the data points are smoothly distributed between roughly 7 and 17 hundredths of a second,
but a small but noticeable cluster of points lies between 30 and 70 hundredths, trailing off to
the right. This second cluster of points was obtained primarily from trials on which the sub-
ject missed the button on the first try. Their inclusion in the data significantly affects the dis-
tribution’s shape. An experimenter who had such a collection of data might seriously
consider treating times greater than some maximum separately, on the grounds that those
times were more a reflection of the accuracy of a psychomotor response than a measure of
the speed of that response. Even if we could somehow make that distribution look better, we
would still have to question whether those missed responses belong in the data we analyze.
It is important to consider the difference between Bradley’s data, shown in Figure 2.11,
and the data that I generated, shown in Figures 2.1 and 2.2. Both distributions are positively
skewed, but my data generally show longer reaction times without the second cluster of
points. One difference was that I was making a decision on which button to press, whereas
Bradley’s subjects only had to press a single button whenever the light came on. Decisions
take time. In addition, the program I was using to present stimuli recorded data only from
correct responses, not from errors. There was no chance to correct and hence nothing
equivalent to missing the button on the first try and having to press it again. I point out
these differences to illustrate that differences in the way in which data are collected can
have noticeable effects on the kinds of data we see.
The last characteristic of a distribution that we will examine is kurtosis. Kurtosishas a
specific mathematical definition, but basically it refers to the relative concentration of
scores in the center, the upper and lower ends (tails), and the shoulders (between the center
and the tails) of a distribution. In Figure 2.10(e) and (f) I have superimposed a normal dis-
tribution on top of the plot of the data to make comparisons clear. A normal distribution
(which will be described in detail in Chapter 3) is called mesokurtic.Its tails are neither
too thin nor too thick, and there are neither too many nor too few scores concentrated in
the center. If you start with a normal distribution and move scores from both the center and
the tails into the shoulders, the curve becomes flatter and is called platykurtic.This is
clearly seen in Figure 2.10(e), where the central portion of the distribution is much too flat.
If, on the other hand, you moved scores from the shoulders into both the center and the
tails, the curve becomes more peaked with thicker tails. Such a curve is calledleptokurtic,
and an example is Figure 2.10(f). Notice in this distribution that there are too many scores
in the center and too many scores in the tails.^5
It is important to recognize that quite large samples of data are needed before we can
have a good idea about the shape of a distribution, especially its kurtosis. With sample sizes
of around 30, the best we can reasonably expect to see is whether the data tend to pile up in
the tails of the distribution or are markedly skewed in one direction or another.
So far in our discussion almost no mention has been made of the numbers themselves.
We have seen how data can be organized and presented in the form of distributions, and we
have discussed a number of ways in which distributions can be characterized: symmetry or
its lack (skewness), kurtosis, and modality. As useful as this information might be in cer-
tain situations, it is inadequate in others. We still do not know the average speed of a sim-
ple decision reaction time nor how alike or dissimilar are the reaction times for individualSection 2.5 Describing Distributions 29(^5) I would like to thank Karl Wuensch of East Carolina University for his helpful suggestions on understanding
skewness and kurtosis. His ideas are reflected here, although I’m not sure that he would be satisfied by my state-
ments on kurtosis. Karl has spent a lot of time thinking about kurtosis and made a good point recently when he
stated in an electronic mail discussion, “I don’t think my students really suffer much from not understanding
kurtosis well, so I don’t make a big deal out of it.” You should have a general sense of what kurtosis is, but you
should focus your attention on other, more important, issues. Except in the extreme, most people, including statis-
ticians, are unlikely to be able to look at a distribution and tell whether it is platykurtic or leptokurtic without
further calculations.
Kurtosis
mesokurtic
platykurtic
leptokurtic