This calculator will compute the confidence interval for a proportion with given sample size, number of successes, and confidence level. It will provide confidence intervals calculated using different methods, including the Normal approximation method, Wilson Score method, and Clopper-Pearson method.
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Critical value: zα/2 = 1.96
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1. Normal Approximation (Wald Interval)
Definition: The simplest and most intuitive method, based on the normal approximation to the binomial distribution.
Where:
- is the sample proportion
- is the critical value from the standard normal distribution
- is the sample size
- When both np ≥ 10 and n(1-p) ≥ 10
- Good for quick calculations and teaching purposes
- Can produce intervals outside [0,1]
- Poor coverage for small samples or extreme proportions
- Symmetric intervals may not reflect the true uncertainty
In a sample of 200 people, 30 are left-handed (p̂ = 0.15). For 95% confidence:
2. Wilson Score Interval
Definition: A more sophisticated method that adds the critical value to both the numerator and denominator, providing better coverage properties.
Where is the same critical value used in the normal approximation method.
- Always produces intervals within [0,1]
- Better coverage properties than normal approximation
- Works well for small samples and extreme proportions
- Accounts for the uncertainty in the variance estimate
- Recommended for most practical situations
- Especially good when normal approximation assumptions aren't met
- Default choice for modern statistical software
Why it's better: The Wilson Score method "pulls" the interval toward 0.5, which reflects the fact that we're less certain about extreme proportions. This results in more honest confidence intervals that better reflect our actual uncertainty.
3. Clopper-Pearson (Exact) Interval
Definition: The "exact" method based on the binomial distribution, using the relationship between binomial and beta distributions.
For x successes in n trials:
Where Beta represents the inverse beta cumulative distribution function.
- Guarantees exact coverage probability
- No approximations involved
- Works for any sample size
- Always produces valid intervals [0,1]
- Conservative (actual coverage often exceeds nominal level)
- Can produce unnecessarily wide intervals
- Computationally more complex
- When you need guaranteed coverage
- For regulatory or safety-critical applications
- When sample sizes are very small
4. Method Comparison and Recommendations
For most situations: Wilson Score
Best balance of accuracy, reliability, and practical properties.
For teaching/quick calculations: Normal Approximation
Easy to understand and calculate by hand, but check assumptions first.
For guaranteed coverage: Clopper-Pearson
Conservative but reliable, especially for critical decisions.
- n < 30: Use Wilson Score or Clopper-Pearson
- 30 ≤ n < 100: Wilson Score recommended, Normal Approximation acceptable if assumptions met
- n ≥ 100: All methods generally work well, Wilson Score still preferred
Important Note: The "exact" Clopper-Pearson method is exact in terms of coverage probability, but no confidence interval method can tell you exactly where the true parameter lies. All methods provide ranges of plausible values based on your sample data.
5. Interpretation
What does a 95% confidence interval mean?
If we repeated our sampling process many times and calculated a 95% confidence interval each time, approximately 95% of those intervals would contain the true population proportion.
Common misconception: A 95% confidence interval does NOT mean there's a 95% probability that the true proportion lies in our specific interval. The true proportion is fixed (but unknown), and our interval either contains it or it doesn't.