How can I explain my logic? - Related to Herfindahl index I've tried to measure something that I have in mind.
My problem is as following:


*

*Let's assume that there is a group with 8 members.

*There are two cases:
First, A group consists of 4 subgroups each with 1,1,2, and 4 members.(1+1+2+4=8)
Second, A group consists of 4 subgroups each with 2 members (2+2+2+2=8)


I want to calculate the relative size of a subgroup with 2 members.
If I calculated it as 2/8, it does not reflect the size of the other subgroups.
In the first case, a subgroup with 2 members is likely to be a minor subgroups because of a major subgroup with 4 members.
But in the latter case, a subgroup with 2 members is not either minor or major subgroup because other subgroups have 2 members.
To reflect this concern, I try to adopt Herfindahl index.
First of all, divide the number of members of subgroups by total members(=8).
For example, in the group with 1,1,2,4 members, the value is 0.125, 0.125, 0.25, 0.5.
Then, I calculate the square value (0.0156,0.0156,0.0625,0.25). The sum of it is Herfindahl index as you know. 
The important things, here, is that I divide the square value by sum of square value. 
In the end, the 2-member subgroup in the group with 1,1,2,4 members have the value-0.181818.
Contrary to it, the 2-member subgroup in the group with 2,2,2,2 members have the value-0.25
I think it can adequately reflect my thought because the 2 member subgroup in the former case (=0.1818) has relatively smaller size than in the latter group (=0.25).
I can understand it intuitively but I cannot explain it logically.
Maybe one reason might be I cannot find the reference.
Calculating Herfindahl index is common, but dividing the fraction by the index is unusual to me.
Does anyone know the similar situation? Does it make sense?
If you know any reference or have any recommendation to my logic, please let me know.
Thanks a lot!
 A: The Herfindahl index, also known as the Herfindahl–Hirschman Index (or HHI), was developed as an economic measure of competitiveness in a market.  It is the sum of squares of market shares for all (or all the largest) firms competing in that market, and it can be understood as a modified measure of average market share.  
As a reference, the paper Concentration in the Property and Liability Insurance Market by Line of Insurance by Nissan and Caveny (J. of Actuarial Practice, 2000) may be useful.
In economic/investment calculations the values are (arbitrarily) scaled up by a factor of 10,000, but we will omit this.
To understand this better, suppose first that we were to calculate "average market share" simply by adding the market shares and dividing by the number of firms $N$.  Since the sum of all market shares is simply $1$ in all cases, this "measure" of average market share would always equal $1/N$, regardless of whether there is rough equality in market shares or great disparity.
Instead the Herfindahl index captures the notion of a value that is $1/N$ when all the competitors have equal market share but increases (with an upper bound of $1$) as the market shares are unequally held by larger and smaller firms.
Letting $s_i$ denote the (fractional) market share held by the $i$th firm (out of $N$), the Herfindahl index can be written:
$$ \sum_{i=1}^N s_i^2 = \frac{1}{N} + \sum_{i=1}^N (s_i - 1/N)^2 $$
Thus the Herfindahl index is always bounded below by $1/N$ and attains this minimum only when all market shares are equal.  Moreover the increase of the index above that lower threshold can be described as $N$ times the variance of these shares.  The Herfindahl index tends toward a maximum of $1$ (but cannot exceed this) as some single firm holds almost the entire market.
You express that it seems unusual to take a ratio of the fraction of one subset size to the Herfindahl index, but the effect of this mathematically is to create a comparison of the particular subset's size to the "average subset size" as measured by the Herfindahl index.  In this way it seems a reasonable approach, one that would have the sort of characteristics I would expect for a comparison that differentiates "major and minor" subgroups (subsets) across a variety of partitions of sets.
In this connection it can be noted that the reciprocal of HHI is often taken as a revised measure of the "effective" number of competing firms, $N_{EFF}$.  A similar use is made in portfolio theory to count the "effective" size of an investment portfolio (distinct number of holdings).  Thus instead of thinking of dividing by HHI, it may be more intuitive to consider that you are multiplying by the reciprocal, $N_{EFF}$, to gauge the size of a subset.
