The Long Run Distribution of a Proportion

For a large sample size, the probability distribution of the possible values of an average is approximately a normal distribution, regardless of the probability distribution of the original individual measurements. In this video, we investigate whether something similar is true when we’re interested in a proportion.

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Notes on the video: The Long Run Distribution of a Proportion

A point to consider for this video:

The Central Limit Theorem is usually stated in terms of averages. Why does it also hold for proportions? A proportion is the count of the number of successes in **n trials divided by n. If the outcome of each trial is thought of as an observation of a Bernoulli random variable which is 1 if the outcome of the trial is a success and 0 otherwise, then the sum of the outcomes of the n Bernoulli random variables is the number of successes, and the average of the n Bernoulli random variables is the proportion of successes.

A natural question to ask is: How many trials are needed so that the Normal distribution is a good approximation for the probability distribution of an estimator of a proportion? For averages, it is difficult to give a general answer because of the wide variety of probability distributions that the individual observations could possibly have. However, it is possible to give a guideline for proportions. Counts of the number of successes in a fixed number of trials are known to have Binomial distributions. Binomial distributions are symmetric when p=0.5 and are more skewed the closer p is to 0 or 1; so smaller sample sizes are needed when p is close to 0.5 than when p is close to 0 or 1. Here is a guideline for a sufficient sample size, n, for the Central Limit Theorem to apply when estimating a proportion: The distribution of an estimator of a proportion will be well approximated by a Normal distribution if np ≥ 10 and n(1-p) ≥ 10.