# Understanding the Central Limit Theorem

In statistics, suppose x is a random variable with normal distribution with mean mu and standard deviation sigma. Now let x-bar be the sample mean from a sample of size n from the x distribution. The following assumptions can now be made:

1. The x-bar distribution is normally distributed, just like the x distribution.
2. The mean of the x-bar distribution is mu.
3. The standard deviation of the x-bar distribution is sigma divided by the square root of n.

This theorem states that the x-bar will be normally distributed if the x distribution is normal, no matter what the sample size is. The mean will always be the same as the mean of the x distribution and the standard deviation is always sigma divided by the square root of n.

What happens if we don't know the shape of the x distribution? The Central Limit Theorem for any probability distribution states that if x has any distribution with mean mu and standard deviation sigma, then x-bar with sample size n will have a distribution that approximates the normal distribution as n gets larger. That means that as a sample size gets larger, the distribution of x-bar will always approach the normal distribution. But how large must the sample size get? Generally speaking, a sample size of 30 or greater will give a reasonable approximation to the normal distribution.

Here's an example using the Central Limit Theorem.

Suppose x has a normal distribution with mean mu = 15 and standard deviation sigma = 4. If you draw random samples of size 5 from the x distribution and x-bar is the sample mean, what can be said about the x-bar distribution?

Even though the sample size is small, much less than 30, you could say the x-bar distribution is approximately normal, since the x distribution is normal. The means mean of x-bar is 15 and standard deviation is 15 divided by square root of 4 which is 7.5.

For another example, suppose the x distribution has mean mu = 100 and standard deviation sigma = 20. But there is no information about the shape of the x distribution. If samples are drawn of size 35 from the x distribution, what can be said about the x-bar distribution? Since the sample size is greater than or equal to 30, the x-bar distribution will be approximately normally distributed with mean = 100 and standard deviation equal to 20 divided by the square root of 35, which is 3.4.

If the sample size were 10 but we did not know the shape of the x distribution, you could not say that x-bar distribution is approximately normal because the sample size is too low.

This guide should help students who are having difficulty understanding the Central Limit Theorem.