Confidence Interval Calculator (Single Proportion)
Estimate a confidence interval for a population proportion using sample data.
What this confidence interval for proportions calculator does
This tool computes a confidence interval for a single population proportion from binomial data: how many “successes” you observed in a sample and how large that sample was. Examples include conversion rates, defect rates, response rates, or survey agreement percentages.
Instead of reporting only one point estimate (like 40%), a confidence interval gives a range of plausible values for the true population proportion. That range is usually much more informative for decision-making.
Inputs explained
- Number of successes (x): how many observations met your condition of interest.
- Sample size (n): total observations in the sample.
- Confidence level: common choices are 90%, 95%, and 99%.
- Method: statistical formula used to construct the interval.
Methods available in this calculator
1) Wilson score interval (recommended)
The Wilson interval adjusts both center and spread to avoid poor behavior near boundaries. It is generally more reliable than the plain normal approximation.
Let p̂ = x/n and z be the critical value. Then:
denom = 1 + z²/n
center = (p̂ + z²/(2n)) / denom
adj = z * sqrt(p̂(1−p̂)/n + z²/(4n²)) / denom
CI = center ± adj
2) Wald interval (normal approximation)
This is the classic textbook interval: p̂ ± z * sqrt(p̂(1−p̂)/n). It is simple but can be inaccurate in edge cases.
3) Agresti-Coull interval
This method uses an adjusted sample size and adjusted successes before applying a normal-style interval. It often improves over the plain Wald interval while remaining easy to interpret.
How to interpret the result
Suppose your output is 95% CI = [0.34, 0.47]. A practical interpretation is: “Based on this sample and method, plausible values for the true proportion are about 34% to 47%.”
It does not mean there is a 95% probability the true value lies in this exact interval after seeing the data. The confidence level describes long-run performance of the method over repeated sampling.
Worked example
Imagine 48 out of 120 customers clicked a new call-to-action button. Here: x = 48, n = 120, and p̂ = 0.40. With a 95% confidence level, the calculator will return an interval around 0.40 that reflects sampling uncertainty.
- If the interval is narrow, your estimate is more precise.
- If the interval is wide, uncertainty is larger and you may need more data.
Common mistakes to avoid
- Using a tiny sample and expecting a narrow confidence interval.
- Treating the interval as proof of causation (it only quantifies uncertainty in a proportion estimate).
- Using the Wald interval when x is near 0 or n (extreme rates).
- Mixing up percentages and proportions (e.g., 0.42 vs 42%).
When this calculator is appropriate
Use it for one-sample binomial-style outcomes: yes/no, pass/fail, clicked/not clicked, defective/not defective. If you need to compare two proportions (A/B test), you should use a two-proportion confidence interval or hypothesis test.
Quick FAQ
What confidence level should I choose?
95% is the standard in many fields. 99% is more conservative (wider interval), while 90% is narrower but less conservative.
Why does my interval width shrink as n grows?
The standard error decreases as sample size increases, so your estimate becomes more precise.
Can the bounds exceed 0% or 100%?
Some formulas can mathematically produce out-of-range bounds; this calculator truncates displayed bounds to [0, 1] for interpretability. Wilson and Agresti-Coull usually behave better than Wald at the extremes.