Normal Approximation for Proportions
When we want to draw conclusions about a population proportion — say, the fraction of voters who support a candidate, or the percentage of a product that is defective — we collect sample data and use it to build confidence intervals or perform hypothesis tests. The mathematics behind these procedures relies on the normal distribution, but that reliance is only valid under certain conditions. Understanding why and when the normal approximation works is the foundation of inference for proportions. The key insight comes from the Central Limit Theorem: as sample size grows, the distribution of the sample proportion p̂ becomes approximately normal regardless of the true shape of the underlying population. Because each observation is essentially a Bernoulli trial (success or failure), the sample proportion is a scaled sum of independent random variables, and the CLT guarantees approximate normality. The quality of this approximation depends critically on the sample size relative to the true proportion p. In practice, statisticians use two main checks — the Large Counts Condition and the Independence Condition — to verify that normal methods are appropriate. Skipping these checks can lead to severely misleading intervals and p-values, especially when proportions are close to 0 or 1. This article explains what those conditions are, where they come from mathematically, and how to apply them correctly.
- The Sampling Distribution of p̂
- The Large Counts Condition
- The Independence Condition
- Standardizing to Use the Normal Table
- Checking Conditions in Practice
The Sampling Distribution of p̂
If we draw a random sample of size n from a large population where the true proportion of successes is p, then each observation is a Bernoulli(p) random variable. The sample proportion p̂ = X/n, where X ~ Binomial(n, p). The exact mean is E[p̂] = p and the exact variance is Var(p̂) = p(1−p)/n. By the Central Limit Theorem, for large n, p̂ is approximately normally distributed: p̂ ~ N(p, p(1−p)/n).
The Large Counts Condition
The normal approximation is only trustworthy when n is large enough that the binomial distribution isn't too skewed. The standard rule is: both np ≥ 10 and n(1−p) ≥ 10 must hold. This ensures there are at least 10 expected successes and 10 expected failures in the sample. Intuitively, if p is close to 0, almost all outcomes are failures and the distribution is heavily right-skewed; the normal approximation fails badly. When p = 0.5 and n = 20, np = 10 exactly, and the binomial is already quite symmetric. In practice, we substitute p̂ for p when p is unknown.
The Independence Condition
The CLT requires that the individual observations are independent (or approximately so). For simple random samples from a finite population, observations are technically dependent, but the 10% Condition rescues us: if the sample size n is no more than 10% of the population size N (i.e., n ≤ 0.10N), then the dependence is negligible and independence is a safe approximation. For truly random samples with replacement, independence holds exactly.
Standardizing to Use the Normal Table
Once conditions are met, we standardize p̂ to obtain a z-score. The standard error of p̂ is SE = √(p(1−p)/n). For a confidence interval, we use SE = √(p̂(1−p̂)/n) because p is unknown. The z-score is z = (p̂ − p₀) / √(p₀(1−p₀)/n) for a hypothesis test where p₀ is the null value. This z-score follows (approximately) the standard normal distribution N(0,1), allowing us to use z-tables or critical values like z* = 1.96 for 95% confidence.
Checking Conditions in Practice
Always state the conditions explicitly before proceeding. (1) Random: Was the data collected via a random sample or randomized experiment? (2) Independence / 10% Condition: Is n ≤ 10% of the population? (3) Large Counts: Is np̂ ≥ 10 and n(1−p̂) ≥ 10? If any condition fails, normal-based procedures are not valid, and exact binomial methods or other techniques should be used instead.
The Key Equation
- \hat{p} — Sample proportion (dimensionless): The proportion of successes observed in the sample, equal to X/n where X is the count of successes.
- p — Population proportion (dimensionless): The true (unknown) fraction of individuals in the population who have the characteristic of interest.
- n — Sample size (count): The number of independent observations drawn from the population.
- \frac{p(1-p)}{n} — Variance of p-hat (dimensionless): Measures how much the sample proportion varies from sample to sample; decreases as n increases.
Worked Example
Problem. A school administrator claims that 30% of students at a large high school (2,400 students) ride the bus. A student council member surveys a random sample of 120 students and finds that 42 ride the bus. (a) Check all conditions for a one-sample z-test for a proportion. (b) Calculate the test statistic and determine whether the sample result is surprising at the α = 0.05 level.
Answer. z ≈ 1.20, p-value ≈ 0.23; fail to reject H₀ at α = 0.05.
Origins · 1 minAbraham de Moivre and how this was discovered
De Moivre first discovered that the binomial distribution could be approximated by a normal (bell-shaped) curve as sample size grows large, publishing this result in 1733 — decades before Gauss or Laplace formalized the normal distribution. His work in 'The Doctrine of Chances' laid the groundwork for the Central Limit Theorem and directly justifies the normal approximation for proportions used in modern statistics. De Moivre reportedly predicted the date of his own death by noticing he was sleeping 15 minutes longer each day; extrapolating arithmetically, he calculated he would die on November 27, 1754 — and he did.
Related Concepts
- One-Sample z-Test for a Proportion — The normal approximation is the direct justification for using a z-test when performing hypothesis tests about a single population proportion.
- Confidence Interval for a Proportion — Constructing a confidence interval for p uses the same normal approximation and SE formula; the conditions checked here must hold before building the interval.
- Central Limit Theorem — The theoretical foundation that guarantees p̂ is approximately normal for large samples, making all normal-based inference for proportions valid.
- Binomial Distribution — The exact distribution of the count of successes X = np̂; the normal approximation replaces the binomial when conditions are met.
- Comparing Two Proportions — Extends the same normal approximation logic to comparing proportions from two independent groups, requiring conditions to be checked for both samples.