Wilcox has done a great deal of work on the problems of trimming, and I certainly re-
spect his well-earned reputation. In addition I think that students need to know about
trimmed means because they are being discussed in the current literature. But I don’t think
that I can go as far as Wilcox in promoting their use. However, I don’t think that my reluc-
tance should dissuade people from considering the issue seriously, and I recommend
Wilcox’s book (Wilcox, 2003).2.8 Measures of Variability
In the previous section we considered several measures related to the center of a distribu-
tion. However, an average value for the distribution (whether it be the mode, the median, or
the mean) fails to give the whole story. We need some additional measure (or measures) to
indicate the degree to which individual observations are clustered about or, equivalently, de-
viate from that average value. The average may reflect the general location of most of the
scores, or the scores may be distributed over a wide range of values, and the “average” may
not be very representative of the full set of observations. Everyone has had experience with
examinations on which all students received approximately the same grade and with those
on which the scores ranged from excellent to dreadful. Measures referring to the differences
between these two situations are what we have in mind when we speak of dispersion,or
variability, around the median, the mode, or any other point. In general, we will refer specif-
ically to dispersion around the mean.
An example to illustrate variability was recommended by Weaver (1999) and is based
on something with which I’m sure you are all familiar—the standard growth chart for in-
fants. Such a chart appears in Figure 2.12, in the bottom half of the chart, where you can
see the normal range of girls’ weights between birth and 36 months. The bold line labeled
“50” through the center represents the mean weight at each age. The two lines on each side
represent the limits within which we expect the middle half of the distribution to fall; the
next two lines as you go each way from the center enclose the middle 80% and the middle
90% of children, respectively. From this figure it is easy to see the increase in dispersion as
children increase in age. The weights of most newborns lie within 1 pound of the mean,
whereas the weights of 3-year-olds are spread out over about 5 pounds on each side of the
mean. Obviously the mean is increasing too, though we are more concerned here with
dispersion.
For our second illustration we will take some interesting data collected by Langlois and
Roggman (1990) on the perceived attractiveness of faces. Think for a moment about some
of the faces you consider attractive. Do they tend to have unusual features (e.g., prominent
noses or unusual eyebrows), or are the features rather ordinary? Langlois and Roggman
were interested in investigating what makes faces attractive. Toward that end, they pre-
sented students with computer-generated pictures of faces. Some of these pictures had been
created by averaging together snapshots of four different people to create a composite. We
will label these photographs Set 4. Other pictures (Set 32) were created by averaging across
snapshots of 32 different people. As you might suspect, when you average across four peo-
ple, there is still room for individuality in the composite. For example, some composites
show thin faces, while others show round ones. However, averaging across 32 people usu-
ally gives results that are very “average.” Noses are neither too long nor too short, ears
don’t stick out too far nor sit too close to the head, and so on. Students were asked to
examine the resulting pictures and rate each one on a 5-point scale of attractiveness. The
authors were primarily interested in determining whether the mean rating of the faces in
Set 4 was less than the mean rating of the faces in Set 32. It was, suggesting that faces with
distinctive characteristics are judged as less attractive than more ordinary faces. In this
section, however, we are more interested in the degree of similarity in the ratings of faces.36 Chapter 2 Describing and Exploring Data
dispersion