# 8: Sampling Distributions

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A statistic, such as the sample mean or the sample standard deviation, is a number computed from a sample. Since a sample is random, every statistic is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. As a random variable it has a mean, a standard deviation, and a probability distribution. The probability distribution of a statistic is called its sampling distribution. Typically sample statistics are not ends in themselves, but are computed in order to estimate the corresponding population parameters. This chapter introduces the concepts of the mean, the standard deviation, and the sampling distribution of a sample statistic, with an emphasis on the sample mean

• 8.1: The Mean and Standard Deviation of the Sample Mean
The sample mean is a random variable and as a random variable, the sample mean has a probability distribution, a mean, and a standard deviation. There are formulas that relate the mean and standard deviation of the sample mean to the mean and standard deviation of the population from which the sample is drawn.
• 8.2: The Sampling Distribution of the Sample Mean
This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general.  The importance of the Central Limit Theorem is that it allows us to make probability statements about the sample mean, specifically in relation to its value in comparison to the population mean, as we will see in the examples
• 8.3: The Sample Proportion
Often sampling is done in order to estimate the proportion of a population that has a specific characteristic.
• 8.4: Using the Central Limit Theorem
The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean <x> gets to μ . The central limit theorem illustrates the law of large numbers.
• 8.E: Sampling Distributions (Exercises)
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.

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