Central Limit Theorem Machine for Proportions

When is the sampling distribution of p̂ approximately normal? Draw samples, or press Play; each sample proportion p̂ is added to the sampling distribution at the bottom.

Population individuals who have a certain characteristic, or do not

p =

One sample

Sampling distribution of p̂

Your sample proportions Normal curve with mean p and standard deviation √p(1 − p)/n Latest p̂ Population proportion p

The Central Limit Theorem for Proportions: Let p̂ be the sample proportion for a sample size of n and population proportion p. If np ≥ 10 and n(1 − p) ≥ 10 then the distribution of p̂ is approximately normal, with mean and standard deviation μp̂ = p and σp̂ = √p(1 − p)/n.

A common rule of thumb is that the distribution may be approximated with a normal curve whenever np and n(1 − p) are both at least 10.

When p is close to 0 or close to 1, a larger sample size is needed before the distribution of p̂ is close to normal. When p = 0.5, the sampling distribution of p̂ is somewhat close to normal even for a small sample size like n = 5.