Page 70 - MATLAB Recipes for Earth Sciences

62 4 Bivariate Statistics

Bivariate Scatter
120 i-th data point ( x i ,y i )
Age of sediment (kyrs) 80 Regression line Regression line:
100

60
age = 6.6 + 5.1 depth
40
20
1 Slope = 5.1 Correlation coefficient:
r = 0.96
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 defines 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 coefficients is overestimated and outliers in the

data set lead to an extremely biased estimator of the population correlation
coeffi cient.

The most popular correlation coefficient is Pearson·s linear product-mo-
ment correlation coefficient ρ (Fig. 4.2). We estimate the population·s cor-

relation coeffi cient ρ from the sample data, i.e., we compute the sample
correlation coeffi cient r, which is defi ned as