# Variance of the Index of Coincidence

This concerns Friedman's Index of Coincidence (IC) in the
form (c/(N(N-1))) sum{n_i(n_i - 1), i = 1..c), as at
http://en.wikipedia.org/wiki/Index_of_coincidence
and in LANAKI at
http://www.und.nodak.edu/org/crypto/crypto/
lanaki.crypt.class/docs/misc/index-of-coincidence-note

Assuming the ciphertext alphabet has c letters that
occur (independently) with probabilities p_1,...,p_c,
the expected value of IC is c * sum(p_i^2, i=1..c).

I would like to know the formula for the variance of IC,
similarly as a function of c and p_1,...,p_c.

.

## Relevant Pages

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