• Background
  • Instructions
  • Illustration


The Central Limit Theorem is about what we can know about the range of possible means that can be found from samples of a certain size.  In a study, you take a sample from a population.  To do an inferential statistical test, it is necessary to be able to describe in some fashion how that mean can vary around the actual population mean.  Does this sample mean have to be close to the population mean or can it be far off?  The central limit theorem allows us to answer this question. 

To understand the Central Limit Theorem, it is important to understand the concept of a sampling distribution.  A sampling distribution describes all the possible values for a given statistic, e.g., a mean, that can happen for a specific sample size.  So the Central Limit Theorem tells us about this distribution for means.

The Central Limit Theorem makes these claims:

  • The mean of the sampling distribution equals the mean of the population.
  • If the population is normally distributed, so will the sampling distribution of the mean.
  • Even if the population is not normally distributed, if the sample is "large enough", the sampling distribution of the mean will still be normally distributed.
  • The standard deviation of the sampling distribution of the mean (called the standard error)=the standard deviation of the population/the square root of the sample size.

With these claims, we can know how and how much the mean of samples will vary from the true population mean allowing us to do inferential statistics comparing means.

See the illustration to see examine the claims of the Central Limit Theorom empirically.


Full Screen Mode

To see the illustration in full screen, which is recommended, press the Full Screen button, which appears at the top of the page.

Illustration Tab


Below is a list of the ways that you can alter the model. The settings include the following:

Sample: Repeatedly generate a sample to plot on the left which will be the sample distribution and when there are enough samples, you will see the shape of the population. You will also generate the sample size number of samples and calculate the mean to plot on the right hand graph. This will be the sampling distribution of the mean.
Sample Size: This value allows you to select different samples sizes for your sampling distribution of the mean. The sample and sampling distribution graphs are cleared when this value is changed.
Population SHape: Choose a different shape for the population to be sampled from. This values is important to change to test the second of the claims of the Central Limit Theorom.
Show Fitted Normal Distribution: click to place a normal curve on the graph that best fits the data. That does not mean that it fits the data well, just that it is the best choice. This distinction will be important for the sample graph which shows the population.


Pressing this button restores the settings to their default values.