Geoscience Reference
In-Depth Information
h e most popular correlation coei cient is
Pearson's linear product-
moment correlation coei cient
ˁ (Pearson 1895) (Fig. 4.2). We estimate the
population's correlation coei cient ˁ from the sample data, i.e., we compute
the sample correlation coei cient
r
, which is dei ned as
where
n
is the number of pairs
xy
of data points,
s
x
and
s
y
are the univariate
standard deviations. h e numerator of Pearson's correlation coei cient is
known as the
corrected sum of products
of the bivariate data set. Dividing the
numerator by (
n
-1) yields the
covariance
which is the summed products of deviations of the data from the sample
means, divided by (
n
-1). h e covariance is a widely-used measure in
bivariate statistics although it has the disadvantage of being dependent on the
dimensions of the data. Dividing the covariance by the univariate standard
deviations removes this dependency and leads to Pearson's correlation
coei cient.
A popular way to test the signii cance of Pearson's correlation coei cient
is to determine the probability of an
r
-value for a random sample from a
population with a ˁ=0. h e signii cance of the correlation coei cient can be
estimated using a
t
-statistic
h e correlation coei cient is signii cant if the calculated
t
is greater than the
critical
t
(
n
-2 degrees of freedom,
ʱ
=0.05). h is test is, however, only valid if
both variables are Gaussian distributed.
Pearson's correlation coei cient is very sensitive to disturbances in
the bivariate data set. Several alternatives exist to Pearson's correlation
coei cient, such as
Spearman's rank correlation coei cient
proposed by the
English psychologist Charles Spearman (1863-1945). Spearman's coei cient
can be used to measure statistical dependence between two variables without
requiring a normality assumption for the underlying population, i.e., it is a
