Compute the histogram entropy estimation of a string

Python 3.3+, 64 bytes

import math
lambda s:sum(math.log2(len(s)/s.count(c))for c in s)

Got math.log2 from mbomb007's solution.

Jelly, 11 8 bytes

ċЀ÷Ll.S

Try it online!

APL, 18 14 bytes

+/2⍟≢÷(+/∘.=⍨)

This is an unnamed, monadic function train that accepts a string on the right and returns a real.

Like all good things in life, this uses xnor's formula. We get a matrix of booleans corresponding to the occurrences of each character in the string using ∘.=⍨, sum this along the first axis (+/) to get the number of occurrences of each character, divide the length of the string by each, then take log base 2 (2⍟) and sum.

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Saved 4 bytes thanks to Dennis!