What the Gini index is and why you need an adjusted one

Say you are looking at these numbers

Metric	Value
Posts	50
Authors	5
Views	1,000,000

Metric	Value
Views per post	20,000
Views per author	200,000

Can you draw conclusions from this? Probably, but they will not be objective.

Per author the numbers look like this:

Author	Posts	Views	Share
@alice	10	990,000	99.00%
@bob	10	2,500	0.25%
@carol	10	2,500	0.25%
@dave	10	2,500	0.25%
@erin	10	2,500	0.25%

Views by author: four bars are invisible - they are the 1%.

For a saner picture we can look at percentiles

Percentile	Views per author
10	2,500
25	2,500
50 (median)	2,500
75	2,500
mean	200,000

Now we know for sure that 75% of authors get 2,500 views or fewer. But what if we want to express this as a single number? That is where the Gini index helps.

The Gini index is a measure of distribution inequality. 0 means every author collects the same, 1 means one author takes everything.

G = \frac{2 \sum_{i=1}^{n} i \, x_i}{n \sum_{i=1}^{n} x_i} - \frac{n+1}{n}

Symbol	Meaning
$n$	number of authors in the community
$x_i$	views (or posts) of the author in $i$ -th place, with all authors sorted ascending: $x_1 \le x_2 \le \dots \le x_n$
$i$	the author's position in that sorted list, from 1 to $n$
$\sum_{i=1}^{n} x_i$	total views of the community

As an example, take this community: The Startup.

The standard formula was meant for large datasets, so when we apply it to X communities with few authors the numbers come out wrong. For example:

Let $x = [0, 0, 0, 0, 100]$ , so $n = 5$ . In the sum $\sum i \, x_i$ only the last term is non-zero, $5 \cdot 100 = 500$ , and total views are 100:

G = \frac{2 \cdot 500}{5 \cdot 100} - \frac{6}{5} = 2 - 1.2 = 0.8

Even with a total skew toward one author, the index comes out at 0.8.

The maximum of the index over $n$ authors is $\frac{n-1}{n}$ , not 1.

Authors	Maximum Gini	Error
2	0.500	50.0%
3	0.667	33.3%
5	0.800	20.0%
10	0.900	10.0%
20	0.950	5.0%
50	0.980	2.0%
100	0.990	1.0%

Shortfall to 1 under a total skew. The error is noticeable for fewer than 50 authors.

So for fewer than 50 authors we use the adjusted Gini index:

G_{adj} = G \cdot \frac{n}{n-1}

On the same example: $G_{adj} = 0.8 \cdot \frac{5}{4} = 1.0$ .

Use x-community.top for proper community analysis.