Graphics Programs Reference
In-Depth Information
Bivariate Scatter
120
i-th data point ( x
i
,y
i
)
100
Regression line
80
60
Regression line:
age = 6.6 + 5.1 depth
40
Correlation coefficient:
r = 0.96
Slope = 5.1
20
1
y-intercept = 6.6
0
0
5
10
15
20
Depth in sediment (meters)
Fig. 4.1
Display of a bivariate data set. The twenty data points represent the
age
of a sediment
(in kiloyears before present) in a certain
depth
(in meters) below the sediment-water interface.
The joint distribution of the two variables suggests a linear relationship between
age
and
depth
,
i.e., the increase of the sediment age with depth is constant. Pearson·s correlation coeffi cient
(explained in the text) of
r
=0.96 supports the strong linear dependency of the two variables.
Linear regression yields the equation
age
=6.6+5.1
depth
. This equation indicates an increase
of the sediment age of 5.1 kyrs per meter sediment depth (the slope of the regression line).
The inverse of the slope is the sedimentation rate of ca. 0.2 meters/kyrs. Furthermore, the
equation defi nes the age of the sediment surface of 6.6 kyrs (the intercept of the regression
line with the
y
-axis). The deviation of the surface age from zero can be attributed either to
the statistical uncertainty of regression or any natural process such as erosion or bioturbation.
Whereas the assessment of the statistical uncertainty will be discussed in this chapter, the
second needs a careful evaluation of the various processes at the sediment-water interface.
statistics. They are only a very rough estimate of a rectilinear trend in the
bivariate data set. Unfortunately the literature is full of examples where the
importance of correlation coeffi cients is overestimated and outliers in the
data set lead to an extremely biased estimator of the population correlation
coeffi cient.
The most popular correlation coeffi cient is
Pearson·s linear product-mo-
ment correlation coeffi cient
ρ
(Fig. 4.2). We estimate the population·s cor-
relation coeffi cient
from the sample data, i.e., we compute the sample
correlation coeffi cient
r
, which is defi ned as
ρ
