Story update 8/27/2009: An error was spotted and corrected by author in paragraph starting with "The population mean for a six-sided die..."

Mark Twain famously quipped that there were three ways to avoid telling the truth: lies, damned lies, and statistics. The joke works because statistics frequently seem like a black box—it can be difficult to understand how statistical theorems make it possible to draw conclusions from data that, on their face, defy easy analysis.

But because data analysis plays a critical role in everything from jet engine reliability to determining the shows we see on television, it’s important to acquire at least a basic understanding of statistics. One of the most important concepts to understand is the central limit theorem.

In this article, we will explain the central limit theorem and show how to demonstrate it using common examples, including the roll of a die and the birthdays of Major League Baseball players.

Defining the central limit theorem

A typical textbook definition of the central limit theorem goes something like this:

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