610 
Journal of Agricultural Research 
Vol. XXX, No. 7 
are usually based on 1-inch diameter 
intervals and 10-foot height intervals. 
For example, a tree 14.7 inches in 
diameter at breast height and 82.2 feet 
tall, belongs in the 15 inch-80 foot class. 
Trees are usually well distributed as to 
diameters within a given inch and height 
class so that the average of all trees 
falling in such given class is very close 
to the assigned even diameter. Of 
course, for each height class there is a 
mean diameter from which the diame¬ 
ter values fall away on each side 
roughly in accord with the laws of 
chance. This curve is long and flat, 
however, for diameters in any height 
class, and the distribution of diameters 
through any single inch class is virtu¬ 
ally even, although, of course, theoret¬ 
ically not quite so in any but the model 
class. Thus, in the 12-inch d. b. h. 
class, individual trees will range from 
11.6 to 12.5 inches d. b. h. Unless 
there are very few trees indeed in the 
class, in practice the mean will always 
fall very close to 12.0 inches. Only 
occasionally in the highest and lowest 
diameter classes will this rule fail, as it 
may indeed by chance in any class 
represented by a very few trees. 
Conditions are not the same, how¬ 
ever, in regard to the height. Trees of 
the same inch class may be tall, short, or 
medium, and a very wide variation of 
height values is possible, ranging from 
two-thirds to one-half of the total 
height of the trees, in lodgepole pine at 
least. This variation is very nearly 
“ normal,” that is, the heights vary 
from the central or modal value 
according to the laws of chance. In 
case, therefore, the entire sample of 
data relating to a single inch class is 
subdivided into five smaller classes on 
the basis of height, the central class will 
have evenly balanced values and the 
actual average will generally represent 
the true mean of the class. The next 
class adjacent both above and below 
will have its values piled up asymmetri¬ 
cally, with a preponderance of values 
nearer the average height for the inch 
class. For example, suppose there are 
five height classes, 100, 90, 80, 70, 60 
feet, as shown in Figure 1. The aver¬ 
age height of all trees in the central 
class, 80 feet, will be 80 feet. In the 
90-foot class there will tend to be more 
86 -foot trees than 94, and the average 
height of the 90-foot class will be lower, 
perhaps about 88 feet. Conversely, 
in the 70-foot class values will tend to 
run high. In the outermost classes 
this effect is still more pronounced and 
the average 100-foot tree may be only 
96 feet tall, while the average 60-foot 
tree may be 64. If the tables were to 
be used only in the region where pre¬ 
pared, this would be of little moment, 
for in the timber estimate 100-foot trees 
would again average 96 feet tall and the 
volume of 96-foot trees would be the 
correct volume to use. In a region 
where sites were better, however, trees 
in the 100-foot class mi|ght be 100 feet 
tall and volumes based on 96-foot trees 
would be low. 
It must be the aim of taper curves 
and volume tables to give the true 
middle values of the class represented, 
or else they will be meaningless. 
Therefore, the values of the 15 inch- 
100 foot tree class must be exactly 
right for trees 15 inches in diameter, 
and 100 feet tall, and not perhaps for a 
tree 14.8 inches d. b. h. and 96.7 feet in 
height. One qualification of any sys¬ 
tem of volume-table construction must 
be its ability to take these values .that 
fall a little from the middle of the class 
they represent and bring them into their 
proper places. 
When diameters taken at intervals 
up the tree are averaged in each diam¬ 
eter and height class, as is the first 
step in the preparation of taper curves, 
there is some question as to the value 
these averages have and how nearly 
they truly represent the midtree of 
the class. This apparently simple 
operation is not without its complexi¬ 
ties. 
Sometimes in getting average diame¬ 
ters well toward the top it will be found 
that certain trees “drop out.” For 
example; if the diameter 78 feet from 
the base in the 80-foot class is being 
sought, some trees will “drop out,” 
being less than 78 feet tall and yet 
over 75 feet tall, and therefore still in 
the 80-foot class. In averaging they 
usually are given a value of zero, but 
it is easily seen that they have a poten¬ 
tially minus value. So, where some of 
the trees have “dropped out” average 
diameters are too high. It is well to 
disregard measurements of diameters 
at heights where all the trees in the 
class are not represented, as they lead 
to error rather than accuracy. 
Aside from these mathematical points 
is the natural fact that the greatest 
variability in trees is toward the tops. 
In any given inch and height class the 
diameter of the trees will vary little 
more than an inch between maximum 
and minimum diameters 8 feet above 
the ground, and perhaps about 2 inches 
at 16 feet up; while 72 feet up in an 
80-foot height class the diameters may 
range over 5 or 6 inches in variance 
even in smooth normal trees. Almost 
every mass of field data contains ab¬ 
normal trees measured too near swell- 
