CHAPTER 9 Summarizing and Graphing Your Data 115

How can you convey a visual picture of what the true distribution may look like by

using just a few summary numbers? By reporting values of measures of some

important characteristics of these distributions, so that the reader can infer the

shape. This is similar to learning that one Olympic ice skater scored an average of

9.0 compared to another who scored an average of 5.0. You will not know what the

skate routines looked like unless you watch them, but the score will already tell

you that if you were to watch them, you would expect to see that the one that

scored 9.0 was executed in a more visually pleasing way than the one that

scored 5.0.

Frequency distributions have names for their important characteristics, including:»

» ^{Center: Where along the distribution of the values do the numbers tend}

to center?»

» ^{Dispersion: How much do these numbers spread out?»}

» ^{Symmetry: If you were to draw a vertical line down the middle of the}

distribution, does the distribution shape appear as if the vertical line is a

mirror, reflecting an identical shape on both sides? Or do the sides look

noticeably different — and if so, how?»

» ^{Shape: Is the top of the distribution nicely rounded, or pointier, or flatter?}

Like using average skating scores to describe the visual appeal of an Olympic skate

routine, to describe a distribution you need to calculate and report numbers that

measure each of these four characteristics. These characteristics are what we

mean by summary statistics for numerical variables.

Locating the center of your data

When you start exploring a set of numbers, an important first step is to determine

what value they tend to center around. This characteristic is called, intuitively

enough, central tendency. Many statistical textbooks describe three measures of

central tendency: mean (which is the same as average), median, and mode. You may

assume these are the three optimal measures to describe a distribution (because

they all begin with m and are easy to remember). But all three have limitations,

especially when dealing with data obtained from samples in human research, as

described in the following sections.

Arithmetic mean

The arithmetic mean, also commonly called the mean (or the average), is the most

familiar and most often quoted measure of central tendency. Throughout this

book, whenever we use the two-word term the mean, we’re referring to the