A STATISTICS PRIMER A–5
and female height shown in Figure A.2 differ by about 1.5 standard deviations.
For some purposes, the standard deviation is a more useful unit than the original
units of measurement. If a species evolves to become 1 mm larger, is that a lot of
change or not? It is a microscopic change for a species of elephant, but an enor-
mous change for a species of ant. In contrast, if a species has evolved to become 5
standard deviations larger, we know immediately that it is very different than what
it used to be, whether it is an elephant or an ant.
We often are interested in the relationship between variation in two or more
traits. Humans with long arms tend also to have long legs, simply because those
individuals are larger than average overall. Covariance is a basic measure of the
association between two measurements. The covariance between variables x and y
is often written as σxy, and is defined as σxy = Mean value of (x – x–)(y – y–) (A.2)where x and y represent the two measurements, and x– and y– are their means. A cova-
riance is positive if the measurements tend to increase and decrease together, and it is
negative if one measurement tends to get smaller as the other becomes larger.
Another way to measure the association between two variables is the correla-
tion, symbolized as r. A correlation is a covariance that has been rescaled so that
it has no units, and ranges from a minimum value of –1 to a maximum value of 1.
The correlation between variables x and y is defined asr =(^) σxy
(^) σxσy^ (A.3)
where σx and σy are the standard deviations of x and y. A positive correlation means
that individuals that are larger for one trait also tend to be larger for the second, as
with arm and leg length. A negative correlation implies that individuals that are
larger than average for one trait tend to be smaller than average for the second trait.
A correlation of r = 0 means there is no simple relation between the two measure-
ments: individuals that are larger than average for the first are equally likely to be
either smaller or larger than average for the second. At the other extreme, a correla-
tion of r = 1 tells us that the value of one measurement is perfectly associated with
the value of the second. Examples of correlations are shown in FIGURE A.5.
A regression predicts the value of one variable from the value of another.
The most common kind of regression fits a line to the points in a plot of the two
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–3
–2
–1
0
1
2
(^3) r = 0
Measurement 2
–2 –1 0 1 2 3
Measurement 1
r = –0.75
–2 –1 0 1 2 3
r = 1
–2 –1 0 1 2 3
(A) (B) (C)
FIGURE A.5 Correlation measures how
two variables vary together. (A) A correlation
of r = 0 means there is no simple relation
between the two variables. (B) A negative
value of r means that larger than average
values of one variable tend to be associ-
ated with smaller than average values of the
other. (C) A positive value of r means that
larger than average values of one variable
are associated with larger than average val-
ues of the other. The largest possible value
for a correlation is r = 1, which means the
two variables are perfectly associated.
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