In this article, you will study means and the central limit theorem.

The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of the statistics.

There are two alternative forms of the theorem, and both alternatives are concerned with drawing finite samples size n from a population with a known mean, μ, and a known standard deviation, σ.

The first alternative says that if we collect samples of size n with a “large enough n,” calculate each sample’s mean, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape.

The second alternative says that if we again collect samples of size n that are “large enough,” calculate the sum of each sample and create a histogram, then the resulting histogram will again tend to have a normal bell-shape.

In either case, it does not matter what the distribution of the original population is, or whether you even need to know it.

The important fact is that the distribution of sample means and the sums tend to follow the normal distribution.

The size of the sample, n, that is required in order to be “large enough” depends on the original population from which the samples are drawn (the sample size should be at least 30 or the data should come from a normal distribution).

If the original population is far from normal, then more observations are needed for the sample means or sums to be normal. Sampling is done with replacement.

### The Central Limit Theorem for Sums

Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:

a. μX = the mean of Χ

b. σΧ = the standard deviation of X

If you draw random samples of size n, then as n increases, the random variable ΣX consisting of sums tends to be normally distributed and ΣΧ ~ N((n)(μΧ), ( n )(σΧ)).

The central limit theorem for sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution), which approaches a normal distribution as the sample size increases.

The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.