The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes how the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the original population distribution.
How to Use
Enter any two of the following values to calculate the third:
- Population Standard Deviation: The standard deviation of the entire population
- Sample Size: The number of observations in the sample
- Sample Standard Deviation: The standard error of the sample mean
Formula
The Central Limit Theorem relates these three values through the following formula:
$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$
Where:
- ฯ (sigma with x-bar subscript) is the standard error of the mean (sample standard deviation)
- ฯ (sigma) is the population standard deviation
- n is the sample size
This can be rearranged to solve for any of the three values:
$$\sigma = \sigma_{\bar{x}} \times \sqrt{n}$$
$$n = \left(\frac{\sigma}{\sigma_{\bar{x}}}\right)^{2}$$
Example
A factory produces widgets with a population standard deviation of 10 grams. If you take a sample of 25 widgets:
$$\sigma_{\bar{x}} = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2$$
The sample standard deviation (standard error) would be 2 grams.
Why the Central Limit Theorem Matters
The CLT is essential for statistical inference because it allows us to:
-
Make predictions about sample means - Even if we do not know the exact distribution of a population, we can predict the behavior of sample means.
-
Calculate confidence intervals - The CLT enables us to construct confidence intervals for population parameters.
-
Perform hypothesis testing - Many statistical tests rely on the assumption that sample means are normally distributed.
-
Determine sample sizes - We can calculate how large a sample we need to achieve a desired level of precision.