Sample Proportion

Consider an infinite (or very large) population, where each observation has a probability p of being a success, and a probability (1-p) of being a failure. Let the set of independent and identically distributed random variables X₁, X₂, ..., X_n represent the observations from a sample of size n, where

X_i = 1 if the ith observation is a success
= 0 if the ith observation is a failure

Then the random variables X₁, X₂, ..., X_n are Bernoulli variables with parameter p, and the sum (X₁ + X₂ + ... + X_n) is a binomially distributed random variable with parameters p and n. Letting X = (X₁ + X₂ + ... + X_n), and utilizing the normal approximation to the binomial distribution,

Now 

Therefore

As n ® Â¥ , the binomial random variable X approximates a normal random variable with m = np and s² = np(1-p), while the random variable (X/n) tends towards a normal random variable with m = p and s² = p(1-p) / n. The random variable (X/n) is also called the sample proportion, defined as the ratio of the number of successes in a sample to the sample size.