Variance of an index of dispersion

I wish to test a hypothesis that neighborhood organizations are more
racially homogeneous than the neighborhoods that they represent.

Because race is a categorical variable I am measuring homogeneity with
an index of dispersion called the Lieberson Index. The formula sums
the squared proportion of each racial group and substracts that figure
from one:

L = 1 - summation[(n(i)/N)^2]

Where n(i) is number of members in group i
N = total number of all members

Essentially it looks at all possible dyads (including dyads with
oneself) in a population, and calculates the proportion of dyads that
include individuals from differing racial categories.

To do a statistical test, I not only need this index, but also its
variance, and this raises some issues.

1. First there is a conceptual issue. Does it make sense to have a
variance of a measure of dispersion, since dispersion is already a type
of measure of variance?

2. If this is not an issue, this is how I would calculate a variance,
but I am not sure that I am correct.

The Lieberson index is essentially looks at a complete network between
all individuals in that group, and calculates the proportion of ties
between individuals with differing characteristic (there are N^2 ties
instead of N*(n-1) ties since it implicitly assumes that the groups are
so large that counting ties to oneself does not alter the statistic
appreciably).

If you just considered the statistic, L, a proportion in a population
of N^2 ties, then the variance would simply by L(1-L). Does it make
sense to use that variance to describe the index L which just describes
a population of N group members?

Any help would be greatly appreciated

Chris

.

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