Wilson Score Interval Calculator
Run a calculation in the Calculator tab first, then return here for the full step-by-step solution.
No data yet — enter values in the Calculator tab first.
What Is a Wilson Score Interval?
A Wilson score interval is a confidence interval for a population proportion, computed from binary (success/failure) data. Proposed by statistician Edwin Bidwell Wilson in 1927, it provides an interval estimate for the true proportion p in a population, based on x successes observed in n independent Bernoulli trials.
The central idea is the same as any confidence interval: you have a sample proportion p̂ = x/n and want to construct a plausible range for the underlying population proportion p. The Wilson interval does this through a method that stays inside [0, 1] by construction and maintains better statistical properties than the textbook Wald approach, particularly when sample sizes are small or when the proportion is close to 0 or 1.
Every input to this calculator corresponds to a real binomial data problem: x successes and n trials with a binary outcome (converted or not, defective or not, responded or not, clicked or not). The confidence level sets how often the procedure produces intervals that capture the true p across many repeated samples.
How to Use This Wilson Score Calculator
Type the count of positive outcomes in the "Number of Successes" field. This must be a whole number between 0 and n. For a conversion rate, this is the number of conversions; for a defect study, the number of defective items.
Type the total number of observations in the "Total Trials" field. Trials must be a positive integer greater than zero, and successes cannot exceed trials.
If you know the number of failures instead of total trials, enter it in the optional "Failures" field. The calculator checks that successes + failures = total trials. Mismatched values produce an error message.
Select 90%, 95%, or 99%. The default is 95%, which is the standard in most research contexts. A higher confidence level produces a wider interval.
The calculator displays the lower and upper bounds of the Wilson confidence interval, the observed proportion and percentage, the CI width, and the z critical value used. Switch to the Step-by-Step tab for a full derivation.
Wilson Score Formula
Let x be the number of successes, n the number of trials, p̂ = x/n the sample proportion, and z the standard normal critical value for the chosen confidence level. The Wilson interval is:
Wilson Center
center = (p̂ + z²/(2n))
/ (1 + z²/n)
Wilson Margin
margin = z × √(p̂(1−p̂)/n
+ z²/(4n²))
/ (1 + z²/n)
Lower Bound
L = center − margin
Upper Bound
U = center + margin
What each symbol means:
| Symbol | Meaning | Notes |
|---|---|---|
| x | Number of successes | Non-negative integer; x ≤ n |
| n | Number of trials | Positive integer; n > 0 |
| p̂ | Sample proportion | p̂ = x/n; ranges from 0 to 1 |
| z | Normal critical value | 1.6449 for 90%, 1.9600 for 95%, 2.5758 for 99% |
| z²/(2n) | Continuity adjustment term | Shifts the center away from the boundary |
| z²/n | Denominator adjustment | Contracts the interval relative to the Wald interval |
| center | Midpoint of the Wilson interval | Not equal to p̂ when z²/n is large |
| margin | Half-width of the Wilson interval | Asymmetric around p̂ |
The Wilson interval is derived by inverting the score test for the binomial proportion rather than plugging the sample proportion into the standard error formula directly. This gives it better mathematical properties than the Wald interval, as described by Wilson (1927) and discussed in Brown, Cai, and DasGupta (2001).
Worked Wilson Score Example
x = 80 successes, n = 100 trials
p̂ = x/n = 80/100 = 0.80 (80%)
For 95% confidence: α = 0.05, z = 1.9600
z² = 1.96² = 3.8416
z²/n = 3.8416/100 = 0.038416
z²/(2n) = 3.8416/200 = 0.019208
center = (0.80 + 0.019208) / (1 + 0.038416) = 0.819208 / 1.038416 = 0.78893
p̂(1−p̂)/n = 0.80 × 0.20 / 100 = 0.0016
z²/(4n²) = 3.8416 / 40000 = 0.000096
Sum = 0.001696
√0.001696 = 0.04118
margin = 1.96 × 0.04118 / 1.038416 = 0.080713 / 1.038416 = 0.07772
Lower = 0.78893 − 0.07772 = 0.7112 (71.12%)
Upper = 0.78893 + 0.07772 = 0.8666 (86.66%)
Interpretation: We are 95% confident that the true conversion rate for this campaign lies between 71.12% and 86.66%. The observed rate was 80%, but the interval reflects the uncertainty from a sample of only 100 visitors. Notice the interval is not symmetric around 80%: it extends 8.88 percentage points downward and 6.66 points upward, because the Wilson interval is asymmetric near the boundaries. You can verify this result using the calculator above.
How to Interpret a Wilson Confidence Interval
Suppose the Wilson 95% confidence interval for your proportion comes out as [0.711, 0.867]. The correct interpretation is:
Two other things to keep in mind when reading Wilson interval results:
The interval is asymmetric. Unlike the Wald interval, the Wilson interval is not centered exactly on p̂. When p̂ is close to 0 or 1, the center shifts toward the interior of [0, 1], which is mathematically appropriate. With small samples, the shift can be noticeable.
Width reflects sample size, not just p̂. A narrow Wilson interval reflects a large sample; a wide one reflects a small sample or a proportion near 0.5. With n = 10 and p̂ = 0.5, the 95% interval spans roughly 0.25 to 0.75. With n = 1000 and p̂ = 0.5, it spans roughly 0.47 to 0.53.
Why Use the Wilson Score Interval?
The simplest confidence interval for a proportion is the Wald interval: p̂ ± z × √(p̂(1−p̂)/n). It is easy to calculate and easy to explain. Yet it has two practical problems that make the Wilson interval a better default in many situations.
Problem 1: The Wald interval can fall outside [0, 1]. With p̂ = 0.02 and n = 30, the Wald lower bound is negative. Negative proportions are meaningless, but nothing in the Wald formula prevents them. The Wilson interval, being derived from a test inversion, is bounded within [0, 1] without any clamping.
Problem 2: The Wald interval has poor coverage near the boundaries. Coverage probability is the frequency with which a procedure's intervals actually contain the true parameter. The Wald interval's coverage drops well below its nominal level when p is close to 0 or 1, especially with small n. The Wilson interval has substantially better (though not perfect) coverage in those regions, as shown by Brown, Cai, and DasGupta (2001).
The Wilson interval is not universally best for every problem. For very small n or for exact guarantees, the Clopper-Pearson exact interval may be preferred. For a fast approximation that is easy to remember, Agresti-Coull is an option. The Wilson interval occupies a practical middle ground: it is computationally simple, does not require exact binomial calculations, and has good coverage across a wide range of conditions.
Wilson Score vs. Wald Interval
The Wald interval is the most commonly taught method for a proportion CI, but direct comparison with the Wilson interval shows where each method is appropriate.
| Property | Wilson Score | Wald (Normal Approximation) |
|---|---|---|
| Formula | Test inversion of score statistic | p̂ ± z × √(p̂(1−p̂)/n) |
| Bounded in [0, 1] | Yes, by construction | No; can go negative or above 1 |
| Coverage near p = 0 or 1 | Generally good | Poor; well below nominal |
| Small sample behavior | Acceptable | Unreliable |
| When p̂ = 0 or p̂ = 1 | Produces meaningful interval | Collapses to point [0,0] or [1,1] |
| Symmetry around p̂ | No; asymmetric | Yes; symmetric |
| Computational complexity | Moderate | Simple |
| Typical use | Small n, extreme p, careful analysis | Large n, p away from boundaries |
For large samples with p̂ well away from 0 and 1, both methods give similar results. With n = 1000 and p̂ = 0.5, the Wald 95% interval is (0.469, 0.531) and the Wilson 95% interval is (0.469, 0.531) to three decimal places. The difference becomes apparent with smaller samples or more extreme proportions.
Wilson Score vs. Other Proportion Confidence Intervals
| Method | Approach | Strengths | Limitations | Best for |
|---|---|---|---|---|
| Wilson Score | Score test inversion | Good coverage; bounded in [0,1]; handles 0/n and n/n | Asymmetric; slightly complex formula | General use; small to moderate n |
| Wald | Normal approximation to p̂ | Simple; symmetric; easy to explain | Poor coverage near boundaries; can leave [0,1] | Large n, p away from 0 and 1 |
| Clopper-Pearson (Exact) | Exact binomial | Conservative; guaranteed coverage | Overly wide; no closed-form inversion | Very small n; regulatory or safety studies |
| Agresti-Coull | Add z²/2 successes and failures, then Wald | Easy to compute; nearly as good as Wilson | Slightly wider than Wilson; still can have coverage gaps | Quick manual calculation; teaching |
| Jeffreys | Bayesian with Jeffreys prior Beta(0.5, 0.5) | Excellent coverage; performs well at boundaries | Bayesian framework; requires understanding priors | Bayesian analyses; very small n |
No single method is best for every situation. The frequentist default for a practical, general-purpose binomial CI is the Wilson interval. For guaranteed minimum coverage, Clopper-Pearson is safer. For Bayesian analyses, the Jeffreys interval is a natural choice. The Agresti-Coull approximation gives most of Wilson's benefits with a slightly simpler formula: add 2 successes and 2 failures (for 95%), then use the Wald formula on the adjusted counts.
Confidence Level, z Critical Values, and Interval Width
The confidence level determines how often the interval-building procedure contains the true proportion across many repeated samples. A 95% confidence level means that if you collected 100 independent samples and computed a Wilson interval for each, roughly 95 of those intervals would contain the true p.
Higher confidence levels produce wider intervals because the z critical value is larger, increasing the Wilson margin. For example, with x = 50, n = 100:
| Confidence Level | z | Lower Bound | Upper Bound | Width |
|---|---|---|---|---|
| 90% | 1.6449 | 40.82% | 59.18% | 18.36 pp |
| 95% | 1.9600 | 39.75% | 60.25% | 20.50 pp |
| 99% | 2.5758 | 37.55% | 62.45% | 24.90 pp |
How Sample Size Affects the Wilson Interval
Larger samples produce narrower Wilson intervals, reflecting greater precision in the estimate of p. The table below holds p̂ = 0.60 constant and shows the 95% Wilson interval at several sample sizes.
| n (trials) | x (successes) | p̂ | Wilson Lower | Wilson Upper | Width |
|---|---|---|---|---|---|
| 10 | 6 | 0.60 | 28.30% | 85.14% | 56.84 pp |
| 30 | 18 | 0.60 | 41.53% | 76.24% | 34.71 pp |
| 100 | 60 | 0.60 | 50.26% | 69.12% | 18.86 pp |
| 500 | 300 | 0.60 | 55.70% | 64.14% | 8.44 pp |
| 1,000 | 600 | 0.60 | 56.96% | 62.96% | 6.00 pp |
With n = 10, the Wilson interval nearly spans the entire plausible range. With n = 1000, the precision is much higher. Quadrupling n roughly halves the interval width, reflecting the 1/√n relationship in the standard error. For studies where a narrow interval matters, planning the required n before collecting data is handled by a sample size calculator.
Wilson Score for Small Samples and Extreme Proportions
One of the practical reasons to use the Wilson method is its behavior in situations where the Wald interval fails.
When successes = 0 (x = 0)
The Wald interval degenerates to [0, 0], which conveys nothing. The Wilson interval still produces a meaningful upper bound. For x = 0, n = 20 at 95% confidence: Wilson upper bound ≈ 0.1611 (16.11%). This tells you the true proportion is plausibly as high as 16%, even though no successes were observed. This matters in safety studies and rare-event monitoring.
When successes = trials (x = n)
The Wald interval degenerates to [1, 1]. The Wilson interval produces a meaningful lower bound. For x = 20, n = 20 at 95% confidence: Wilson lower bound ≈ 0.8389. This is a sensible reflection of the uncertainty when you observe a perfect success rate.
When n is very small
For n < 10, all approximate methods including Wilson can have coverage issues. Wilson generally outperforms Wald, but for very small n the Clopper-Pearson exact interval may be more appropriate when guaranteed minimum coverage is needed.
When p̂ is near 0.5
Near p̂ = 0.5, all approximate methods perform reasonably well, and the difference between Wilson and Wald is small. The maximum standard error for any proportion occurs at p = 0.5, so intervals are widest there for a given n.
Wilson Score Lower Bound and Ranking
A distinct application of the Wilson interval is using the lower confidence bound as a ranking score, rather than using the observed proportion directly. This idea was popularized by Evan Miller in the context of rating systems.
The basic problem it addresses: an item with 5 positive ratings out of 5 total has a 100% observed rate, but an item with 450 positive out of 500 has a 90% observed rate. Ranked by observed proportion, the first item appears better. Ranked by Wilson lower bound (at 95% confidence), the second item scores higher because 450 observations provide much stronger evidence for a high underlying rate.
Wilson lower-bound ranking has been described as useful for ranking items in rating systems, review platforms, and recommendation engines, where both the observed success rate and the number of observations should influence the ranking. It does not claim to reflect any specific real-world platform's algorithm; its value is as a principled way to incorporate uncertainty into a ranking score.
| Item | Positives | Total | Observed Rate | Wilson Lower (95%) | Rank by Observed | Rank by Wilson Lower |
|---|---|---|---|---|---|---|
| A | 5 | 5 | 100% | 47.8% | 1st | 3rd |
| B | 9 | 10 | 90% | 55.5% | 2nd | 2nd |
| C | 450 | 500 | 90% | 87.2% | 2nd | 1st |
This illustrates why the lower bound better represents long-run expected performance when the number of observations varies across items.
Business and Real-World Applications
Wilson intervals appear across any domain where binary outcome data is measured and a proportion needs to be estimated with uncertainty quantified. Here are ten concrete hypothetical examples.
Website Conversion Rate
A landing page converts 120 out of 800 visitors. Wilson 95% CI: [12.87%, 17.17%]. The team can decide whether the true rate plausibly meets the 15% target.
Email Click Rate
A campaign sends 5,000 emails; 350 recipients click. Wilson 95% CI: [6.33%, 7.80%]. This range guides expected performance for future similar campaigns.
Product Defect Rate
A sample of 200 units finds 8 defective. Wilson 95% CI: [1.78%, 7.70%]. The interval determines whether the process is within the 5% defect tolerance.
Customer Satisfaction
A survey of 150 customers shows 132 satisfied. Wilson 95% CI: [80.75%, 92.33%]. Management can assess whether satisfaction meets the 85% threshold.
A/B Test Conversion
Variant B converts 42 of 300 visitors. Wilson 95% CI: [10.67%, 18.72%]. Comparing non-overlapping intervals for variants A and B suggests a meaningful difference.
Treatment Response Rate
A trial with 60 patients sees 45 respond to treatment. Wilson 95% CI: [62.12%, 84.43%]. This is compared with the known response rate for the standard treatment.
Survey Response Rate
A mail survey receives 63 responses from 250 sent. Wilson 95% CI: [19.85%, 31.46%]. The interval sets realistic expectations for response rate in similar surveys.
Classification Accuracy
A model correctly classifies 87 of 100 test examples. Wilson 95% CI: [79.04%, 92.64%]. This accounts for the sampling variability in the test set evaluation.
Approval Rate
A poll of 400 residents finds 230 support a proposed development. Wilson 95% CI: [52.52%, 62.47%]. The lower bound stays above 50%, suggesting majority support.
Customer Retention
Of 500 customers from Q1, 380 are still active in Q4. Wilson 95% CI: [71.76%, 79.72%]. This range informs churn modeling and lifetime value projections.
In every case the mathematical calculation is the same; what changes is the interpretation within the specific business context. The Wilson interval provides the uncertainty range; whether a decision follows depends on domain knowledge, tolerances, and what the data represents. A statistically significant result is not automatically a practically meaningful one.
Wilson Score Interval and the Binomial Distribution
Wilson intervals are designed for binomial data: situations where each observation is a Bernoulli trial with two possible outcomes (success or failure), observations are independent, and the probability of success p is constant across trials. This setup underlies the binomial distribution.
The Wilson interval uses the score test for the binomial proportion. The score test statistic is (p̂ − p) / √(p(1−p)/n), and inverting the rejection region of this test at level α gives exactly the Wilson confidence interval. This derivation produces an interval that uses p under the null rather than p̂, which is the key difference from the Wald approach and the source of the Wilson interval's better boundary behavior.
Not every proportion problem follows a strict binomial model. In clustered samples, observations within a cluster may be correlated. In survey data, complex sampling designs violate the independence assumption. In time-series data, successive outcomes may depend on each other. The Wilson interval calculation is straightforward, but the appropriateness of the result depends on whether the underlying assumptions are reasonably met.
Assumptions and Limitations
The Wilson score interval rests on several assumptions. Violations affect the validity of the result regardless of whether the arithmetic is correct.
The Wilson interval addresses sampling uncertainty from a finite sample. It does not address bias, confounding, poor data quality, or the correctness of the study design. A Wilson interval from biased data describes the biased proportion with good precision; it does not describe the underlying truth.
Common Mistakes
If your data shows "60% success rate from 100 trials," enter x = 60, not x = 0.60. The calculator expects whole number counts, not decimals.
x > n is impossible. The calculator returns an error for this input. Check your data for typos or unit mismatches.
The 95% refers to the procedure's long-run performance, not to the probability the true p is inside any single computed interval. The true p is fixed; the interval either contains it or it does not.
95% is a convention, not a universal standard. Some regulatory contexts require 99%. Some exploratory analyses use 90%. Match the confidence level to the decision being made.
Two confidence intervals that barely overlap do not establish that the proportions are statistically equal. Proper comparison of two proportions requires a two-sample test, not visual inspection of intervals.
The Wilson confidence interval is a two-sided statistical interval. Wilson lower-bound ranking uses only the lower bound as a scoring heuristic. They use the same formula but serve different purposes.
Wilson intervals accurately quantify sampling uncertainty from unbiased sampling. If the data collection process itself is biased (self-selection, convenience sampling), the Wilson interval reflects that bias faithfully rather than correcting it.
Related Tools and Guides at Statistics Fundamentals
The Wilson interval connects to many topics in statistical inference. These resources cover the broader context.
Sources and Further Reading
Authority sources on Wilson score intervals and binomial proportion confidence intervals:
- Wilson, E.B. (1927). "Probable inference, the law of succession, and statistical inference." Journal of the American Statistical Association, 22(158), 209–212. jstor.org
- Brown, L.D., Cai, T.T., & DasGupta, A. (2001). "Interval estimation for a binomial proportion." Statistical Science, 16(2), 101–133. jstor.org
- Newcombe, R.G. (1998). "Two-sided confidence intervals for the single proportion." Statistics in Medicine, 17(8), 857–872. onlinelibrary.wiley.com
- Penn State STAT 415. Confidence Intervals for Proportions. online.stat.psu.edu
- NIST Engineering Statistics Handbook. Confidence Intervals for the Proportion. itl.nist.gov
- Agresti, A. & Coull, B.A. (1998). "Approximate is better than exact for interval estimation of binomial proportions." The American Statistician, 52(2), 119–126.
Frequently Asked Questions
A Wilson score interval is a confidence interval for a binomial proportion, proposed by Edwin B. Wilson in 1927. Given x successes in n trials, it estimates the true underlying proportion p with a specified level of confidence (typically 95%). Unlike the simpler Wald interval, it stays bounded within [0, 1] by construction and maintains better coverage probability when the sample size is small or the proportion is close to 0 or 1.
The Wilson interval is calculated as: center = (p̂ + z²/(2n)) / (1 + z²/n), and margin = z × √(p̂(1−p̂)/n + z²/(4n²)) / (1 + z²/n), where p̂ = x/n is the sample proportion and z is the standard normal critical value (1.96 for 95%). Lower bound = center − margin; upper bound = center + margin.
The Wald interval (p̂ ± z × √(p̂(1−p̂)/n)) can produce boundaries below 0 or above 1, and has poor coverage probability when p is near 0 or 1 or when n is small. The Wilson interval avoids both problems. It is derived by inverting the score test rather than plugging p̂ directly into the standard error, which gives it better statistical properties. For large n and moderate p, the two methods give nearly identical results.
The 95% refers to the procedure's long-run behavior: if you repeated the same sampling process many times and computed a Wilson interval each time, approximately 95% of those intervals would contain the true population proportion p. It does not mean there is a 95% probability that the true p is inside the specific interval you calculated. The true p is a fixed number; the interval either contains it or it does not.
Yes. When x = 0, the Wilson interval produces a meaningful upper bound, unlike the Wald interval which degenerates to [0, 0]. For example, with x = 0, n = 20 at 95% confidence, the Wilson upper bound is approximately 0.161 (16.1%). This tells you the true proportion could be as high as 16% even though no successes were observed, which is useful in rare-event monitoring and safety studies.
The Wilson lower bound is the smallest value in the Wilson confidence interval. It can be used as a conservative ranking score: instead of ranking items by observed proportion, you rank by the lower bound of the Wilson interval. This penalizes items with few observations even if they show a high observed rate. For example, an item with 5/5 positive reviews scores lower on Wilson lower bound than one with 450/500, even though both have the same or similar observed rates. The approach is described in Evan Miller's essay "How Not To Sort By Average Rating."
The Clopper-Pearson interval uses exact binomial probabilities rather than a normal approximation. It guarantees that coverage probability is at least the nominal level (conservative), but tends to produce wider intervals than Wilson. Wilson intervals are based on an asymptotic approximation that has good but not exact coverage. For very small n or when exact guarantees are needed (regulatory submissions, safety studies), Clopper-Pearson may be preferred. For general use, Wilson is more efficient.
In A/B testing, Wilson intervals can be computed separately for the control and variant conversion rates. If the intervals do not overlap, there is informal evidence of a difference. For a formal test, a two-proportion z-test or a dedicated A/B test calculator is more appropriate, since non-overlapping CIs are a conservative heuristic rather than an exact significance test. Wilson intervals are also useful in early experiment monitoring, where small samples make Wald intervals unreliable.
95% is the most common convention in academic research, business analytics, and clinical reporting. Use 99% when a false conclusion is particularly costly (regulatory filings, safety decisions) or when multiple tests are run simultaneously (reducing the family-wise error rate). Use 90% in exploratory analysis where you accept more uncertainty to gain a narrower interval. The choice is a judgment call that should match the stakes of the decision being made.
A proportion is a number between 0 and 1 (e.g., 0.75). A percentage is the proportion multiplied by 100 (e.g., 75%). They convey the same information; the Wilson formula works in proportions. The calculator displays both forms: "Observed Proportion: 0.75" and "Observed Percentage: 75%." A probability is also between 0 and 1 but refers to the likelihood of an event rather than an observed fraction of a sample.
Larger samples produce narrower Wilson intervals because the standard error term p̂(1−p̂)/n shrinks as n grows. Roughly, to halve the width of the interval you need to quadruple the sample size. At very small n (say, n = 5 or n = 10), Wilson intervals are wide and all approximate methods have coverage issues. At large n (n ≥ 500), differences between Wilson, Wald, and Agresti-Coull become negligible for proportions not extremely close to 0 or 1.