SOME POSSIBLE ERRORS IN THE USE OF CURVES 1 
By Donald Bruce 
Silviculturist t Forest Service , United States Department of Agriculture 
Although the analysis of data by means of graphs and curves has 
been one of the most common procedures in forestry, little investiga¬ 
tion apparently has been made of the limitations of the method or of the 
seriousness of possible errors which may result from ignorance or 
disregard thereof. The basic idea of plotting measurements on 
coordinate paper and of fitting curves thereto is derived from analyt¬ 
ical geometry, and there has been a tendency to assume that any of 
the processes of the algebraic basis for this branch of mathematical 
science are equally permissible in the case of empirical curves. Pure 
mathematics, however, assumes exact equations between the vari¬ 
ables under consideration, and these variables are therefore perfectly 
correlated. Empirical data are never perfectly correlated and are 
usually (at least in the case of forestry) very loosely correlated. As 
the degree of correlation becomes less, it can be shown that certain 
mathematical processes become increasingly inapplicable until in 
many instances gross errors are introduced by their use. 
Probably the best way to make this possibility of error clear is to 
take a single data series and subject it to apparently plausible trans¬ 
formations, comparing the results in each case with those obtained by 
a more direct handling. The material selected is taken from measure¬ 
ments of a series of 524 second-growth long-leaf pine trees and, for 
simplicity, is limited to the three measurements: Diameter (breast 
high), height, and volume in cubic feet. This is a case in which the 
variables are comparatively closely correlated and in which, therefore, 
no abnormally* serious discrepancies should occur. Few forestry 
problems afford better material, and if any important errors are dis¬ 
covered in this instance, difficulties at least as great may be expected 
wherever parallel processes are used. 
Let us consider first the possibility of interchanging the independent 
and the dependent variables. Texts in forest mensuration have 
warned against this, but without sufficiently emphasizing the kind 
and degree of error involved therein. At first glance, it seems quite 
obvious that if a table or curve showing average heights of trees of 
given diameters is available, then the average diameters of trees of 
given heights should be obtainable therefrom. To do so would be 
analogous to transforming an algebraic equation such as y = ax + b into 
x= ^ ■■ -> and it is probably because of the correctness of such an 
algebraic transformation that the temptation to perform a similar 
process with empirical curves is so strong. But let us see the result. 
i Received for publication Jan. 29. 1925: issued January, 1926. 
Journal of Agricultural Research, 
Washington, D. C. 
( 923 ) 
Vol. XXXI, No. 10 
Nov. 15, 1925 
Key No. F-33 
