Confidence Interval for a Proportion Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/9/2026
The confidence interval for a proportion is calculated with p̂ ± z × √(p̂(1−p̂) ÷ n). For an observed proportion of 40% in a sample of 200 people, at 95% confidence, the interval runs from 33.2% to 46.8%.
Explanation
The confidence interval for a proportion gives a plausible range for the true proportion in the overall population, from a single observed proportion in a sample. It's built by adding and subtracting a multiple of a standard error to the observed proportion (p̂), the same general logic our standard error of the mean calculator applies to a mean instead of a proportion: this calculator computes the proportion's own standard error directly to build the full interval around p̂, rather than stopping at the margin of error alone. The multiplier z depends on the chosen confidence level: 1.96 for a 95% interval (the most commonly used), 1.645 for 90%, 2.576 for 99% — the same critical scores already used in our z-score calculator, since both calculations share the same theoretical basis (the normal approximation to a proportion's sampling distribution). This calculator is distinct from our confidence interval calculator, which covers a MEAN rather than a proportion: the two formulas look similar in structure (an estimate ± z × standard error), but the standard error itself is calculated differently in each case. This method, called the Wald interval, remains an approximation that loses reliability for a small sample or a proportion close to 0 or 1, where other more robust methods (like the Wilson interval) are generally preferred by statisticians.
Example: a 40% proportion in a sample of 200 people
Inputs
Observed proportion: 0.4 (40%). Sample size: 200. Confidence level: 95% (z = 1.96).
Calculation
Standard error = √(0.4 × 0.6 ÷ 200) = √0.0012 ≈ 0.0346. Interval = 0.4 ± 1.96 × 0.0346 = 0.4 ± 0.0679, or 0.3321 to 0.4679.
Result
The 95% confidence interval for this proportion runs from 33.21% to 46.79%.
Frequently asked questions
Why is the interval wider with a higher confidence level?
Because a higher confidence level (99% rather than 90%, for example) requires being sure to a greater degree that the true proportion actually falls within the given interval, which mechanically requires a wider interval to offer this extra guarantee. It's an unavoidable trade-off between the precision of the estimate (a narrow interval) and the certainty that it's correct (a high confidence level).
Is this method reliable for a small sample or an extreme proportion?
No, the Wald interval used here rests on a normal approximation that becomes unreliable for a small sample (generally under about thirty observations) or a proportion very close to 0% or 100%, where the interval can even exceed these impossible bounds. In these cases, more robust methods like the Wilson interval or the exact Clopper-Pearson interval are generally preferred by statisticians.
What is the difference with the standard error alone?
The standard error only measures the expected spread of the estimate from one sample to another, while the confidence interval translates this standard error into a concrete, interpretable range around the observed proportion, at a chosen confidence level. See our standard error of the mean calculator for the equivalent idea applied to a mean rather than a proportion.