What Is Probability of a Sample Proportion?
A sample proportion p̂ is the share of a sample that has some trait. When the sample is large enough, p̂ is approximately normal with mean p and standard error √(p(1 − p) ÷ n), so you can find probabilities with a z-score.
Enter the true proportion, the sample size and the sample proportion or range. The calculator returns the normal-approximation probability, the exact binomial probability for comparison, the standard error and a shaded curve. You can turn on a continuity correction to bring the approximation closer.
Check that np and n(1 − p) are both at least 10. If not, the approximation is poor and the exact binomial probability is the better answer.
How to Use the Probability of a Sample Proportion Calculator
- Choose below, above or between.
- Enter the true proportion p, the sample size n and the sample proportion value or values.
- Compare the normal approximation with the exact binomial probability.
Probability of a Sample Proportion Formula
Standard error
z for a proportion
Probability of a Sample Proportion Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| p | True population proportion | |
| p̂ | Sample proportion | |
| n | Sample size | |
| SE | Standard error, √(p(1 − p)/n) | |
| z | Standard score of the sample proportion | (p̂ − p)/SE |
Probability of a Sample Proportion Worked Example
With the example values loaded in the calculator above, the steps are:
- The sample proportion has mean p = 0.4 and standard error √(0.4 × 0.6 ÷ 100) = 0.0489898.
- z = (0.5 − 0.4) ÷ 0.0489898 = 2.0412.
- P(p̂ > 0.5) ≈ 0.0206134 by the normal approximation; the exact binomial value is 0.0167617.
Result: P(p̂ > 0.5) = 0.0206134; Exact binomial probability = 0.0167617.
Worked example
If the true proportion is 0.40 and n = 100, the standard error is √(0.4 × 0.6 ÷ 100) = 0.04899. For P(p̂ > 0.50), z = (0.50 − 0.40) ÷ 0.04899 = 2.041.
The normal approximation gives P = 0.0206. The exact binomial probability of more than 50 successes is 0.0168, so the plain approximation runs a little high. With a continuity correction it gives 0.0160, much closer.
When the approximation works
Check that np and n(1 − p) are both at least 10. If not, the sampling distribution is noticeably skewed and the exact binomial value is the safer choice.
Where Probability of a Sample Proportion Is Used
Polling and surveys
Find how likely a poll is to show a proportion above or below a value.
Statistics coursework
Solve sampling distribution of p̂ problems.
Quality control
Estimate the chance a sample shows too many defects.
Common Probability of a Sample Proportion Mistakes to Avoid
- Use the true population proportion p, not an estimate from the sample.
- A continuity correction improves the approximation for modest n.
Probability of a Sample Proportion Terms and Definitions
- Sample proportion
- The fraction of a sample that has a given trait.
Probability of a Sample Proportion FAQs
How do you find the probability of a sample proportion?
Find the standard error √(p(1 − p)/n), compute the z-score of the sample proportion and read the normal probability.
What are the conditions for using the normal approximation?
np and n(1 − p) should both be at least 10, and the sample should be random and less than 10% of the population.
What is the difference between p and p̂?
p is the true population proportion. p̂ is the proportion in your sample.
Why does the calculator also show an exact value?
The exact binomial probability does not rely on an approximation, so you can see how good the normal answer is.
What does a continuity correction do?
It adjusts for the fact that a discrete count is being approximated by a continuous curve, usually by moving 0.5 successes.
How does sample size change the answer?
A larger n shrinks the standard error, so p̂ stays closer to p and extreme sample proportions become less likely.