Geoscience Reference
In-Depth Information
DID YOU KNOW?
The computed value of a statistic such as the correlation efficient depends on which particular
units were selected for the sample. Such estimates will vary from sample to sample. More
important, they will usually vary from the population value which we try to estimate.
Then
()() ( (
86
395
12
)
()()()()
420940
+
+
+
11 40
2960
2830 83
11
.
s xy =
=
=
11 74
.
1
221
The positive covariance is consistent with the well-known and economically unfortunate fact that
larger diameters tend to be associated with older ages.
7.9 SIMPLE CORRELATION COEFFICIENT
The magnitude of the covariance, like that of the standard deviation, is often related to the size
of the variables themselves. Units with large X and Y values tend to have larger covariances than
units with small X and Y values. Also, the magnitude of the covariance depends on the scale of
measurement; in the previous example, if the diameter had been expressed in millimeters instead of
inches, the covariance would have been 298.196 instead of 11.74. The simple correlation coefficient,
a measure of the degree of linear association between two variables, is free of the effects of scale of
measurement. It can vary from between -1 and +1. A correlation of 0 indicates that there is no linear
association (although there may be a very strong nonlinear association). A correlation of +1 or -1
would suggest a perfect linear association. As for the covariance, a positive correlation implies that
the large values of X are associated with the large values of Y . If the large values of X are associated
with the small values of Y , then the correlation is negative.
The population correlation coefficient is commonly symbolized by ρ (rho) and the sample-based
estimate r . The population correlation coefficient is defined to be
Covariance of
XY
and
ρ=
(Varianceof
X
)(Vari
ance of
Y
)
For a simple random sample, the sample correlation coefficient is computed as follows:
∑∑
xy
s
ss
xy
r
=
=
(
)(
)
2
2
x
y
xy
where
s xy = Sample covariance of X and Y.
s x = Sample standard deviation of X.
s y = Sample standard deviation of Y.
xy = Corrected sum of XY products:
(
)(
)
∑∑
XY
n
XY
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