Mean, variance, covariance and regression
.
The mean of a trait,
z
, is simply the sum of values of
z
in the population, divided by the
number of observations,
n
.
Let
i
represent anyone of the individuals in the population, so
it varies from 1 to n.
The mean is then.
z
=
1
ν
ζ
ι
=1
∑
where
z
i
is the measure for
z
in the
i
th
individual in the population.
The variance is simply the average of the squared deviations between
z
i
and
z
.
V
z
=
σ
2
=
1
(

29
2
=1
∑
Squaring the difference ensures that all the values are positive.
Suppose we have two variables,
z
and
w
.
We can estimate the covariance between these
two variables as the average of the products of
z
i

ανδ
ϖ

as
cov
w
,
z
=
,
=
1
(

29
=1
∑
(

29
Note that the product here need not be positive, unless the differences in parentheses have
the same sign.
Also note that as the differences become “correlated,” the covariance
increases.
Now, if one variable,
z
, is say femur length, and the other variable,
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 Fall '10
 Curtis
 Genetics, Standard Deviation, Evolution, Artificial selection

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