Sampling Distribution of Proportions

When we want to know what proportion of a large population has some characteristic — say, the fraction of students who walk to school — it is rarely practical to survey everyone. Instead, statisticians take a random sample of size n and compute the sample proportion p̂ (read 'p-hat'), the fraction of sampled individuals who have the characteristic. Because every new random sample would give a slightly different value of p̂, the sample proportion is itself a random variable with its own probability distribution, called the sampling distribution of the sample proportion. Understanding this distribution is essential for statistical inference. It tells us how far a single observed p̂ is likely to stray from the true population proportion p, and it allows us to build confidence intervals and perform hypothesis tests. The key results — that the distribution is approximately normal, centred on p, and has a spread that shrinks as n grows — follow directly from the Central Limit Theorem applied to proportions. This concept sits at the heart of opinion polling, quality control in manufacturing, medical clinical trials, and virtually every field that draws conclusions about a population from a sample. Mastering the sampling distribution of proportions gives you the mathematical language to say precisely how confident you should be in any percentage-based claim.

Key takeaways
  • Define the Population Proportion
  • Define the Sample Proportion p̂
  • Mean of the Sampling Distribution
  • Standard Error of p̂
  • Normal Approximation (Central Limit Theorem)
  • Finite Population Correction

Define the Population Proportion

Let p be the true (but usually unknown) proportion of individuals in the population that have the characteristic of interest, where 0 < p < 1. The complementary proportion is q = 1 − p. For example, if 40% of all teenagers own a bicycle, then p = 0.40 and q = 0.60.

Define the Sample Proportion p̂

Draw a simple random sample of size n. Count the number of successes X (individuals with the characteristic). The sample proportion is p̂ = X/n. Because X can be modelled as a binomial random variable with parameters n and p, we can derive all properties of p̂ directly from the binomial distribution.

Mean of the Sampling Distribution

The expected value of p̂ equals the population proportion: E(p̂) = p. This means p̂ is an unbiased estimator of p — on average, it hits the target. The proof is immediate: E(p̂) = E(X/n) = (1/n)·E(X) = (1/n)·np = p.

Standard Error of p̂

The standard deviation of p̂, called the standard error, measures the typical spread of sample proportions around p. Starting from Var(X) = npq for the binomial, we get Var(p̂) = Var(X/n) = (1/n²)·npq = pq/n. Taking the square root gives the standard error: SE(p̂) = √(pq/n) = √(p(1−p)/n). This decreases as n increases, confirming that larger samples give more precise estimates.

Normal Approximation (Central Limit Theorem)

When the sample is large enough, the sampling distribution of p̂ is approximately normal: p̂ ~ N(p, p(1−p)/n). The rule of thumb requires both np ≥ 10 and n(1−p) ≥ 10 (sometimes written as np ≥ 5 and nq ≥ 5 in some textbooks). Once normality is established, we can standardise: Z = (p̂ − p) / √(p(1−p)/n) follows an approximate standard normal distribution, enabling probability calculations and inference.

Finite Population Correction

If the population is finite with size N and the sample is more than 5% of the population (n/N > 0.05), a finite population correction factor applies: SE(p̂) = √(p(1−p)/n) · √((N−n)/(N−1)). For most real-world surveys where n is small relative to N, this correction is negligible and is often omitted.

The Key Equation

SE(p^)=p(1−p)nSE(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}
  • SE(\hat{p}) — Standard Error of p̂ (dimensionless): The standard deviation of the sampling distribution of p̂; measures how much p̂ varies from sample to sample.
  • p — Population proportion (dimensionless): The true fraction of individuals in the population with the characteristic of interest; satisfies 0 < p < 1.
  • 1 - p — Complementary proportion (q) (dimensionless): The fraction of the population without the characteristic; equals q.
  • n — Sample size (count (individuals)): The number of individuals randomly selected from the population; larger n produces smaller standard error.

Worked Example

Problem. A school administrator claims that 65% of students at a large high school (N = 2000) bring a packed lunch. A student council member randomly samples n = 100 students and finds that 58 of them bring a packed lunch. (a) What are the mean and standard error of the sampling distribution of p̂? (b) Assuming the claim is true, what is the probability of observing p̂ ≤ 0.58? (c) Do the conditions for normal approximation hold?

E(p^)=0.65,SE(p^)≈0.0477,P(p^≤0.58)≈0.071E(\hat{p})=0.65,\quad SE(\hat{p})\approx 0.0477,\quad P(\hat{p}\leq 0.58)\approx 0.071

Answer. Mean = 0.65; SE ≈ 0.0477; P(p̂ ≤ 0.58) ≈ 0.071 (about 7.1%).

Figure
Diagram
Sampling Distribution of p̂ (p=0.65, n=100)Sample proportion p̂Densityμ = p = 0.65p̂=0.58P(p̂ ≤ 0.58) ≈ 7.1%0.700.580.65SE = √(pq/n) ≈ 0.0477±1 SE
Origins · 1 minAbraham de Moivre and how this was discovered
AD
Abraham de Moivre
1667–1754 · French-British

De Moivre was the first mathematician to derive the normal distribution as an approximation to the binomial distribution in 1733, a result published in his book 'The Doctrine of Chances'. This discovery laid the groundwork for the Central Limit Theorem, which justifies the normal approximation used in sampling distributions of proportions. His formula showed that as n grows large, binomial probabilities cluster around a bell-shaped curve — exactly the property that makes the sampling distribution of p̂ tractable. De Moivre allegedly predicted the exact date of his own death by noting that he was sleeping 15 more minutes each night; extrapolating arithmetically, he calculated he would die on 27 November 1754 — and he did.

  • Binomial Distribution — The sample count X in a proportion problem follows a binomial distribution; the properties of p̂ are derived directly from binomial mean and variance.
  • Central Limit Theorem — The CLT justifies why the sampling distribution of p̂ is approximately normal for large n, enabling probability calculations using the standard normal table.
  • Sampling Distribution of the Mean — The sampling distribution of p̂ is an analogue of the sampling distribution of x̄; both use the same CLT logic but for proportions versus means.
  • Confidence Intervals for Proportions — Confidence intervals for a population proportion are constructed directly using the standard error formula SE(p̂), making this concept a prerequisite.
  • Hypothesis Testing for Proportions — One-proportion z-tests use the standardised form of p̂ to decide whether observed data are consistent with a claimed population proportion.