Find the sample proportion (p̂) — the point estimate of a population proportion based on observed sample data.
How it works
The formula is p̂ = successes ÷ sample size. Observing 45 successes in a sample of 200 gives p̂ = 0.225, or 22.5%.
What this does not include
This reports the point estimate only. It doesn’t calculate a confidence interval around that estimate, which would need the sample size and a chosen confidence level to compute a margin of error.
How to use this calculator
- Enter the number of successes observed.
- Enter the total sample size.
A worked example
45 successes out of a sample of 200: p̂ = 45 ÷ 200 = 0.225.
120 successes out of 500: p̂ = 0.24.
What the variables mean
| Variable | Meaning |
|---|---|
| Successes | Count of the outcome being measured |
| Sample size | Total number of observations |
Edge cases worth knowing
p̂ (p-hat) is a sample proportion, an estimate of the true population proportion — not the actual population value itself, which is why it’s marked with a hat rather than written as a plain p.
Successes can’t exceed the sample size — more successes than observations is logically impossible, so the calculator declines to show a result for that combination.
Why is it called “p-hat” instead of just “p”?
The hat symbol (p̂) signals that this is an estimate calculated from sample data, distinguishing it from p, which typically denotes the true, usually unknown, population proportion.
Does p-hat get more accurate with a larger sample?
Generally yes — a larger sample size tends to produce a p-hat closer to the true population proportion, and it also narrows the margin of error in any confidence interval built around it.
Can p-hat ever equal exactly 0 or 1?
Yes — if no successes occur in the sample, p-hat is 0; if every observation is a success, p-hat is 1, both valid (if sometimes statistically awkward) results.
