simpler normal-approximation result.
Common Use Cases
- Researchers and analysts working with small samples, such as pilot studies or limited survey responses, who need a confidence interval that will not misbehave near 0% or 100%.
- Quality assurance teams tracking rare defect or failure rates where the observed proportion is naturally close to zero.
- Anyone auditing A/B test or conversion-rate results with a modest number of observed events, where the standard interval formula is known to understate uncertainty.