Sampling Distribution of Proportion

Model the behavior of sample proportions under the Central Limit Theorem. Compute z‑scores, probabilities, and interval estimates for inference.

Central Limit Theorem: For large n, p̂ ~ Normal( mean = p, SE = √[p(1‑p)/n] ).
True proportion (0 < p < 1)
Number of observations (≥1, integer)
Observed proportion (0–1)
For interval
P(L < p̂ < U)
p=0.5, n=100, p̂=0.55 p=0.3, n=200, p̂=0.35 p=0.8, n=50, p̂=0.7 Election poll Rare event Interval [0.4,0.6]
Computing normal approximation...

Understanding Sampling Distribution of p̂

The sample proportion p̂ is a random variable. Its sampling distribution describes how p̂ varies from sample to sample. Under the Central Limit Theorem, for large n, p̂ is approximately normal with mean p and standard error √[p(1‑p)/n].

Standard Error:

SE = √[ p(1‑p) / n ]

z‑score for p̂: z = (p̂ − p) / SE

CDF (normal approximation): P(p̂ ≤ x) ≈ Φ( (x−p)/SE )

1
Population proportion (p) — the true parameter (unknown, but hypothesized).
2
Sample size (n) — larger n ⇒ smaller SE ⇒ more precise estimates.
3
Sample proportion (p̂) — point estimate from data. We compute its probability under the sampling distribution.
4
Interval bounds — probability that p̂ falls between two values.

When is the Normal Approximation Valid?

  • Success‑failure condition: np ≥ 10 and n(1‑p) ≥ 10.
  • Random sampling; independent observations.
  • If conditions are not met, consider binomial exact methods.
Real‑world example: In a poll with p = 0.6 (60% support) and n = 250, the SE ≈ 0.031. The probability of obtaining p̂ ≤ 0.64 is about 0.90. This calculator quantifies such sampling variability.

Frequently Asked Questions

The binomial distribution is exact but can be cumbersome for large n. The normal approximation (CLT) is simple, intuitive, and works well when np ≥ 10 and n(1‑p) ≥ 10. It forms the basis for confidence intervals and hypothesis tests.

The normal approximation is less accurate near boundaries. However, the calculator still returns a z‑score and probability; interpret with caution. For extreme proportions, consider exact binomial methods.

Larger n reduces the standard error, making the sampling distribution narrower and more concentrated around p. This means p̂ is more likely to be close to p.

Yes! Use the tail probability P(p̂ > p̂_obs) for upper‑tail, or P(p̂ < p̂_obs) for lower‑tail. For two‑tailed, double the one‑sided tail probability (if symmetric about p).