Confidence Interval for Proportion Calculator
Calculate population proportion confidence intervals ($\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$) at 90%, 95%, or 99% confidence levels, margin of error %, and sample size.
Calculate Population Proportion Confidence Interval
Enter success count (x), total sample size (n), and confidence level.
Calculation Results
Calculated using Wald proportion confidence interval equations: \text{CI} = \hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
Quick Summary
The Confidence Interval for Proportion Calculator evaluates population proportion bounds ($\hat{p} \pm z^* \cdot \text{SE}$), percentage margin of error, and standard error.
Formula Explanation
\text{CI} = \hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\text{Margin of Error ME} = z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}How It Works
A proportion confidence interval estimates the true population percentage $p$ for binary survey responses. The Confidence Interval for Proportion Calculator computes sample rate $\hat{p} = x/n$, standard error $\text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, and margin of error.
Step-by-Step Worked Example
Practical Problem: Out of $n = 100$ respondents, $x = 60$ approve ($\hat{p} = 0.60$). Calculate 95% confidence interval for true population approval rate.
- Step 1: Calculate sample proportion ($\hat{p}$): $\hat{p} = 60 / 100 = \mathbf{0.6000\text{ (60.00\%)}}.$
- Step 2: Calculate Standard Error ($\text{SE}$): $\text{SE} = \sqrt{\frac{0.60 \times 0.40}{100}} = \sqrt{0.0024} = \mathbf{0.04899}$.
- Step 3: Calculate Margin of Error ($\text{ME}$): $\text{ME} = 1.960 \times 0.04899 = \mathbf{0.09602\text{ (9.60\%)}}.$
- Step 4: Calculate lower bound ($60.0\% - 9.60\%$): $0.6000 - 0.0960 = \mathbf{50.40\%}$.
- Step 5: Calculate upper bound ($60.0\% + 9.60\%$): $0.6000 + 0.0960 = \mathbf{69.60\%\text{ (95\% CI: [50.40\%, 69.60\%])}}.$
Real-World Calculation Examples
Scenario 1: Survey Approval (x = 60, n = 100)
Parameters: p̂ = 60.0%
Result: [50.40%, 69.60%] (ME = ±9.60%).
Scenario 2: Election Political Poll (x = 520, n = 1000)
Parameters: p̂ = 52.0%
Result: [48.90%, 55.10%] (ME = ±3.10%).
Scenario 3: Factory Defect Rate Quality Audit
Parameters: x = 12 defects, n = 500 units
Result: 95% CI: [1.06%, 3.74%] defect rate.
Scenario 4: Clinical Vaccine Protection Rate
Parameters: Medical efficacy proportion
Result: 99% confidence interval estimation.
Key Benefits of Using This Calculator
Poll & Survey Standard Tool
Essential calculator for election polling and public opinion research.
Margin of Error Percentage
Outputs exact margin of error percentage $\text{ME} = z^* \cdot \text{SE}$.
Multi-Confidence Level Support
Supports 90%, 95%, and 99% confidence levels.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a confidence interval for a proportion?
It estimates the plausible range for an unknown population proportion $p$ based on sample success count $x$ and size $n$.
What are normal approximation criteria for proportion CI?
$n \hat{p} \ge 10$ and $n(1 - \hat{p}) \ge 10$ (at least 10 successes and 10 failures).
How do I calculate a proportion confidence interval in Excel?
Calculate `pHat = x / n` and `ME = 1.96 * SQRT(pHat * (1 - pHat) / n)`.
What is the Wilson Score Interval vs Wald Interval?
The Wald interval ($\hat{p} \pm z^* \text{SE}$) can perform poorly near $0\%$ or $100\%$; the Wilson score interval provides superior coverage near boundaries.
What is Agresti-Coull "Plus Four" method?
Adds 2 successes and 2 failures ($x+2, n+4$), producing reliable confidence intervals even for small samples.
Why do political polls quote a ±3% margin of error?
For $n \approx 1,000$ respondents and $\hat{p} \approx 50\%$, $\text{ME} = 1.96 \sqrt{\frac{0.25}{1000}} \approx 3.1\%$.
What proportion $\hat{p}$ yields maximum margin of error?
$\hat{p} = 0.50$ (50%) maximizes $\hat{p}(1-\hat{p}) = 0.25$, yielding the worst-case maximum margin of error.
What happens if sample size $n$ quadruples?
Quadrupling sample size $n$ cuts margin of error IN HALF ($\text{ME} \propto 1/\sqrt{n}$).
Can proportion confidence interval exceed 0% or 100%?
Theoretical Wald bounds can overshoot; the calculator clips lower bounds to $0\%$ and upper bounds to $100\%$.
What is Clopper-Pearson exact interval?
The Clopper-Pearson interval is a non-asymptotic exact confidence interval based on the binomial distribution.