A proportion confidence interval gives a range of plausible values for a population proportion, based on a sample. Enter the number of successes x and the sample size n, choose a confidence level, and the calculator gives the sample proportion, the margin of error and the interval — with the formula and every step shown.
Proportion Confidence Interval Calculator
p̂ ± z* × SE — find the interval for a population proportion.
Conditions: random sample, np̂ ≥ 10 and n(1 − p̂) ≥ 10, and the sample is less than 10% of the population.
What Is a Confidence Interval for a Proportion?
A confidence interval for a proportion is a range built from a sample that probably contains the true population proportion. A 95% confidence interval does not mean there is a 95% chance the true value is inside this one interval — it means that if you repeated the sampling many times, about 95% of the intervals you built would contain the true value. Wider intervals come from smaller samples and higher confidence levels.
The Confidence Interval Formula
The interval is p̂ ± z* × √(p̂(1 − p̂) ÷ n), where p̂ = x/n is the sample proportion, z* is the critical value (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), and the square-root term is the standard error. The part after ± is the margin of error. Notice the standard error here uses p̂, not a claimed p0 — a confidence interval does not assume a null hypothesis.
How to Interpret a Proportion Confidence Interval
Always state the interval with its confidence level and in context: “We are 95% confident that the true proportion of voters who support the policy is between 0.502 and 0.698.” A narrower interval is more precise, and precision improves with a larger sample. Doubling the sample size shrinks the margin of error by a factor of about 1.41, not by half — precision grows with the square root of n.
Worked Example
In a survey, 60 of 100 people said yes. p̂ = 0.60, SE = √(0.60 × 0.40 ÷ 100) = 0.049, margin of error = 1.96 × 0.049 = 0.096. The 95% confidence interval is 0.60 ± 0.096, or (0.504, 0.696). We are 95% confident the true share of “yes” in the population is between 50.4% and 69.6%.
Proportion Confidence Interval FAQs
What is the margin of error?
The margin of error is z* × standard error. It is the distance from the sample proportion to each end of the confidence interval.
Why is the 99% interval wider than the 95% interval?
Higher confidence needs a larger critical value (2.576 instead of 1.96), which makes the margin of error larger and the interval wider.
What does 95% confident actually mean?
If you took many random samples and built a 95% interval from each one, about 95% of those intervals would contain the true population proportion.
Related tools: One-Proportion Z-Test · Two-Proportion Confidence Interval · Sample Size for Proportion · Statistics