P Hat Calculator
Calculate sample proportion p-hat from the number of successes and sample size for polls, surveys, and inferential statistics.
What Is p-hat?
The sample proportion $\hat{p}$ (p-hat) is the fraction of successes in a sample. It estimates the true population proportion from survey, poll, or experiment data. Related tools: Confidence Interval for Proportion Calculator and Sample Size Calculator.
p-hat Formula
Divide the number of successes $x$ by the sample size $n$.
$$\hat{p} = \frac{x}{n}$$
The standard error of the sample proportion is $\sqrt{\hat{p}(1-\hat{p})/n}$. This measures typical sampling variability when estimating a population proportion.
Example
In a poll of 700 students, 450 support a menu change. Then $\hat{p} = 450/700 \approx 0.643$, or about 64.3%.
Frequently Asked Questions
Can p-hat equal exactly 0 or 1?
Yes, if all sample observations are failures or all are successes. Both are valid sample proportions.
How is p-hat different from population proportion p?
p-hat is computed from a sample and varies from sample to sample. The population proportion p is the fixed true fraction in the entire population.
What if successes exceed sample size?
That is invalid input. Successes must be between 0 and the sample size inclusive.
Why report standard error?
Standard error shows how much p-hat might vary across repeated samples of the same size, which is essential for confidence intervals and hypothesis tests.
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