Business Statistics For Dummies
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In statistics, sampling distributions are the probability distributions of any given statistic based on a random sample, and are important because they provide a major simplification on the route to statistical inference. More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.

The value of a sample statistic such as the sample mean (X) is likely to be different for each sample that is drawn from a population. It can, therefore, be thought of as a random variable, whose properties can be described with a probability distribution. The probability distribution of a sample statistic is known as a sampling distribution.

According to a key result in statistics known as the Central Limit Theorem, the sampling distribution of the sample mean is normal if one of two things is true:

  • The underlying population is normal

  • The sample size is at least 30

Two moments are needed to compute probabilities for the sample mean; the mean of the sampling distribution equals:


The standard deviation of the sampling distribution (also known as the standard error) can take on one of two possible values:


This is the appropriate choice for a "small" sample; for example, the sample size is less than or equal to 5 percent of the population size.

If the sample is "large," the standard error becomes:


Probabilities may be computed for the sample mean directly from the standard normal table by applying the following formula:


About This Article

This article is from the book:

About the book author:

Alan Anderson, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.

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