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
Instructions
- 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.
About This Tool
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.
Common 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.