Recall the observation made in class
that the distribution of size in a rather
startlingly diverse collection of sets of objects
follows an approximate pattern where if the largest object in the set
is assigned size 1, the next largest is about size 1/2, the next 1/3,
then 1/4, 1/5, 1/6, ...
asteroid diameters, mineral deposits, company sizes, wealth distribution,
country sizes, speakers of various languages, sizes of lakes ...
. Sets that follow this pattern tend to have the following properties:
1) There is no strong process that controls size or sets upper or lower bounds (near the range of interest) for the objects involved, so that the sizes range over at least a few orders of magnitude
2) The number of known objects in the set is fairly large compared to the sample of top members being considered.
So, for example, the height of people, which is strongly constrained by biology about a mean value, does not follow this pattern.
The sequence has the property that an unbounded value may be obtained by summing elements, but it takes an exponentially increasing number of elements to get each incremental increase. Specifically, the number of elements needed increases by a factor of about e (2.7) for each unit increase in the sum. Thus the sum of the first N elements of the sequence can be approximated by ln(N) (or more precisely, ln(N) + 1), where ln(N) is log to the base e or natural log. This can be useful if you want to make a guess at the total amount of a resource available from the top N sources, if you only know the size of the largest source.
You will be assigned one of the following classes.
Research the sizes of the top 10-20 objects in
your set, and scale them so that the largest has size 1.
Compare the esquence of numbers to the sequence
1.0, 0.5., 0.33, 0.25, 0.20, 0.167, 0.143, ... and the sequence
1.0, 0.66, 0.50, 0.40, 0.33, 0.286, 0.250, ...
which is the same sequence starting at .5 instead of 1 and scaled up.
You can look at some of the other examples if you want.
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