Central Limit Theorem

Central Limit Theorem

The central limit theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend towards the normal distribution as the sample size gets larger. This may be restated as follows:

Given a set of independent and identically distributed random variables
X₁, X₂, ..., X_n, where E(X_i) = m and
Var(X_i) = s², i = 1, 2, ..., n, and the mean of the random variables given by

the distribution of approaches the normal distribution, i.e.

The central limit theorem is one of the most important theorems in the field of probability as well as statistical inference, as it justifies the use of the normal curve in a wide range of statistical applications, both theoretical and practical. The approximate normality of the sampling distribution of the mean is usually achieved when n ³ 30 (except in certain cases where the probability distribution of the population has a very unusual shape). A Standard Normal Distribution Table can then be used in these situations to solve for probabilities involving the sample mean.

The normal approximation to the binomial distribution is a special case of the central limit theorem, where the independent random variables are Bernoulli variables with parameter p.