2.5 Sampling and Sampling Distributions
This chapter concludes with the hypothetical concept of a sampling distribution. Understanding this concept is crucial to understanding the entire point of inferential statistics.
Recall that we want to make statistical inferences that use statistics calculated from samples to estimate parameters of the population.
Plain statistics draws conclusions about the sample (those are facts) while statistical inference draws conclusions about the population.
In practice, a single sample is all that is selected. In other words, once you construct your sample it is all of the observations you have to use.
Since the actual observations inside your sample were selected at random, then the sample you constructed is in fact a random sample.
If the random observations were drawn from a sample frame, what was the resulting random sample drawn from? The answer is a sampling distribution.
2.5.1 An Application
Consider the scenario discussed above where we want to determine the population average human body temperature. At a particular point in time, the population is every human. As the particular points in time change, new births implies that the population is changing as well! Clearly the overall population is unobtainable - so we need to draw a sample.
Suppose we decide on a sample size of 10,000 adults. Regardless of the sampling method chosen from the list above, we arrive at a data sample of 10,000 observations of human body temperatures. Since these individuals were selected randomly, then the sample mean calculated from the random sample is itself random. If we randomly draw another sample of 10,000 observations, we can get another sample average. We can do this repeatedly, getting a different sample average for every sample randomly drawn.
Note that this is purely hypothetical because we would never draw numerous samples… but bear with me.
We have established that our sample was a random draw from our population. Therefore, the sample mean calculated from our random sample is itself a random draw from a sampling distribution.
Think of a sampling distribution as a histogram showing you the outcomes of all possible sample means and their frequency of appearing. This distribution will have characteristics of its own. The mean of this distribution would be the mean value of all possible sample means. The standard deviation would be the amount of average dispersion all individual sample means around the overall mean.
What we will soon see is that this sampling distribution will be the foundation to inferential statistics. To see this, we will combine this concept of a sampling distribution with something called the Central Limit Theorem (CLT). The CLT is so important, it deserves its own chapter. However, before we get to that conceptual stuff, we will first get into the practical stuff. Namely, an introduction to the R project for Statistical Computing.