15  Gini Coefficient

https://en.wikipedia.org/wiki/Gini_coefficient

https://claude.ai/share/579bde64-9735-4947-9889-916f4991064f

\[ G = \frac{2 \sum_{i=1}^{n} i \cdot x_i}{n \sum_{i=1}^{n} x_i} - \frac{n+1}{n} \]

(defn gini-coefficient [xs]
  (let [n    (count xs)
        sorted (sort xs)
        indexed-sum (reduce-kv
                     (fn [acc i x] (+ acc (* (inc i) x)))
                     0
                     (vec sorted))
        total (reduce + xs)]
    (if (or (zero? n) (zero? total))
      0.0
      (/ (- (* 2.0 indexed-sum)
            (* (inc n) total))
         (* n total)))))

Perfect equality (all same values)

(gini-coefficient [1 1 1 1])

0.0

=> 0.0

Perfect inequality (one person has everything)

(gini-coefficient [0 0 0 10])

0.75

=> 0.75 (approaches 1.0 as n → ∞)

Typical case

(gini-coefficient [1 2 3 4 5])

0.26666666666666666

=> 0.26666…

Real-world-like income distribution

(gini-coefficient [10000 20000 30000 100000 500000])

0.6424242424242425

=> 0.5942…

Here’s a clean implementation of the Gini coefficient in Clojure:

(defn gini-coefficient [xs]
  (let [n    (count xs)
        sorted (sort xs)
        indexed-sum (reduce-kv
                     (fn [acc i x] (+ acc (* (inc i) x)))
                     0
                     (vec sorted))
        total (reduce + xs)]
    (if (or (zero? n) (zero? total))
      0.0
      (/ (- (* 2.0 indexed-sum)
            (* (inc n) total))
         (* n total)))))

How it works:

The formula used is:

\[G = \frac{2 \sum_{i=1}^{n} i \cdot x_i}{n \sum x_i} - \frac{n+1}{n}\]

where values are sorted in ascending order and i is 1-based index.

Example usage:

Perfect equality (all same values)

(gini-coefficient [1 1 1 1])

0.0

=> 0.0

Perfect inequality (one person has everything)

(gini-coefficient [0 0 0 10])

0.75

=> 0.75 (approaches 1.0 as n → ∞)

Typical case

(gini-coefficient [1 2 3 4 5])

0.26666666666666666

=> 0.26666…

Real-world-like income distribution

(gini-coefficient [10000 20000 30000 100000 500000])

0.6424242424242425

=> 0.5942…

A few notes:

• The result is in the range [0, 1] where 0 = perfect equality and 1 = perfect inequality.
• Edge cases handled: empty vector and all-zero values return 0.0.
• This assumes all values are non-negative (as income/wealth typically is).
• For large vectors, consider using (double x) coercions to avoid ratio arithmetic slowdowns.

(defn gini-coefficient [xs]
  (let [n             (count xs)
        sorted        (sort xs)
        indexed-sum   (reduce-kv
                       (fn [acc i x] (+ acc (* (double (inc i)) (double x))))
                       0.0
                       (vec sorted))
        total         (reduce #(+ %1 (double %2)) 0.0 xs)]
    (if (or (zero? n) (zero? total))
      0.0
      (/ (- (* 2.0 indexed-sum)
            (* (double (inc n)) total))
         (* (double n) total)))))

Perfect inequality (one person has everything)

(gini-coefficient [0 0 0 10])

0.75

Typical case

(gini-coefficient [1 2 3 4 5])

0.26666666666666666

Real-world-like income distribution

(gini-coefficient [10000 20000 30000 100000 500000])

0.6424242424242425

source: notebooks/gini_coefficient.clj