Page 319 - Becoming Metric Wise

311
The Informetric Laws

Finally, we come to the most important requirement, known as
Dalton’s transfer principle (Dalton, 1920). In terms of monetary values
this requirement states that if one takes money from a poorer person and
gives it to a richer one (a so-called elementary transfer) then the concen-
tration of richness increases. Mathematically this requirement can be
expressed as follows:
If x i # x j and 0 , k # x i then

Fx 1 ; .. .; x i ; .. .; x j ; .. .; x N , Fx 1 ; .. .; x i 2 k; .. .; x j 1 k; .. .; x N
(9.22)
The transformation from the left-hand side of (9.22) to the right-hand
side is called an elementary transfer. The fact that the Lorenz curve is the
correct tool to describe concentration follows from the following
theorem.

Theorem
(Muirhead, 1903): Let X 5 (x 1 ,.. .,x N ) and Y 5 (y 1 ,.. .,y N ) be decreasing
arrays and let L(X) and L(Y) be their corresponding Lorenz curves. If
L(X) is situated above L(Y) (but not coinciding) then X can be generated
from Y by a finite number of elementary transfers. The reverse is also
true.
Besides the Gini index (see Subsection 4.10.3) also the coefficient of
variation defined as the standard deviation divided by the arithmetic
mean is a measure which, for fixed N, satisfies all the requirements men-
tioned above. Another one is the Simpson index, also known as the
Hirschmann-Herfindahl index. This measure, denoted as λ, is defined as:

N N N
X 2 X X
x i
2
λðXÞ 5 5 p with M 5 x i the total number of itemsÞ
ð
M i
i51 i51 i51
(9.23)
9.6 MEASURING DIVERSITY FOR A VARIABLE NUMBER OF
CELLS

If F is an acceptable measure of concentration which never takes the
value 0, then G 5 1/F is an acceptable measure of diversity. Similarly, if F
is an acceptable measure of concentration, bounded by 1, then G 5 1 2 F
is an acceptable measure of diversity.