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# How to Construct a Lorenz Curve and calculate Gini coefficient of a village

In this article,  a Lorenz curve is constructed from the income data of a typical village of Bangladesh. Besides, Gini coefficient is determined from that Lorenz curve and with normal calculation.

Contents

## What is Lorenz Curve:

The Lorenz curve is nothing but a graphical representation of the cumulative distribution function of a probability distribution. Lorenz curve can also be used to show distribution of assets. In such use, many economists consider it to be a measure of social inequality and distribution of income. Lorenz curve can also be used to show distribution of assets. In such use, many economists consider it to be a measure of social inequality. Max O Lorenz developed Lorenz curve in 1905 for the representation of income distribution. A perfectly equal income distribution would be one in which every person in the society has the same income. In this case, the bottom N% people of society would always have N% of the income. This can be shown by the straight line y=x called the line of perfect equality or the 450 line.

## What is Gini Coefficient:

The Gini coefficient is simply the area between the line of perfect equality and the observed Lorenz curve as a percentage of the area between the line of perfect equality and the line of perfect inequality. The Gini coefficient was discovered by the Italian statistician Corrando Gini and published on his famous  paper “Variability and Mutability” in 1912. The Gini coefficient is only a measure of statistical dispersion basically used as a measure of inequality of income distribution or inequality of wealth distribution.  It is well-defined as a ratio with values between 0 and 1.

Thus a low Gini coefficient indicates more equal income or wealth distribution, where a high Gini coefficient indicates more unequal distribution. 0 corresponds to perfect equality (everyone having exactly the same income) and 1 corresponds to perfect equality (where one person has all the income, while everyone else has zero income).

This paper covers how Lorenz curve is constructed from the income data of a particular area and how Gini coefficient is found from that Lorenz curve and without Lorenz Curve. The result is showing the income inequality of the village.

## Materials and Methods for constructing a Lorenz Curve:

The following materials and method is followed to built a Lorenz curve.

### Data Collection for Lorenz Curve:

A survey is conducted in the village of sharaf bhata(kazi Para) to know the income of the households. The sample intensity of data collection was 0.05. Data sorted in ascending order.

### Methods Calculating for Lorenz Curve and Gini Coefficient:

Lorenz Curve is used to find out the Gini coefficient. Gini co efficient indicates how incomes are equally distributed.  The Gini coefficient can be defined from the Lorenz curve diagram. If the area between the line of perfect equality and Lorenz curve is A and the area under the Lorenz curve is B, then the Gini coefficient will be as follows:

Gini coefficient = A/(A+B)

The Gini index is the Gini coefficient expressed as a percentage and is equal to the Gini coefficient multiplied by 100.

From the Lorenz Curve, we find that, A+B=0.5

Now, G=A/0.5

Or, 0.5G=A

Or, (1/2) G=A

Or, G=2A

Now, A+B=0.5=1/2

Or, 2A+2B=1

Or, 2A=1-2B

Or, G=1-2B

Gini coefficient can be calculated without Lorenz Curve which is stated below:

Step 1: The surveyed data is organized into a table with the category head mentioned below.

 Fraction of Income Fraction of Population % of Population that is richer Score

Step 2: Percent of population is to be filled that is richer column by adding all terms in Fraction of Population.

Step 3: The score is to be calculated for each of the rows. The formula for Score is:

Score = Fraction of Income*(Fraction of population + 2* % of Population that is richer)

Step 4: Next, all the terms is to be added in the Score column.

Step 5: The Gini coefficient is to be calculated using this formula: = 1- Sum

## Calculation for Constructing Lorenz Curve:

The income found from the surveyed data is given in the following table which is organized from the lowest to the highest.

 Kazi Noyon: 5000 Kazi Ilias: 30000 Kazi Rafiq: 12000 Kazi Jamshed : 55000 Kazi Liton: 22000 Kazi Bappi: 65000 Kazi Rashed: 22000 Kazi Isa: 72000 Kazi Junayed: 25000 Kazi Arif: 75000

The Total income of the surveyed household is 3,83,000 Taka.

The household is to be divided into quintiles. 10/5= 2 household in each quintile. Noyon and Rafiq compose the lowest quintile or 20% of income earner. Liton and Rashed compose the second quintile or cumulative of 40% of income earners. Junayed and Ilias compose the third quintile or a cumulative of 60% of income earners. Jamshed and Bappi compose the fourth quintile or a cumulative of 80% of income earners and Isa and Arif are the fifth quintile or a cumulative of 100% of income earners.

• The total income is calculated in each quintile

20:17000

40:44000

60:55000

80:120000

100:147000

• The percent of total income is calculated in each quintile

20:17000: 0.044386423

40:44000: 0.114882507

60:55000: 0.143603133

80:120000: 0.313315927

100:147000: 0.38381201

• The percentages approximately taken for easier graphing

20:17000: 0.04

40:44000: 0.12

60:55000: 0.14

80:120000: 0.31

100:147000: 0.38

• The cumulative percentage of household income is calculated

20:17000: 0.04:0.04

40:44000: 0.16

60:55000: 0.30

80:120000: 0.61

100:147000: 1

• Graph quintiles, cumulative percent of income and line of perfect equality
 0 0 0 0.2 0.04 0.2 0.4 0.16 0.4 0.6 0.3 0.6 0.8 0.61 0.8 1 1 1

A graphical presentation is found By using the Chart Wizard select “Scatter Plot” is given below Figure 01: Lorenz Curve of income of a typical village of Bangladesh

• The area under the Lorenz curve (B) is calculated using the properties of a trapezoid.

Area of B= ½(0+O.04)*0.2+ ½(0.04+0.16)*0.2+ ½(0.16+0.30)*0.2+ ½(0.30+0.61)*0.2+ ½(0.61+1)*0.2

=0.322

• Now area of A= 0.5-.322= .178

Gini coefficient= A/(A+B)

= 0.178/0.5

=0.356

We can find the Gini coefficient without Lorenz Curve

Score = Fraction of Income*(Fraction of population + 2* % of Population that is richer)

 Fraction of Income Fraction of Population % of Population that is richer Score 0.01 0.1 0.9 0.019 0.21 0.4 0.5 0.294 0.58 0.4 0.1 0.348 0.20 0.1 0 0.02

Gini Coefficient is (1-0.681) = 0.319

From the value of Gini Coefficient it can be said that the income equality of our village is not much better than other areas. If the value increases, the income inequality will increase.

## Conclusion

Bangladesh is a developing country. The development is not equal in every area of Bangladesh. Gini coefficient of Bangladesh (0.482) indicates that the development have to be distributed equally across the country. The countries with the highest Gini coefficients are South Africa (.625), Haiti (.608), Namibia (.589) etc. On the other hand the lowest Gini coefficients are Sweden (.25), Ukraine (.26), Belgium (.27). It clearly shows that most of the European countries which are economically developed belongs lower Gini coefficient. Developing country like Bangladesh (.482) belong comparatively higher Gini Coefficient. Our economic growth is happened but our economic development haven’t ensured yet. The result found in this article is also shows the same results. So, the development activities of Bangladesh should be equally distributed.

However, by reading this article properly, I think you can make a Lorenz curve and determine Gini coefficient easily.  If you face any problem please feel free to comment below.