The Canonical Distribution of Commonness and Rarity: Part I F. W ...

ECOLOGY
iro~. 43 SPRIKG 1962 No. 2
THE CXNONICPlL DISTRIBUTION OF COMhiONNESS AND RARITY: PART I
F. Ji. PRESTOS
Prc sfoli Lal~oriifi~rlis, Blrtlrr. Prrirts~ l~~riiio
In an earlier paper ( Preston 1948 ) we found
that, in a sufficiently large aggregation of individuals
of many species, the individuals often
tended to be distributed among the 5pecies according
to a lognormal law. We plotted as abscissa
equal increments in the logarithms of the number
of individuals representing a species, and as orclinate
the number of species falling into each of
these increments. We found it collvenient to use
as such increments the "octave," that is the interval
in which representation donbled, so that
our abscissae became simply a scale of "octaves,"
but this choice of unit is arbitrary. IVhatever
logarithmic unit is used, the graph tended to talie
the form ot a iiornlal or Gaussian curve, so that
the d~str~bution was "logi~orn~al." 11-e called this
the "Sprcies Curve."
In the present paper we take up a point nierely
nientioiied in 194s that not only is the distribution
log~~ormal. but the constants or parameters seein
to he restricted in a peculiar way. They are not
fixed, but they are interlocked. The nature of
this restriction and iriterlocking 1s the main theme
of the present paper.
I11 the earlier paper we graduated the experimental
results with curves of the form
= e-':'~"
0
(1)
where y is the number of species falling into the
Rth "octave" to the right or left of the mode, yo
is the number in the modal octave, and a was
treated as an arbitrary constant, to be found from
the experimental evidence. This constant is related
to the logarithmic standard deviation a by
the formula
a2 = Ma2 (2)
and we noted that it had a pronoutlced tendency
to come out at a figure not far from 0.2, so that a
would have a value of about 3.5 octaves, and, since
there are 3.3 octaves to an "order of magnitude,"
a would be a little more than an order of magnitude.
If we now make a 2nd graph in which, using
the same abscissae, we plot as ordinate not the
number of species (y) that fall in each interval
but the nunlber of individuals which those y species
comprise, we get another lognormal cnrve
with the same standard deviation as the first
graph, but with its mode or peak displaced to the
right. This we call the "Individuals Curve."
Though we can use a Gaussian curve to "graduate"
the observed points of the species curve, the
curve extends infinitely far to left and right, while
the number of observed points is necessarily finite.
This is a common situation in statistical worlc and
usuallj causes no complications, hut in our problem
there results an additional piece of information.
The Individuals Curve ~iecessarily lacks at
least part of the descending limb. It terminates
over the last observed point of the Species Curve,
and this is long before the Individuals Curve begins
to l~ecorne asymptotic to the horizontal axis.
111 the earlier paper we noted that, as a matter of
obser~.ation,it seenls to terminate at its crest, so
that, in effect, only half of the curve is present.
\Yhat I failed to observe in 1938 was that when
the Individuals Curve terminates at its crest or
very close to it, the value of "a" in equation (1)
and of the standard deviation "a," is fixed within
narrow limits, and this value is in fact the one
actually observed. This does not mean that "a"
is a true constant, but only that it is not iudependent
of y, or of the total number of species, N.
It may be said that "a" is a function of yo, so that
given one, the other is settled. Thus we are reduced
from 2 seemingly disposable parameters or
constants to one. More generally, given any
one piece of information about our collection, for
instance given either the total number of species