Confidence Interval for a Proportion Simulator

Draw repeated samples and watch how Wilson and Wald confidence intervals for a proportion behave differently, especially when the true proportion is small.

Set up the simulation

True proportion p = 0.10

Interval method

Confidence level

Sample size (n)

Each line is one sample's 95% wilson interval, on a 0-to-1 scale

Press a Draw button to build some intervals

How to use this tool

Set the true proportion, sample size, and confidence level.
Choose the Wilson or Wald method for building each interval.
Press a Draw button to take that many samples, each producing its own confidence interval.
With a small true proportion and the Wald method, watch for intervals that collapse to a single point — then switch to Wilson and see the difference.

Formula Reference

Wald interval

p̂ ± z × √[p̂(1-p̂)/n]

Wilson interval

a rescaled interval that stays valid near 0 and 1

When they diverge most

small n, or p near 0 or 1

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Frequently Asked Questions

Why does the Wald interval sometimes give exactly (0, 0)?+

When a sample happens to have zero successes, the estimated proportion is 0, which makes the standard error in the Wald formula also exactly 0, collapsing the interval to a single point even though the true proportion is almost certainly not exactly 0.

Is the Wilson interval always better?+

For proportions near 0.5 with reasonably large samples, the two methods give very similar intervals. Wilson's advantage shows up specifically when the proportion is close to 0 or 1, or the sample size is small, which is exactly where the Wald interval performs poorly.

What does coverage mean here?+

It is the share of the simulated intervals that actually contain the true proportion. A well-behaved method should have its observed coverage close to the stated confidence level, such as 95%, once enough intervals have been drawn.

Why use a small true proportion like 0.1 by default?+

It is where the difference between the two methods is easiest to see. With a proportion near 0.5, both methods usually look reasonable even with modest samples.

How to use this tool

Set the true proportion, sample size, and confidence level.

Choose the Wilson or Wald method for building each interval.

Press a Draw button to take that many samples, each producing its own confidence interval.

With a small true proportion and the Wald method, watch for intervals that collapse to a single point — then switch to Wilson and see the difference.