Question 5.20 Choose the correct answerThe formal calculation of the Gini coefficient is as follows: (i) Find the difference in income between every possible pair in the population, (ii) Take the mean of these differences, (iii) Divide (ii) by the mean income of the population and (iv) The Gini coefficient is (iii) divided by 2. Consider a population with three members and an aggregate income of 100. Read the following statements and select the correct one.

Question

BlackTom AI · Accepted Answer

Gini coefficient equals half the relative mean absolute difference: G = (mean of all pairwise absolute differences) / (2 × mean income).

1) Relevant concepts
- Mean income μ = total income / n.
- Mean absolute difference = (1/n^2) Σ_{i,j} |x_i − x_j| (summing over all ordered pairs).
- Gini = (mean absolute difference) / (2μ). This matches the steps given in the question.

2) Step-by-step calculation for (100, 0, 0)
- n = 3, total = 100 ⇒ μ = 100/3 ≈ 33.3333.
- Compute |x_i − x_j| for all ordered pairs (3×3 = 9):
  (100,100):0; (100,0):100; (100,0):100;
  (0,100):100; (0,0):0; (0,0):0;
  (0,100):100; (0,0):0; (0,0):0.
- Sum of differences = 100+100+100+100 = 400.
- Mean absolute difference = 400 / 9 ≈ 44.4444.
- Relative mean difference = 44.4444 / 33.3333 = 1.333333.
- Gini = 1.333333 / 2 = 0.6666667 ≈ 0.67.

3) Intermediate steps shown above: sums, division by 9, division by μ, then by 2.

4) Why this answer is correct
- The numeric steps directly follow the formal procedure given in the question. For (100,0,0) the pairwise differences are large, producing a relative mean difference of 4/3 and thus Gini = 2/3 ≈ 0.67. This matches option b.

5) Why the other options are incorrect
- (a) (50,50,0): Sum of ordered differences = 200 ⇒ mean difference = 200/9 ≈ 22.2222; relative = 22.2222/33.3333 = 0.6667; Gini = 0.3333, not 0.5.
- (c) (60,20,20): Sum of ordered differences = 160 ⇒ mean = 160/9 ≈ 17.7778; relative = 17.7778/33.3333 = 0.5333; Gini = 0.2667. So its Gini is lower, not “higher.”
- (d) (100/3,100/3,100/3): all equal ⇒ all pairwise differences = 0 ⇒ Gini = 0 (perfect equality), not 1.

6) Conclusion
Calculated Gini for (100,0,0) is 0.67, so option b is correct — this matches the provided answer.

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