Binomial 95% Confidence Interval Calculator

A binomial 95% confidence interval calculator estimates a range for the true proportion of successes from a sample. Enter the number of successes and the total number of trials; the calculator returns a 95% interval using the Wilson score method, a reliable option even when the sample is small or the observed proportion is near 0 or 1.

Formula

This calculator uses the two-sided Wilson score interval for a binomial proportion. Let x be the number of successes and n the total number of independent trials. The observed proportion is p̂ = x/n. For a 95% interval, the standard normal critical value is z = 1.96 (approximately).

The Wilson center and half-width are:

• Center: (p̂ + z²/(2n)) / (1 + z²/n)
• Half-width: z × √[p̂(1 − p̂)/n + z²/(4n²)] / (1 + z²/n)

The lower bound is the center minus the half-width; the upper bound is the center plus the half-width. The calculator displays the two bounds as proportions rounded to four decimal places, and also reports the observed proportion as a percentage. Multiply a bound by 100 to express it as a percentage. Successes must be a whole number from 0 through the number of trials, and the number of trials must be a positive whole number.

Worked Example

With the default inputs, x = 50 successes and n = 100 trials, so the observed proportion is p̂ = 50/100 = 0.5. Using z = 1.96:

1. Compute z² = 3.8416 and the denominator 1 + z²/n = 1.038416.
2. The center is (0.5 + 3.8416/200) / 1.038416 ≈ 0.5.
3. The square-root term is √[0.5 × 0.5/100 + 3.8416/(4 × 100²)] ≈ 0.05096.
4. The half-width is 1.96 × 0.05096 / 1.038416 ≈ 0.0962.

The Wilson interval is approximately 0.4038 to 0.5962, or 40.38% to 59.62%. Rounded as in the calculator, the lower bound is 0.4038, the upper bound is 0.5962, and the observed proportion is 50.00%.

What the Interval Means

A confidence interval is a method for expressing sampling uncertainty, not a promise that the particular interval has a 95% chance of containing the fixed true proportion. In repeated comparable samples, intervals constructed by this method capture the underlying proportion about 95% of the time, subject to the model assumptions and the method’s approximation.

For example, if 50 of 100 sampled items have a feature, the estimate is 50%, while the interval describes the uncertainty around that estimate. It does not mean that 40.38% to 59.62% of the individual items are uncertain, nor does it account for bias in how the sample was collected.

When to Use a Binomial Interval

Use a binomial proportion interval when each observation can be classified into one of two outcomes, such as success/failure, yes/no, or defective/not defective. The trials should be independent, and the probability of success should be reasonably consistent across trials. Typical applications include estimating a survey response proportion or the fraction of inspected parts that meet a criterion.

The input “success” is simply the outcome being counted; it does not have to mean a favorable result. Choose the category before counting, then make sure the total trial count includes every observation in the sample that belongs in the analysis.

Why the Wilson Method

The familiar Wald interval uses p̂ ± 1.96√[p̂(1 − p̂)/n]. It is simple, but can perform poorly for small samples and near the extremes, producing intervals with weak coverage or bounds below 0 or above 1. Wilson’s method adjusts the center and width and generally behaves better in those cases.

Other methods are appropriate in some settings. The exact Clopper–Pearson interval is based directly on binomial probabilities and is often conservative, meaning its actual coverage can exceed the nominal 95%. A normal approximation may be adequate in sufficiently large, balanced samples, but “large enough” depends on the proportion and the desired accuracy. If a result must match a particular study protocol or software, use that specified method rather than assuming all calculators use the same one.

Common Input and Interpretation Mistakes

• Entering a percentage instead of a count: enter 50 successes and 100 trials, not 50 and 100 as percentages or 0.50 and 1.
• Reversing the counts: successes cannot exceed trials. If the category of interest is absent, enter zero successes rather than changing the denominator.
• Ignoring the sampling design: clustered, repeated, weighted, or dependent observations may not follow the simple independent-binomial model. A basic calculator cannot correct for those features.
• Overreading precision: the displayed digits are rounded calculations, not evidence that the estimate is known to that many decimal places. Report a sensible number of digits for the sample and context.
• Confusing uncertainty with bias: an interval quantifies sampling variation under assumptions. It does not repair a nonrepresentative sample, measurement errors, or missing responses.

Frequently Asked Questions

How do I calculate a 95% confidence interval for a proportion?
Count the successes and total trials, then apply a proportion-interval method such as Wilson’s. This calculator uses Wilson’s score formula with the 95% critical value 1.96.

Can I use this calculator when there are zero successes?
Yes. The Wilson method can return a meaningful interval when the observed count is zero or equals the total, unlike the simple Wald formula, whose estimated standard error becomes zero at those extremes. The result still depends on the binomial assumptions.

Why does my result differ from another calculator?
Different tools may use Wilson, Wald, exact Clopper–Pearson, or another interval method. They may also use different confidence levels or rounding rules. Check the method and settings before comparing results.

Does a 95% confidence interval mean there is a 95% chance the true proportion is inside?
Not in the usual frequentist interpretation. The true proportion is treated as fixed; the 95% refers to the long-run performance of the interval procedure across repeated samples.

What if my observations are not independent?
A simple binomial interval may not provide valid uncertainty estimates for dependent or clustered observations. Consider an analysis that accounts for the sampling design, or consult a statistician familiar with that design.

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in x | Successes | - | 50 | 0
in n | Trials | - | 100 | 1
out lower | Lower bound | proportion | (x / n + 1.96^2 / (2 * n) - 1.96 * sqrt((x / n) * (1 - x / n) / n + 1.96^2 / (4 * n^2))) / (1 + 1.96^2 / n) | 4
out upper | Upper bound | proportion | (x / n + 1.96^2 / (2 * n) + 1.96 * sqrt((x / n) * (1 - x / n) / n + 1.96^2 / (4 * n^2))) / (1 + 1.96^2 / n) | 4
out percent | Estimated proportion | % | 100 * x / n | 2