Explore the sampling distribution of p̂. Calculate mean, standard error, probabilities, and quantiles. Visualize the bell curve with normal approximation conditions.
When you take a random sample of size n from a population where the true proportion is p, the sample proportion p̂ varies from sample to sample. The sampling distribution of p̂ describes this variability: it's approximately normal with mean p and standard error √(p(1−p)/n) when the sample is large enough.
This calculator lets you specify the population proportion p and sample size n, then shows the complete sampling distribution: its mean, standard error, key quantiles, and probabilities. Enter an observed p̂ to find the probability of getting a sample proportion that extreme.
Understanding the sampling distribution is fundamental to survey design, hypothesis testing, and confidence interval construction. It answers questions like: "If the true proportion is 50%, what's the probability of getting 53% or more in my sample of 1,000?". Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
The sampling distribution connects population parameters to sample statistics. This calculator makes the connection concrete by showing exact probabilities, visualizing the bell curve, and checking whether the normal approximation is valid for your data. The comparison tables help you understand how sample size affects precision. Keep these notes focused on your operational context.
Sampling Distribution of p̂: Mean: μₚ̂ = p Standard Error: SE = √(p(1−p)/n) With FPC: SE_adj = SE × √((N−n)/(N−1)) Normal Approximation Conditions: np ≥ 10 and n(1−p) ≥ 10 Z-score for observed p̂: z = (p̂ − p) / SE P(p̂ < x) = Φ(z)
Result: P(p̂ > 0.53) = 0.0287
With p = 0.5 and n = 1,000, the standard error is 0.0158. An observed p̂ = 0.53 has z = 1.90, giving P(p̂ > 0.53) = 0.029. There's about a 2.9% chance of seeing 53% or more purely by sampling variability if the true proportion is 50%.
Each observation in the sample is a Bernoulli trial with probability p. The count of successes follows a Binomial(n, p) distribution, and p̂ = count/n. As n grows, the binomial distribution approaches a normal distribution (the CLT). The rule of thumb np ≥ 10 ensures the approximation is adequate for practical purposes.
The formula SE = √(p(1−p)/n) has p(1−p) maximized at p = 0.5 (the product is 0.25). At extreme proportions like p = 0.01, p(1−p) = 0.0099, giving much smaller SE. This is why polls often use p = 0.5 for conservative sample size calculations.
Knowing the sampling distribution lets you: (1) calculate how large a sample you need for a given precision, (2) determine the probability of getting a misleading sample, (3) construct confidence intervals, and (4) perform hypothesis tests about population proportions. It's the bridge between your sample data and conclusions about the population.
It describes the probability distribution of sample proportions across all possible samples of size n from the population. By the Central Limit Theorem, it's approximately N(p, p(1−p)/n) for large samples.
Because p̂ is an unbiased estimator of p. On average, across all possible samples, the sample proportion equals the population proportion. Individual samples vary, but the expected value is p.
The standard error SE = √(p(1−p)/n) determines the spread. It depends on the population proportion (maximum spread at p = 0.5) and sample size (larger n = less spread). Population size matters only through FPC.
When np < 10 or n(1−p) < 10, the sampling distribution is noticeably skewed and the normal approximation is poor. For example, with p = 0.01 and n = 100, np = 1 (too small). You'd need n ≥ 1,000 for this proportion.
A 95% confidence interval for p is p̂ ± 1.96×SE. This comes directly from the sampling distribution: 95% of sample proportions fall within 1.96 standard errors of the mean.
When sampling without replacement from a finite population of size N, the variability of p̂ is slightly less than with replacement. The factor √((N−n)/(N−1)) adjusts the SE downward. It's important when n is a substantial fraction of N.