The Central Limit Theorem and Sampling Distributions

In the previous chapters, we looked at calculating probabilities for individual members from a population.  For example, finding the probability that a randomly selected person is taller than [latex]180[/latex] cm.  But, we are typically less concerned about studying individual elements of a population.  Instead, we are more interested in a sample taken from a population under study.  So, we are probably more interested in finding the probability that the mean height of a sample of individuals is more than [latex]180[/latex] cm, rather than the probability that an individual’s height is more than [latex]180[/latex] cm.

Why are we concerned about studying a sample?  Why are probabilities associated with sample statistics so important?  In statistics, we want to study and analyze data about a population so that we can draw conclusions about that population.  But, populations are generally very large, and it costs time and money to gather data about populations.  Instead, we take a random sample from the population, study and analyze the sample, and use the results from the sample to draw conclusions about the wider population.  This process is called statistical inference.  In order for this process to work correctly and give us reliable conclusions about the population, we have to calculate probabilities associated with sample statistics, such as sample means or sample proportions.

In this chapter, we will study sample means, sample 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 distribution.  Because the distribution of the sample means follows a normal distribution, under the right conditions, we can use the normal distribution to calculate probabilities about sample means.

“6.1 Introduction to Sampling Distributions and the Central Limit Theorem” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.