# Level 1 CFA® Exam:

Central Limit Theorem

In this lesson, we're going to focus on one of the most useful statistical theorems, which is the central limit theorem.

The central limit theorem states that the distribution of the sample mean approaches a normal distribution irrespective of the distribution of the population from which the sample was drawn.

But let's take it step by step:

Imagine that we have a population. This population is characterized by finite variance and described by any distribution. So, for example, the population distribution can be normal, binomial, uniform, or any other.

According to the central limit theorem, if we draw samples of the same size from the population, the distribution of a sample mean calculated from these samples will be approximately normal if the size of the samples is large enough.

How big a sample must be to allow making an assumption that the sample mean is approximately normally distributed?

(...)

The standard deviation of a sample statistic, which is equal to the square root of the variance of the sample mean, is known as the standard error of the sample mean. For sample mean (\(\bar{X}\)), the standard error of the sample mean may be expressed as follows:

\(\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}\text{ OR }s_{\bar{X}}=\frac{s}{\sqrt{n}}\)

- \(\sigma\) - population standard deviation
- \(s\) - sample standard deviation
- \(n\) - sample size

The higher the sample size, the lower the standard error.

Standard error equals population standard deviation divided by the square root of the size of the sample. In case we don't know the population standard deviation, we may use the sample standard deviation in the numerator of the formula.

Very often we use the sample standard deviation while calculating standard error because we usually don't know the population standard deviation. In order to calculate \(s\), we can use the formula below:

\(s=\sqrt{\sum_{i=1}^n\frac{(X_i-\bar{X})^2}{n-1}}\)

- \(s\) - sample standard deviation
- \(n\) - number of observations in the sample
- \(X_i\) - observation \(i\) value
- \(\bar{X}\) - sample mean

Standard deviation is the square root of variance.

Note: when the sample size increases, the standard error of the sample mean decreases, and when the sample size decreases, the standard error of the sample mean increases. In other words, the greater the sample size, the more certain we are about the estimation of the population mean based on the central limit theorem.

Let's assume that the population mean and variance are equal to 12,540 and 4,300, respectively. The sample size is 100. What is the value of the standard error of the sample mean?

Since we've got the data on the population, let's use the following formula:

\(\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\sqrt{\frac{\sigma^2}{n}}=\\=\sqrt{\frac{4300}{100}}=\sqrt{43}=6.5574\)

- If we draw samples of the same size from the population, the distribution of a sample mean calculated from these samples will be approximately normal if the size of the samples is large enough (at least \(n=30\)).
- The central limit theorem states that the distribution of the sample mean approaches a normal distribution irrespective of the distribution of the population from which the sample was drawn.
- The standard deviation of a sample mean is known as the standard error of the sample mean.