THE STATISTICAL STUDY OF VARIATION 37 
data may be compared with other similar data some sort of precise de- 
scrtption must be prepared. Graphical representation is good as far as it 
goes; a frequency polygon conveys to the eye more knowledge than one 
would have without it. But in order to secure the best description of 
organisms with reference to specified characters, some mathematical 
expression for the degree of variation must be deduced from the data. 
This process involves two essential steps: (1) To obtain a measure of 
type for the group under observation; (2) to derive an expression for the 
amount of variation from the type. There are three measures of type, 
the median, the mode and the mean, and we have seen that in the theo- 
retical normal curve they always coincide. In actual cases they may be 
widely separated. There are three commonly used measures of variation 
from type, viz., the range, or the distance from one extreme to the other, 
the quartile, and the standard deviation. These expressions and others 
derived from them are known as the constants of the normal curve. In 
practical work the mean, or arithmetical average, is commonly used as 
the measure of type and the standard deviation as the absolute measure 
of variation. A relative index of degree of variation is derived by divid- 
ing the standard deviation by the mean; this is called the coefficient of 
variation. These three constants are the indispensable mathematical 
tools of the biometrician. Some knowledge of their calculation and 
significance is necessary for an intelligent appreciation of considerable 
important biological and agricultural literature. Before proceeding to 
discuss these constants it will be necessary to present a few technical 
terms and methods. 
Some Biometrical Terms.—An Individual may be either an entire 
organism or only a single part as the leaves of a tree or seeds of aplant. 
Individuals are also called variables. 
A Sample is any group of individuals which are measured or com- 
pared with a standard. Samples may be divided into sub-samples for 
definite reasons; for example, corn from different parts of a field. 
The Population is the general mass or entire group from which sam- 
ples are taken. 
A Variate is a single magnitude-determination of a character. 
A Class includes variates of the same or nearly the same magnitude. 
The class range gives the limits between which the variates of any class 
fall. 
Requisites to Reliability—1. Biological Souwndness.—Three great 
sources of untrustworthiness in biological work are: 
(a) Differences due to age; different ages must not be lumped to- 
gether without taking account of it. 
(b) Heterogeneity due to conditions of environment; for example, 
corn from a field in which the soil is definitely heterogeneous. 
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