A–2 APPENDIXProbability Distributions
Physics tells us that all protons in the universe are identical. There
is absolutely no uncertainty about the properties of the next proton
you will encounter. But the same is not true of kangaroos.
A probability distribution describes how frequent different kinds
of things are, or how likely the different outcomes for some future
event are. About 90 percent of people are right-handed, and 10 per-
cent are left-handed. This is an example of a discrete distribution
because it describes the frequencies of distinct (discrete) categories.
A discrete distribution can have more than two categories. The fre-
quencies of people with blue, dark brown, brown, green, and hazel
eyes are described by a distribution with five categories. From the
distribution of eye colors shown in FIGURE A.1, we see that the fre-
quency in the United States of individuals with dark brown eyes is
0.25, or one-quarter of the population. The frequencies of all of the
outcomes in a distribution must sum to 1.
A different kind of distribution is needed to describe traits such
as body height that do not fall into discrete categories. These are
described by continuous distributions. A familiar example is the
normal distribution, sometimes called a Gaussian distribution or bell
curve (FIGURE A.2). We saw an example that was visualized with
living students in Figure 6.2. A normal distribution has a single peak,
and it falls off symmetrically to the left and right of the peak. The
distribution shown in Figure A.2C tells us (for example) that there
are many more women who are about 165 cm tall than individu-
als who are about 180 cm tall. The normal distribution is just one
type of continuous distribution. A continuous distribution can be
asymmetrical, and it can have more than one peak. In some human
populations, the distribution of heights has two peaks because it is a
mixture of females, who tend to be shorter, and males, who tend to
be taller.
Continuous distributions are interpreted differently than discrete distributions.
In Figure A.2C, we see that the value of the probability density for female heights
at 164 cm is 0.055. That does not imply, however, that 5.5 percent of females are
that height. In fact, no females are exactly 164.0000 cm tall. What the distribution
conveys is the relative probability that a height will be close to a given value. For
example, the probability is roughly twice as large that a female’s height is close
to 165 cm than that it is close to 155 cm. The distribution can be used to find the
probability that the height of a randomly chosen female will fall in a given range.
For example, the probability that her height will be between 164 cm and 170 cm
tall is given by the corresponding area beneath the curve (that is, the integral from
164 cm to 170 cm). For females in the United States, that probability is 0.29 (or 29
percent; see Figure A.2C). The total area under the curve must again sum to 1.Descriptive Statistics
A distribution often has more information than we really need. We might want to
know how large male elephants typically are, but not care how often the weight
of a male elephant falls between 7123 kg and 7125 kg. A descriptive statistic is a
number that summarizes a useful fact about a distribution.
The most common (and most familiar) descriptive statistic is the mean, also
called the arithmetic mean or average. Dutch men are the tallest in Europe, with a
mean height of about 183 cm (6 feet). That statistic is enough to immediately convey
the fact that many of us will spend a lot of time looking up at tall people if we visitFIGURE A.1 The frequencies of eye colors can be rep-
resented by a discrete distribution, shown here for eye
color in the United States.Futuyma Kirkpatrick Evolution, 4e
Sinauer Associates
Troutt Visual Services
Evolution4e_A.01.ai Date 01-18-2017 03.01.2017Blue 32%Dark
brown 25%Brown 16%Green 12%Hazel 15%Blue Dark brown Brown Green Hazel0.050.100.150.200.250.300
Eye colorFrequency (%)23_EVOL4E_APP.indd 2 3/22/17 1:52 PM