Part 1:
Suppose you're taking a class with five tests and the entire grade of the class is based on performance on those tests. Here are the grades of two students:
Student A: 85, 85, 19, 85, 85
Student B: 66, 76, 79, 62, 70
What letter grade do you think each student will make? Is that letter grade a good summary of each of these students' performance? (There is not just one correct answer to this second question - this is one for you to think about.)
Part 2:
Do you suppose it would be useful to measure not only the averages (center) of each of their distributions of grades, but also the "spread" of them? If you want to do that, one way is to look at the range of the data. The range is the maximum score minus the minimum score. Find the range for each of the student's test scores. What do these ranges tell you?
Part 3:
According to the 2000 US Census, the average number of children per family was 1.86 (for families with any children.) Why do you suppose they chose to report it with a decimal number? Clearly, all the numbers that went into that average were whole numbers: 1 child, 2 children, etc.
Part 1. I expect that you computed the average grade for each of those two students and found that both averages were 71.8. In most classes, that would be a letter grade of C. I expect that most students would say that a C is a good summary of the grades for Student B. But some of you might have had some hesitation about saying that a C is a good summary for Student A. (I expect that each of you, if you were Student A, would want to try to convince the teacher that a B is a more appropriate grade.)
Part 2. For Student A, the range is 85 - 19 = 66 points. For Student B, the range is 79 - 62 = 17 points.
These two summary statistics illustrate that the overall set of grades for Student B is much more consistent than it is for Student A. (Which was glaringly obvious just by looking at the grades!)
Using the two summary statistics for each student of the average and the range allows us to give a numerical summary that illustrates a very substantial difference in the overall distribution of the grades.
The point of these two activities is that we must learn about interpreting summary statistics and this helps motivate why we will learn about different types of averages and different types of measures of spread of a distribution.
Part 3. When you compute the average, if you round it off to the appropriate level for an individual value, you will lose information that could be useful. So it is more correct to say "the average is 1.86 children per family" than to say "the average is about 2 children per family." One important use for an average is to compute a total. For instance, if, in a particular state, it was know that there were about 5 million families, then you could multiply and estimate that there are about 9.3 million children in the state. That's quite different from multiplying by the rounded number, which would give you an estimate of 10 million children in the state. The people thinking about how to get the resources needed to educate the children would prefer to have as precise an estimate as is reasonable from the data available, so they'd want to be told 1.86 children per family rather than 2 children per family.
While it is important to learn to compute various summary statistics for data, it is even more important to learn to interpret them, and to realize what they aren't telling you as well as realizing what they are telling you.
In Elementary Statistics, you will learn to compute more than one measure of center and more than one measure of spread and you will see how the features of each of these lead to ideas about what situations are appropriate to use each of these.