Physics for Geologists, Second edition

134 Some dangers of mathematical statistics
Table 13.1 Experimental data
Figure 13.2 Plot of data in Table 13.1 with the linear regression line. The sys-
tematic differences are evident although statistical analysis indicates
a very significant linear relationship and implies a random error
about the regression line.
In this case, if you compute the regression of the natural logarithms,
you obtain In y = 2.0 Inx - 0.69, and the correlation coefficient is 1.00
(Figure 13.3). Restoring the natural logarithms (Iny = mlnx + Inb to
y = bxm), this suggests that the true relationship is y = x2/2, which indeed
it is. The result of the test of significance, that it is extremely unlikely that
the association of points arose by chance, must not be interpreted as con-
firmation that it is linear. The test of significance indicates that it is most
unlikely that the association arose by chance, and indeed, the association
did not arise by chance.
The point is, and a very important point it is, that any gentle curve, and
any short part of a curve, will give good linear correlation. You must do the
thinking, because statistics does not think for you. In particular, you must
consider the boundary conditions. Take another example.
You determine the acoustic or sonic velocity (V) and fractional porosity
(f) in sandstones of different porosity so that you may determine the poros-
ity by measuring the acoustic velocity. It is much easier to measure acoustic
velocity than porosity in a borehole. You can measure both with considerable
accuracy when you have a sample, so how would you set about the task of
Copyright 2002 by Richard E. Chapman