Confidence Interval Calculator for a Single Proportion

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Input Data

How to Use This Calculator

1

Enter Your Sample Data

Fill in the three required values:

Number of Successes (x)

The count of observations with the characteristic you're studying (must be ≥ 0)

Sample Size (n)

The total number of observations in your sample (must be ≥ 1)

Significance Level (α)

The probability of Type I error (typically 0.05 for 95% confidence)

Example: If you surveyed 200 people and 45 preferred Product A, then x=45 and n=200.

2

Calculate

Click the "Calculate Confidence Interval" button to generate results.

The calculator automatically validates your inputs and will show error messages if:

  • Successes exceed sample size
  • Sample size is too small
  • Alpha is outside valid range (0.001 to 0.999)
3

Interpret the Results

Review the comprehensive output:

Basic Statistics

  • Sample Proportion (p̂)
  • Confidence Level
  • Critical Z-value

Wald Interval

  • Standard Error
  • Margin of Error
  • Lower & Upper Bounds
  • Validity Check

Wilson Score Interval

  • Lower & Upper Bounds
  • Detailed Calculation

Visualization

  • Interval Sketch
  • Interpretation Guide
4

Choose the Right Interval Method

This calculator provides two different confidence interval methods:

Wald Interval

Use when: Large sample size (n×p̂ ≥ 10 and n×(1-p̂) ≥ 10)

Advantage: Simple formula, easy to calculate

Limitation: Can be inaccurate for small samples or extreme proportions

Wilson Score Interval

Use when: Any sample size, especially small samples or proportions near 0 or 1

Advantage: More accurate, always produces valid intervals

Recommendation: Default choice for most applications

5

Export and Share Results

Save or share your analysis using these options:

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Copy all results as formatted text for reports or presentations

Export to PDF

Download complete results including formulas and visualizations as PDF

Statistical Tip

When to use this calculator: Perfect for survey analysis, quality control, medical studies, or any situation where you need to estimate a population proportion from sample data.

Sample size guidance: For reliable results, aim for at least 30 observations. The Wilson Score interval provides better accuracy when sample sizes are small.

Interpretation example: "We are 95% confident that the true proportion of customers who prefer our product is between 38% and 52%."

Theoretical Background: Confidence Interval for a Single Proportion

A confidence interval for a population proportion (\(p\)) provides an estimated range of values which is likely to include the unknown population proportion based on sample data. This is common in scenarios like opinion polls (proportion of voters favoring a candidate) or quality control (proportion of defective items).

Key Inputs and Point Estimate

The point estimate for the population proportion \(p\) is the sample proportion \(\hat{p}\):

\( \hat{p} = \frac{x}{n} \)

1. Wald Interval (Normal Approximation Method)

The Wald interval is the most traditional method and relies on the normal approximation to the binomial distribution. It works well when the sample size is large and the proportion is not too close to 0 or 1.

The standard error of the sample proportion is estimated as:

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

The confidence interval is then calculated as:

\( CI = \hat{p} \pm Z_{\alpha/2} \cdot SE_{\hat{p}} \)

Where \(Z_{\alpha/2}\) is the critical value from the standard normal distribution corresponding to the desired confidence level (e.g., for 95% CI, \(Z_{0.025} \approx 1.96\)).

Conditions for Normal Approximation: For the Wald interval to be reliable, it's generally recommended that both \(n\hat{p} \ge 10\) and \(n(1-\hat{p}) \ge 10\). If these conditions are not met, the Wald interval can be inaccurate, potentially yielding bounds outside [0, 1] or having poor coverage probability.

2. Wilson Score Interval

The Wilson score interval is an improvement over the Wald interval, especially for small sample sizes or when the proportion is close to 0 or 1. It has better coverage properties and always produces an interval within [0, 1].

The Wilson score interval is derived by solving the quadratic equation from \( \frac{|\hat{p} - p|}{\sqrt{p(1-p)/n}} \le Z_{\alpha/2} \). Its formula is more complex:

\( CI = \frac{\hat{p} + \frac{Z_{\alpha/2}^2}{2n} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n} + \frac{Z_{\alpha/2}^2}{4n^2}}}{1 + \frac{Z_{\alpha/2}^2}{n}} \)

Alternatively, it can be written by first defining a modified sample proportion \(\tilde{p}\) and modified sample size \(\tilde{n}\):

\( \tilde{x} = x + \frac{Z_{\alpha/2}^2}{2}, \quad \tilde{n} = n + Z_{\alpha/2}^2 \)
\( \tilde{p} = \frac{\tilde{x}}{\tilde{n}} = \frac{x + Z_{\alpha/2}^2/2}{n + Z_{\alpha/2}^2} \)

The center of the Wilson interval is \(\tilde{p}\). The interval limits are:

\( \left( \tilde{p} - Z_{\alpha/2} \frac{\sqrt{n \hat{p}(1-\hat{p}) + Z_{\alpha/2}^2/4}}{n+Z_{\alpha/2}^2}, \quad \tilde{p} + Z_{\alpha/2} \frac{\sqrt{n \hat{p}(1-\hat{p}) + Z_{\alpha/2}^2/4}}{n+Z_{\alpha/2}^2} \right) \)

The formula presented in the calculator uses the first, more direct form.

The Wilson interval is generally recommended as a good default choice due to its better performance across a wider range of scenarios compared to the Wald interval.

Interpreting the Confidence Interval

A \( (1-\alpha) \times 100\% \) confidence interval for \(p\) is interpreted as: "We are \( (1-\alpha) \times 100\% \) confident that the true population proportion \(p\) lies between the lower bound and the upper bound of the interval."

For instance, if a 95% confidence interval for the proportion of voters supporting a candidate is [0.52, 0.58], it means we are 95% confident that the true proportion of all voters supporting that candidate is between 52% and 58%.