On this page, we discuss the Central Limit Theorem, which forms the theoretical fundament of much of inferential statistics.

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • Through experimentation, get an intuitive understanding that sampling distributions of the sample mean with large sample size tend towards having the shape of a normal distribution.

  • Understand the statement and conditions of the central limit theorem, and be able to apply it to compute the probability that a sum or average of many independent random variables drawn from the same distribution falls into any interval.

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Know that the central limit theorem is a statement about the asymptotic behaviour of the sampling distribution of the sample mean (how it changes as the sample size grows without bound). For any finite sample size, using a Gaussian to approximate the sampling distribution of the sample mean may or may not yield precise results.

To prepare for class

  • Play with the following interactive webapp (by Online Stat Book), which allows to select a distribution for the population at the top, and then to take repeated samples and calculate the corresponding sample means, building a histogram showing (an approximation) of the sampling distribution (make sure to ready the instructions, then click on “begin”):


  • Watch the following video (by jbstatistics) which explains the Central Limit Theorem and how to use it in practice:

  • Watch the following video (by Brendan Cordy) which goes into more detail and shows another detailed example of using the Central Limit Theorem:

After class

  • Watch the following video by Kevin deLaplante which illustrates the importance of large samples by explains the Small Sample Fallacy:


Brendan Cordy Avatar Brendan Cordy
Gabriel Indurskis Avatar Gabriel Indurskis


Last Updated





Please click here if you find a mistake or broken link/video, or if you have any other suggestions to improve this page!