Sampling Distribution of Sample Proportion
Enter population proportion, sample size, and interval bounds to compute z-scores and probabilities for the sample proportion.
Mean of p̂: 0.3500
Standard error: 0.0337
Z(lower): -1.4825
Z(upper): 1.4825
P(lower ≤ p̂ ≤ upper): 0.861792
How to Use This Calculator
- Enter the true population proportion and sample size.
- Specify the interval for the sample proportion you want to evaluate.
- Confirm that the success-failure rule (np ≥ 10, n(1 − p) ≥ 10) holds for normal approximation validity.
- Use the results to interpret sampling variability, margin of error, or probability of observing certain sample proportions.
Formula
Mean of p̂ = p
Standard error = √(p(1 − p) / n)
Z = (p̂ − p) / √(p(1 − p) / n)
P(L ≤ p̂ ≤ U) ≈ Φ((U − p)/SE) − Φ((L − p)/SE)
Φ denotes the standard normal cumulative distribution function. The approximation improves as n grows large and p is not too close to 0 or 1.
Full Description
Sampling distributions provide a probabilistic view of estimator behavior. For proportions, the sampling distribution of p̂ is approximately normal when samples are sufficiently large and independent. Understanding this distribution is key to constructing confidence intervals, performing hypothesis tests, and planning surveys.
This calculator applies the central limit theorem to estimate probabilities that the observed sample proportion falls within specified bounds. It highlights the role of sample size and population proportion in determining variability.
Frequently Asked Questions
When does the normal approximation fail?
When np or n(1 − p) is small (< 10), the distribution of p̂ is skewed; consider exact binomial methods instead.
What if p is unknown?
Use the sample proportion as an estimate, or apply conservative bounds (e.g., p = 0.5) for planning.
How is this different from a binomial probability?
Binomial probabilities focus on counts of successes, whereas this calculator works on the proportion scale with normal approximation.