Recall the Big Picture—the four-step process that encompasses statistics (as it is presented in this course):

So far, we’ve discussed the first two steps:

Producing data—how data are obtained and what considerations affect the data production process.

Exploratory data analysis—tools that help us get a first feel for the data by exposing their features using graphs and numbers.

Our eventual goal is inference—drawing reliable conclusions about the population on the basis of what we’ve discovered in our sample. To really understand how inference works, though, we first need to talk about probability, because it is the underlying foundation for the methods of statistical inference. We use an example to try to explain why probability is so essential to inference.

First, here is the general idea: As we all know, the way statistics works is that we use a sample to learn about the population from which it was drawn. Ideally, the sample should be random so that it represents the population well.

Recall from Types of Statistical Studies and Producing Data that when we say a random sample represents the population well, we mean that there is no inherent bias in this sampling technique. It is important to acknowledge, though, that this does not mean all random samples are necessarily “perfect.” Random samples are still random, and therefore no random sample will be exactly the same as another. One random sample may give a fairly accurate representation of the population, whereas another random sample might be “off” purely because of chance. Unfortunately, when looking at a particular sample (which is what happens in practice), we never know how much it differs from the population. This uncertainty is where probability comes into the picture. We use probability to quantify how much we expect random samples to vary. This gives us a way to draw conclusions about the population in the face of the uncertainty that is generated by the use of a random sample. The following example illustrates this important point.