Calculate sampling distribution sample proportion with statistical precision, sample distributions, and inference data.
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\mu_{\hat{p}} = p, \quad \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}, \quad Z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}pnσ_p̂p̂Z“In an election where 52% of all voters favor candidate A (p = 0.52), a random exit poll of n = 400 voters is conducted. What is the probability that sample support exceeds 55% (k = 0.55)?”
Both expected successes and failures exceed 10; normal approximation is justified.
Compute the standard deviation of the sampling distribution.
There is an 11.49% probability that the exit poll will measure 55% or higher.
Because np and n(1-p) both comfortably exceed 10, the normal approximation is robust; there is an 11.49% probability that the exit poll will yield 55% or higher.
| Input Scale | Standard Output Range | Variance Band | Recommendation |
|---|---|---|---|
| Low Baseline | 0.00 – 25.00 | ±1.5% | Optimal for conservative estimations |
| Median / Standard | 25.00 – 100.00 | ±0.8% | Benchmark standard for commercial workflows |
| Elevated / High | 100.00 – 500.00 | ±0.4% | Recommended for high-throughput modeling |
| Extreme Maximum | > 500.00 | ±0.1% | Verify input constraints for boundary saturation |
Provide your numeric inputs into the designated fields in Sampling Distribution Sample Proportion Calculator.
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The sample proportion p̂ is an unbiased estimator of the true population proportion p. Across all possible theoretical samples of size n, the expected value E(p̂) strictly equals p.
To ensure the sampling distribution of proportions is reasonably symmetric and normal, the sample must expect at least 10 successes (np ≥ 10) and at least 10 failures (n(1 - p) ≥ 10).
Increasing sample size n narrows the distribution by decreasing the standard error (σ_p̂ = √[p(1-p)/n]). The sample proportion becomes more tightly clustered around the true population proportion.
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