95% Confidence Interval Calculator (Proportion)
Enter the number of successes and total sample size to calculate a 95% confidence interval for a population proportion.
A confidence interval for a proportion gives you a plausible range for the true population percentage based on your sample data. This is useful in surveys, A/B testing, quality control, and clinical studies where outcomes are binary (yes/no, success/failure, clicked/did not click).
What this calculator does
This tool returns a 95% confidence interval from your observed data:
- Sample proportion p̂ = x / n
- Standard error based on p̂
- Margin of error using z = 1.96
- Wald interval (classic textbook method)
- Wilson interval (often more reliable, especially for smaller samples)
Formula behind the 95% proportion interval
Wald (normal approximation):
p̂ ± 1.96 × √(p̂(1 - p̂) / n)
Wilson (recommended in many practical settings):
((p̂ + z2/(2n)) ± z √(p̂(1-p̂)/n + z2/(4n2))) / (1 + z2/n), where z = 1.96
Symbol reference
| Symbol | Meaning |
|---|---|
| x | Number of successes observed |
| n | Total sample size |
| p̂ | Sample proportion, x / n |
| z | Critical value for 95% confidence (1.96) |
How to interpret your result
If your calculator output is, for example, [61.8%, 78.2%], you would say:
"Based on this sample, we are 95% confident the true population proportion lies between 61.8% and 78.2%."
This does not mean there is a 95% probability the fixed true proportion is in this one computed interval. It means this interval method captures the true value in about 95% of repeated samples.
Step-by-step example
Suppose 84 out of 120 respondents say "yes" to a survey question.
- Compute p̂ = 84/120 = 0.70
- Compute standard error = √(0.70×0.30/120) = 0.0418
- Compute margin of error = 1.96×0.0418 = 0.0820
- Wald 95% CI = 0.70 ± 0.0820 = [0.618, 0.782]
That corresponds to 61.8% to 78.2%.
When to be cautious
If either x or (n - x) is small (common rule: less than 10), the normal approximation can be unstable. In those cases, Wilson intervals are usually preferred.
- Very small samples can produce wide intervals.
- Proportions near 0% or 100% can make Wald intervals less accurate.
- Confidence intervals do not fix sampling bias from poor data collection.
Practical use cases
A/B testing
Estimate uncertainty around conversion rates before declaring a winning variation.
Polling and surveys
Report public opinion estimates with an uncertainty range instead of only a single percentage.
Quality and manufacturing
Measure defect rates and communicate likely bounds for true process performance.
Quick FAQ
Why 95% confidence?
95% is a widely used convention balancing precision and uncertainty. You can use other levels (90%, 99%), but this page is specifically tuned to 95%.
Can I use percentages directly?
Enter counts, not percentages. Use successes (x) and total sample size (n) for correct interval estimation.
Should I report Wald or Wilson?
Wilson is generally more robust. Wald is still useful for learning and for large, well-behaved samples.