Evolution, 4th Edition

A STATISTICS PRIMER A–3

Amsterdam. To calculate the mean of a set of measurements, we simply add them

together and then divide that total by the number of measurements. By convention,

we often symbolize the mean of a distribution by putting a bar over the symbol used

for the measurement. Say that we use x to represent the height of an individual. In

a group of Dutch men, if half of them have a height of x = 180 cm and the other half

have a height of x = 186 cm, then their mean height is – x= 183 cm.

The variance is a statistic that measures the spread of a distribution around the

mean. Evolution depends critically on variation, and so variance plays a central role

in evolutionary biology. (In fact, the word “variance” was invented by R. A. Fisher

in the first scientific publication in population genetics [1].) A variance is often

symbolized by σ^2. Variance is defined as

    σ^2 = Mean value of (x – –x)^2 (A.1)

Continuing with the Dutch men, for the shorter individuals we have (x – x–)^2 =

(180 cm – 183 cm)^2 = 9 cm^2. For the taller individuals, we have (x – x–)^2 = (180 cm –

183 cm)^2 = 9 cm^2. By taking the mean of 9 cm^2 and 9 cm^2 , we find that the variance

in height for this group of men is σ^2 = 9 cm^2.

Futuyma Kirkpatrick Evolution, 4e

Sinauer Associates

Troutt Visual Services

Evolution4e_A.02.ai Date 02-02-2017 03-01-2017

Q: Msp cut off the x-axis scale for the top graph. Is it the same as the bottom 2? I left off

tic-marks as what was showing appeared to be different labeling.

Area = 0.29

Females Males

150 160 170 180 190 200

0

0.01

0.02

0.03

0.04

0.05

Height (cm)

Probability density

Probability density

0

0.01

0.02

0.03

0.04

0.05

Females Males

Frequency

0

0.05

0.10

0.15

0.20

(A)

(B)

(C)

FIGURE A.2 The frequencies of height in humans

can be represented by a continuous distribu-

tion. (A) Histograms for the heights of a sample

of women and men ages 20–29 in the United

States. Here the data are represented as a discrete

distribution, with the height of each bar show-

ing the frequency of individuals that fell within a

range of heights. (B) The distributions of heights

of women and men in the U.S. population are well

represented by normal distributions. Here the

y-axis represents the relative probabilities that an

individual has a height very close to a given value

on the x-axis. The general equation for a normal

distribution with mean x

_

and variance σ^2 is

In women, the mean height is 164 cm, and the

variance is 54 cm^2. In men, the mean height is 176

cm, and the variance is 58 cm^2. (C) The area under

the female distribution between 164 cm and 170

cm equals 0.29. This shows that 29 percent of

females in this population have heights within that

range.

f (x) =exp (^2) /m 2 {}
1
(^2) m 2
–(x – x_)^2
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