Confidence Interval for Proportion Calculator

What is a Confidence Interval for a Proportion?

A confidence interval for a population proportion represents the range of values that is likely to contain the true proportion of a population based on sample data. For example, if a poll shows that 55% of 1,000 respondents support a certain policy, this calculator computes the margin of error and the resulting range (such as 52% to 58%) where the true population support is expected to lie, at a selected level of statistical confidence (such as 95%).

The Mathematical Formulas

This calculator computes confidence intervals using two primary methods: the Wald Normal Approximation and the Wilson Score Interval.

1. Wald Interval (Normal Approximation)

The sample proportion is calculated as:

$$\hat{p} = \frac{x}{n}$$

The standard error ($SE$) of the sample proportion is:

$$SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$$

The margin of error ($ME$) is calculated using the critical Z-score ($z^*$) corresponding to the desired confidence level:

$$ME = z^* \times SE$$

The final confidence interval is:

$$CI = \hat{p} \pm ME = \left[\hat{p} - z^* \times SE, \; \hat{p} + z^* \times SE\right]$$

Wald vs. Wilson Score Interval

The Wald interval is the standard textbook method. However, it relies on the normal approximation, which is only valid if both the number of successes ($n\hat{p}$) and failures ($n(1-\hat{p})$) are at least 10. When the sample size is small or the sample proportion is very close to 0 or 1, the Wald interval becomes highly inaccurate. The **Wilson Score Interval** corrects for this by adjusting the center and width of the interval based on the variance of the binomial distribution, providing reliable bounds even for small samples or extreme proportions. For other statistical interval tools, try our Confidence Interval Calculator, Z Score Calculator, or Sample Size Calculator.

Frequently Asked Questions

What does a 95% confidence level mean?

A 95% confidence level means that if you were to repeat the sampling process and construct confidence intervals hundreds of times, approximately 95% of those intervals would contain the true population proportion.

When should I use the Wilson Score interval instead of the Wald interval?

You should use the Wilson Score interval when your sample size is small (less than 30) or when your sample proportion is very close to 0% or 100%. The Wald interval can produce impossible bounds (e.g. below 0% or above 100%) in these cases, while the Wilson Score interval remains mathematically sound.

What are the assumptions for the normal approximation interval?

The main assumptions are: the sample is randomly selected from the population, the observations are independent, and both the number of successes ($x \ge 10$) and failures ($n-x \ge 10$) in the sample are at least 10.

How does sample size affect the width of the confidence interval?

A larger sample size reduces the standard error, which results in a smaller margin of error and a narrower (more precise) confidence interval. To double the precision (halve the width), you must quadruple the sample size.

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