Nov. 15, 1925 
Possible Errors in the Use of Curves 
927 
In every instance large discrepancies were found, which are so simi¬ 
lar in character to those illustrated in Figure 5 that to present them 
here would introduce no new element. 
In both the instances which have been described the way in which 
the erroneous results are obtained can best be understood by actually 
working out an example, and carefully noting how the different indi¬ 
vidual values combine with other values and exert their influence on 
different parts of the curve. In the last instance, for example, it will 
readily be seen by inspecting the curves that for any given diameter 
the particular trees which determine the average height (curve 1) are 
for the most part entirely different from those which determine the 
corresponding average volume. The possibility that diversity in 
manner of grouping is the prime cause of the difficulties encountered 
suggests an alternative procedure. 
Suppose measurements are sorted by height class, and that the 
average diameter at breast height and average volume for each class 
are determined. By plotting these average volumes over these 
average diameters the curve which has been sought can apparently be 
D. B. H.“ INCHES 
Fig. 5—Curve of volume over diameter, obtained indirectly from curves of Figures 4 and 1, compared 
with actual averages 
obtained in still a third way. Furthermore, it is immaterial whether 
actual averages be used or whether advantage be taken of the fact 
that curves have already been drawn for each of these variables over 
height by using corresponding curve values. The algebraic analogy 
would be almost identical with that just described. 
For example, we may find from Figure 4 that the average volume 
of 30-foot trees is 1.3 cubic feet, and from Figure 2 that the average 
diameter at breast height for 30-foot trees is 4.8 inches. We may 
then plot 1.3 over 4.8. Repeating the process for other height classes, 
we produce the volume-diameter curve illustrated in Figure 6. When 
the actual average values are plotted, however, as has been done in 
the figure, this curve in turn is seen to be a very poor expression of 
the data. Particularly noticeable is the fact that the curve covers 
only about two-thirds of the range of the points and gives no values 
at all for the higher diameters. In the present instance the result of 
this last process is perhaps less erroneous than that previously 
illustrated (fig. 5); but when similar procedures were tried with the 
other five possible curves, it was found that this superiority was 
accidental, and that in general there was little or no choice. 
