6.1 Introduction to Sampling Distributions and the Central Limit Theorem

This is a photo of change a set of keys in a pile. There appear to be five pennies, three quarters, four dimes, and two nickels. The key ring has a bronze whale on it and holds eleven keys.
If you want to figure out the distribution of the change people carry in their pockets, using the central limit theorem and assuming your sample is large enough, you will find that the distribution is normal and bell-shaped. Photo by John Lodder, CC BY 4.0.

Why are we so concerned with means? Two reasons are that they give us a middle ground for comparison, and they are easy to calculate.  In this chapter, we will study means, proportions and their relationship to the central limit theorem.

The central limit theorem is one of the most powerful and useful ideas in all of statistics.  The central limit theorem basically says that if we collect samples of size [latex]n[/latex] from a population with mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex], calculate each sample’s mean, and create a histogram of those means, then, under the right conditions, the resulting histogram will tend to have an approximate normal bell shape.

Watch this video: Central limit theorem | Inferential statistics | Probability and Statistics | Khan Academy by Khan Academy [9:45]

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