The thing that makes statistics work is that statisticians have discovered how samples are related to populations. This means that statisticians (and, by the end of the course, you) know that if all of the possible samples from a population are taken and something (generically called a “statistic”) is computed for each sample, something is known about how the new population of statistics computed from each sample is related to the original population. For example, if all of the samples of a given size are taken from a population, the mean of each sample is computed, and then the mean of those sample means is found, statisticians know that the mean of the sample means is equal to the mean of the original population.
There are many possible sampling distributions. Many different statistics can be computed from the samples, and each different original population will generate a different set of samples. The amazing thing, and the thing that makes it possible to make inferences about populations from samples, is that there are a few statistics which all have about the same sampling distribution when computed from the samples from many different populations.
You are probably still a little confused about what a sampling distribution is. It will be discussed more in the chapter on the Normal and t-distributions. An example here will help. Imagine that you have a population —the sock sizes of all of the volleyball players in the South Atlantic Conference. You take a sample of a certain size, say six, and find the mean of that sample. Then take another sample of six sock sizes, and find the mean of that sample. Keep taking different samples until you've found the mean of all of the possible samples of six. You will have generated a new population, the population of sample means. This population is the sampling distribution. Because statisticians often can find what proportion of members of this new population will take on certain values if they know certain things about the original population, we will be able to make certain inferences about the original population from a single sample.
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