Evolution, 4th Edition

A–4 APPENDIX

Just what does a variance tell us? A variance equal to zero says that there

is no spread around the mean: all the measurements are identical. If one

distribution has a larger variance than another, then it has more dispersion

around the mean (FIGURE A.3). Notice the units in the example above: they

are the units of measurement squared. A variance therefore can never be

negative.

Often several factors contribute to the variance of a variable. Whenever

we work with measurements, the data inevitably include measurement

error, which will be small if the measurements are accurate, but large

if not. The variance in our measurements then has contributions from

two sources: the variance in the actual variable, and the variance in the

errors. Statistics provides methods to estimate and correct for the errors

(for example, by remeasuring some of the individuals). In Chapter 6 we

discuss another situation in which multiple factors contribute to a vari-

ance: the phenotypic variance for a quantitative trait has both genetic and

environmental components.

A second useful statistic that describes variation is the standard devia-

tion. It is the square root of the variance and is often symbolized as σ.

The standard deviation in the example of Dutch men is σ = 9 cm^2 = 3

cm. A standard deviation has the same units as the original measurements. For a

normal distribution, about two-thirds (68 percent) of the distribution falls within

1 standard deviation of the mean, that is, between (x– – σ) and (x– + σ), and 95

percent falls within 2 standard deviations of the mean, between (– x– 2σ) and (x– +

2 σ) (FIGURE A.4A).

The standard deviation is useful for measuring the difference between the

means of two distributions. If the difference is much less than a standard devia-

tion of one of the distributions, then it can be difficult for us to see by eye that

the distributions really are different (FIGURE A.4B). A difference of more than 1

standard deviation is typically easy to see. The means of the distributions for male

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σ^2 = 1

σ^2 = 1.5

σ^2 = 3

–5

0

0.1

0.2

0.3

0.4

0 5

Measurement

Probability density

FIGURE A.3 These three distributions have the

same mean but different variances. As the variance

increases, the spread of the distribution around the

mean increases.

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σ

Area = 0.68

x

(A)

2 σ

Area = 0.95

x

Measurement Measurement

Probability density

σ/2

x 1

(B)

Probability density

x 2

2σ

x 1 x 2

FIGURE A.4 The standard deviation, σ,

is a measure of variation. (A) For normal

distributions, about 2/3 (68 percent) of the

distribution falls within 1 standard deviation

above and below the mean (top), and 95

percent falls within 2 standard deviations

of the mean (bottom). (B) The difference

between the means of two distributions

can be measured in terms of their standard

deviations. When the difference is much

less than σ, the distributions overlap greatly

(top). When the difference in their means is

much greater than σ, much of the distribu-

tions do not overlap (bottom).

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