Evolution__3rd_Edition

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Box 9.1
Some Statistical Terms Used in Quantitative Genetics
The text mentions three main statistical terms. This box explains
variance, but serves mainly as a reminder of the exact definitions of
covariance and regression; reference should be made to a statistical
text for fuller explanation.
Variance
The variability of a set of numbers can be expressed as a variance.
Take a set of numbers, such as 4,3,7,2,9. Here is how to calculate
their variance.
1. Calculate the mean:
2. For each number, calculate the square of its deviation from the
mean. For the first number, 4, it is (5 − 4) 2 = 1. We do likewise
for all five numbers.
3. Add up the sum of the squared deviations from the mean. For
the five numbers, it is 1 + 4 + 4 + 9 + 16 = 34.
4. Divide the sum by n − 1; n = 5 in this case.
V
Mean =
Variance = 34/4 = 8.5
The general formula for the variance of a character X is:
n
=
1
− 1
X i
4 + 3 + 7 + 2 + 9
= 5
5
∑
( x − E)
where E is the mean and x i is a standard notation for a set of
numbers. Here we have five numbers. In terms of the notation, that
means that i can have any value from 1 to 5 and is the value of the
character for each i. Thus x 1 = 4, x 2 = 3, and x 5 = 9. The summation
in the general formula is for all values of i: here it is for all five
numbers. If there had been 50 numbers, i would have varied from
1 to 50 and we should proceed as in the example for all 50 numbers.
The variance describes how variable the set of numbers is: the
Figure B9.1
The relation between
two variables (x and y):
(a) negative regression
coefficient (b < 0);
(b) no relation (b = 0);
and (c) positive regression
coefficient (b > 0).
2
y
higher the variance, the greater the differences among the
numbers. If all the numbers were the same (all x i = E) then their
variance is zero.
Standard deviation
This is the square root of the variance.
Covariance
Now imagine the individuals have been measured for two
characters, X and Y. The covariance between the two is defined as:
cov XY i i
Covariance measures whether, if an individual has a large value of
X, it also has a large value for Y. If the x i and y i of an individual are
both large, the product x i y i will also be large, but if y i is not large
when x i is, then the product will be smaller. Generally, if X and Y
covary, the product (and so the covariance) is large, and if they do
not, the sum of the products will come to zero.
Regression
The regression, symbolized by bXY , between characters X and Y is
their covariance divided by the variance of X:
b
XY
n
=
1
− 1
V
cov
=
X
XY
∑
( x − E)( y − F)
Regressions are used to describe the slopes of graphs and are
therefore useful in describing the resemblance between classes of
relatives. If X and Y are unrelated, cov XY = 0 and b XY = 0; if they are
related, the covariance and regression can be positive or negative
(Figure B9.1).
(a) (b) (c)
x
y
x
y
x