P-hat (Sample Proportion) Calculator
Enter the number of successes and sample size to calculate p̂, standard error, and a confidence interval for the true population proportion.
Number of observed successes.
Total observations in the sample.
Enter as decimal (e.g., 0.95). Common levels snap to nearest z-score.
Sample Proportion
37.50%
p̂ = x / n
Standard Error
0.0442
√[p̂(1−p̂)/n]
Margin of Error
8.66%
z · SE
Confidence Interval
28.84% to 46.16%
95.0% CI
How to Use This Calculator
- Enter the count of successes observed in your sample.
- Enter the total number of trials or observations.
- Select a confidence level to compute a proportion confidence interval.
- Review p̂, standard error, and the interval to interpret the population proportion estimate.
Formula
p̂ = x / n
SE = √[p̂(1 − p̂) / n]
CI = p̂ ± z · SE
The z-score corresponds to the desired confidence level (e.g., 1.96 for 95%). The normal approximation is accurate when both np̂ and n(1−p̂) are at least 10.
Full Description
The sample proportion p̂ summarizes binary outcomes — successes divided by total trials. It is a point estimate of the true population proportion. By combining p̂ with standard error and a z-score, you obtain a confidence interval that communicates uncertainty in your estimate.
This calculator is useful for survey results, A/B testing conversions, defect rates, or any Bernoulli process. For small samples or extreme proportions, consider exact (Clopper–Pearson) intervals instead of the normal approximation.
Frequently Asked Questions
What is p̂?
p̂ (p-hat) is the sample proportion: the fraction of successes in your sample. It estimates the true population proportion p.
When is the normal approximation valid?
When np̂ ≥ 10 and n(1 − p̂) ≥ 10. If these conditions fail, consider using an exact confidence interval.
Can I change the z-score directly?
Adjust the confidence level. The calculator automatically chooses the closest standard z-score (90%, 95%, 99%, etc.).
How does this relate to hypothesis testing?
The same standard error powers z-tests for proportions. Compare p̂ against a hypothesized p₀ to compute z = (p̂ − p₀) / SE.