Question

The following table shows income distribution data for an economy in a particular year. Household Group Share of Aggregate Income One-fifth with lowest income 4.7% Next lowest one-fifth 8.5% Middle one-fifth 19.4% Next highest one-fifth 27.6% One-fifth with highest income 39.8% What is the Gini coefficient?

Answer 1

The Gini index or Gini coefficient is a statistical measure of distribution which measures income distribution among a population. The coefficient ranges from 0 (or 0%) to 1 (or 100%), with 0 representing perfect equality and 1 representing perfect inequality. The Gini coefficient is calculated as the measure of deviation from the perfect equality.

The mathematical calculation for the same –

The Gini coefficient is the ratio of the area that lies between the line of equality and the Lorenz curve. G = A/ (A+B) is the mathematical formula for the calculation of Gini Coefficient.

The data provided in the question is as follows –

· The lowest 1/5^th of population i.e. 20% makes an income of 4.7%

· Next 1/5^th of population i.e. 20% makes an income of 8.5%

· Next 20% population makes an income of 19.4%

· Next 20% population makes an income of 27.6%

· Highest 20% population makes an income of 39.8%

We would be calculating the Gini Coefficient in an arithmetic manner –

Step 1 – Organising the data into the table as follows –

Fraction of Income	Fraction of Population	% of population that is Richer	Score
4.7/100 = 0.047	0.20
8.5/100 = 0.085	0.20
19.4/100 = 0.194	0.20
27.6/100 = 0.276	0.20
39.8/100 = 0.398	0.20

Step 2 - Fill in the “% of Population that is Richer” column by adding all the “Fraction of Population” terms below that row

Fraction of Income	Fraction of Population	% of population that is Richer	Score
0.047	0.20	0.80 (0.20+0.20+0.20+0.20)
0.085	0.20	0.60 (0.20+0.20+0.20)
0.194	0.20	0.40 (0.20+0.20)
0.276	0.20	0.20 (the last bracket which is earning higher)
0.398	0.20	0 (as this percentage is earning highest income)

Step 3 - Score = Fraction of Income * (Fraction of Population + 2 * % of Population Richer)

1. 0.047 * (0.20 + 2*0.80)

0.047 * (0.20 + 1.6)

0.047 * 1.8

0.0846

2. 0.085 * (0.20 + 2*0.60) = 0.199

3. 0.194 * (0.20 + 2*0.40) = 0.194

4. 0.276 * (0.20 + 2*0.20) = 0.1656

5. 0.398 * (0.20 + 2*0) = 0.0796

Fraction of Income	Fraction of Population	% of population that is Richer	Score
0.047	0.20	0.80 (0.20+0.20+0.20+0.20)	0.0846
0.085	0.20	0.60 (0.20+0.20+0.20)	0.199
0.194	0.20	0.40 (0.20+0.20)	0.194
0.276	0.20	0.20 (the last bracket which is earning higher)	0.1656
0.398	0.20	0 (as this percentage is earning highest income)	0.0796

Step 4 - Add up all the terms in the “Score” column. Call that “Total”.

Here, we get – 0.0846+0.199+0.194+0.1656+0.0796 = 0.7228

Step 5 - Gini Coefficient = 1 – Total.

Gini Coefficient = 1-0.7228 = 0.2772

The Gini index or Gini coefficient is a statistical measure of distribution which measures income distribution among a population. The coefficient ranges from 0 (or 0%) to 1 (or 100%), with 0 representing perfect equality and 1 representing perfect inequality. The Gini coefficient is calculated as the measure of deviation from the perfect equality.

The mathematical calculation for the same –

The Gini coefficient is the ratio of the area that lies between the line of equality and the Lorenz curve. G = A/ (A+B) is the mathematical formula for the calculation of Gini Coefficient.

The data provided in the question is as follows –

· The lowest 1/5^th of population i.e. 20% makes an income of 4.7%

· Next 1/5^th of population i.e. 20% makes an income of 8.5%

· Next 20% population makes an income of 19.4%

· Next 20% population makes an income of 27.6%

· Highest 20% population makes an income of 39.8%

We would be calculating the Gini Coefficient in an arithmetic manner –

Step 1 – Organising the data into the table as follows –

Fraction of Income	Fraction of Population	% of population that is Richer	Score
4.7/100 = 0.047	0.20
8.5/100 = 0.085	0.20
19.4/100 = 0.194	0.20
27.6/100 = 0.276	0.20
39.8/100 = 0.398	0.20

Step 2 - Fill in the “% of Population that is Richer” column by adding all the “Fraction of Population” terms below that row

Fraction of Income	Fraction of Population	% of population that is Richer	Score
0.047	0.20	0.80 (0.20+0.20+0.20+0.20)
0.085	0.20	0.60 (0.20+0.20+0.20)
0.194	0.20	0.40 (0.20+0.20)
0.276	0.20	0.20 (the last bracket which is earning higher)
0.398	0.20	0 (as this percentage is earning highest income)

Step 3 - Score = Fraction of Income * (Fraction of Population + 2 * % of Population Richer)

1. 0.047 * (0.20 + 2*0.80)

0.047 * (0.20 + 1.6)

0.047 * 1.8

0.0846

2. 0.085 * (0.20 + 2*0.60) = 0.199

3. 0.194 * (0.20 + 2*0.40) = 0.194

4. 0.276 * (0.20 + 2*0.20) = 0.1656

5. 0.398 * (0.20 + 2*0) = 0.0796

Fraction of Income	Fraction of Population	% of population that is Richer	Score
0.047	0.20	0.80 (0.20+0.20+0.20+0.20)	0.0846
0.085	0.20	0.60 (0.20+0.20+0.20)	0.199
0.194	0.20	0.40 (0.20+0.20)	0.194
0.276	0.20	0.20 (the last bracket which is earning higher)	0.1656
0.398	0.20	0 (as this percentage is earning highest income)	0.0796

Step 4 - Add up all the terms in the “Score” column. Call that “Total”.

Here, we get – 0.0846+0.199+0.194+0.1656+0.0796 = 0.7228

Step 5 - Gini Coefficient = 1 – Total.

Gini Coefficient = 1-0.7228 = 0.2772

Gini Coefficient = 0.2772