# 3. The Central Limit Theorem

## e. explain the central limit theorem and its importance;

What is the central limit theorem?

The central limit theorem states that, given a distribution with a mean μ and variance σ2, the sampling distribution of the mean x-bar approaches a normal distribution with a mean (μ) and a variance σ2/N as N, the sample size, increases.

If the shape of a population distribution is non-normal, what is the shape of the sampling distribution of the mean x-bar?

No matter the population distribution, the sampling distribution will approach a normal distribution as `N >= 30`

.

- In this case, `N`

is the sample size, and not the number of samples.

In a sampling distribution, what number of samples is assumed?

A sampling distribution assumes an infinite number of samples.

What two facts should be noted as you increase the size of the sample `N`

(i.e., the number of things you're sampling from the original population)?

1. The distribution becomes more and more normal

2. The spread of the distribution decreases

Because of the central limit theorem, what two things can you assume with a large sample size?

1. The sample mean can infer the population mean

2. You can construct confidence intervals for the population mean based on the normal distribution

##### Source:

- CFA