Chapter 3 – Measures of Center

In the last chapter, we covered our first descriptive statistics, using frequency distributions to organize our data. In this chapter, we will cover some new descriptive statistics that we will use to describe a group of scores. Specifically, we will learn how to measure the center and the spread of a group of scores.

Measures of Center

Imagine your instructor told you that you earned a 65 on a test. How would you feel?

If you are like most people, your first reaction would be some negative feeling (e.g., sad, angry, etc.). But after that, you might think to ask some questions to put your score in context. First, you should probably ask about the total number of points on the test. If there were only 65 points, you earned a perfect score! If there were 100 total points, you might then be concerned.

Let’s say that it was a test with 100 possible points, and again you earned a 65. Most people would be unhappy with that score. But many people will then think to ask how did everyone else in the class perform? If everyone else didn’t do very well, then there might be some relief. If everyone else did much better, then there might be some concern.

This question about how everyone else performed tries to get at how the group performed, rather than an individual. Essentially, it is a question about the “center” of the group, what we will refer to as a “measure of center.”

Measures of center are descriptive statistics that describe how a group of individuals scored. Because the individuals in a group will oftentimes vary in terms of their scores, describing the group’s score can be difficult. Essentially, what we will try to do is to identify the most “typical” or “average” score. Take for example the following set of scores from a group of ten individuals:

4, 4, 4, 5, 5, 5, 5, 6, 6, 6

Most people would agree that the most typical or average score would be 5. In fact, 4 out of the 10 individuals actually had a score of 5, and the remaining 6 individuals are only one point above or below the score of 5. In this case, the typical or average score is pretty clearly in the “center” or “middle” of the group of scores.

However, it is more difficult to find the most typical or average score for some groups of scores. Take for example the following set of scores from another group of ten individuals:

1, 2, 2, 20, 20, 20, 20, 20, 20, 20

The “center” or “middle” of the scores is less obvious, and finding a typical or average score is not clear either. Because of situations like this, we will learn three different measures of center: (1) mean, (2) median, and (3) mode.

Measures of Spread

Measures of spread gauge how spread out the scores are in a distribution. In other words, they measure how different the scores are from each other. If the individuals in a group all scored very close to each other, then the spread is relatively small. If the individuals in a group scored very differently from one another, then the spread is relatively large.

So let’s suppose that you earned a 65 out of 100 on the test, and the average for the class was 65. Your feelings about your score have a lot to do with how you feel about being average in the class. If you wanted to be one of the higher performers in the class, you might be a bit bummed. But what if everyone in the class earned a 65 out of 100 on the test? In other words, there was absolutely no spread in the distribution because everyone scored exactly the same. It might cause you to feel a little less upset. In this way, it is important to have a measure of the spread of the scores (in addition to a measure of center) in order to really understand how the group performed.

Like measures of center, there are a couple of different measures of spread that we can use: range, variance, and standard deviation.

Types of Measures

As it turns out, measuring center and measuring spread is not always straightforward, and thus statisticians have created a number of different ways to measure center and spread. In the following sections, we will explore some of the more commonly used approaches:

• Measures of Center
  • Mean
  • Median
  • Mode
• Measures of Spread
  • Range
  • Variance
  • Standard Deviation