Understanding the quality of the KMeans algorithm

Question

After reading Unbalanced factor of KMeans, I am trying to understand how this works. I mean, from my examples, I can see that the less the value of the factor, the better the quality of KMeans clustering, i.e. the more balanced are its clusters. But what is the naked mathematical interpretation of this factor? Is this a known quantity or something?

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Here are my examples:

C1 = 10
C2 = 100

pdd = [(C1,10), (C2, 100)]
n = 2        <-- #clusters
total = 110  <-- #points
uf = 10 * 10 + 100 * 100
uf = 100100 * 2 / 12100 = 16.5

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C1 = 50
C2 = 60

pdd = [(C1, 50), (C2, 60)]
n = 2        
total = 110  
uf = 2500 + 3600
uf = 6100 * 2 / 12100 = 1.008

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C1 = 1
C2 = 1

pdd = [(C1, 1), (C2, 1)]
n = 2       
total = 2
uf = 2
uf = 2 * 2 / 2 * 2 = 1

Answer 1

It appears to be related to Gini index, a measure of entropy, which also uses the sum of squared counts.

as said in Cross Validated: Understanding the quality of the KMeans algorithm.