Understanding Confidence Intervals for Proportions
Educational Tool
When you measure a proportion from a sample (like survey responses or success rates), there's uncertainty about the true population proportion. A confidence interval quantifies this uncertainty by providing a range of plausible values. This tool demonstrates two common methods: the simple Wald interval and the more robust Wilson score interval.
Wald (Normal Approximation) Interval
The Wald interval is the simplest method, using the normal approximation to the binomial distribution. It's widely taught but can perform poorly for small samples or extreme proportions (near 0 or 1).
Rule of thumb: Reliable when n·p̂ ≥ 5 and n·(1-p̂) ≥ 5
Wilson (Score) Interval
± z × √(p̂(1-p̂)/n + z²/(4n²)) / (1 + z²/n)
The Wilson score interval is more robust, especially for small samples and extreme proportions. It never extends outside [0,1] naturally and has better coverage properties than Wald.
Generally recommended as the default choice
Key Concepts
- Sample proportion (p̂): x/n, the observed fraction of successes
- Standard error: Measures precision of p̂ as an estimate
- z-critical: Standard normal cutoff (1.96 for 95% CI)
- Margin of error: Half-width of the interval
- Coverage: Proportion of intervals that capture the true value
Interpreting Results
- A 95% CI does NOT mean 95% probability the true value is inside
- Correct interpretation: 95% of similarly constructed intervals would contain the true value
- Wider intervals indicate more uncertainty; narrower intervals more precision
- Larger samples generally produce narrower intervals
- If methods disagree substantially, trust Wilson for better coverage
Assumptions & Limitations
- Simple random sampling: Each unit has equal probability of selection. Complex survey designs may require different methods.
- Independence: Each observation is independent. Clustered or correlated data violates this assumption.
- Fixed sample size: The sample size n is determined before data collection, not based on results.
- Binomial model: Assumes constant success probability across all trials.
- Normal approximation: Both methods use normal approximations that may be imperfect for very small samples.
Frequently Asked Questions
Related Tools
Confidence Interval (Means)
Build confidence intervals for population means from sample data
Z-Score & P-Value
Convert between z-scores and p-values for hypothesis testing
Normal Distribution
Calculate probabilities and quantiles under the normal curve
Binomial Distribution
Compute binomial probabilities for success/failure outcomes
Descriptive Statistics
Calculate mean, median, standard deviation, and more
Poisson Distribution
Calculate probabilities for count-based rare events
Error Propagation
Propagate measurement uncertainties through mathematical formulas
Sample Size for Proportions
Calculate required sample size for proportion-based statistical tests