Chapter 4 - Sampling Distributions

Abstract

Sampling distributions play a very important role in statistical analysis and decision making. We begin with studying the distribution of a statistic computed from a random sample. Based on the probabilistic foundation of Chapters 2 and 3, the present study marks the beginning of our learning of statistics beyond the descriptive phase. Because a sample is a set of random variables X1, …, Xn, it follows that a sample statistic that is a function of the sample is also random. We call the probability distribution of a sample statistic its sampling distribution. Sampling distributions provide the link between probability theory and statistical inference. The ability to determine the distribution of a statistic is a critical part in the construction and evaluation of statistical procedures. It is important to observe that there is a difference between the distribution of population from which the sample was taken and the distribution of the sample statistic. In general, a population has a distribution called a population distribution, which is usually unknown, whereas a statistic has a sampling distribution, which is usually different from the population distribution. The sampling distribution of a statistic provides a theoretical model of the relative frequency histogram for the likely values of the statistic that one would observe through repeated sampling. Even though some of the terms in this section have already been defined in Chapter 1, we now present these definitions in terms of random variables. These abstractions are introduced to develop scientifically based methods of analyzing the data, and one should always keep in mind the underlying population.