Lorenz Curve Calculator Formula

Understand the math behind the lorenz curve calculator. Each variable explained with a worked example.

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Formulas Used

Bottom 20% Share (%)

cum_20 = (v1 / total) * 100

Bottom 40% Share (%)

cum_40 = ((v1 + v2) / total) * 100

Bottom 60% Share (%)

cum_60 = ((v1 + v2 + v3) / total) * 100

Bottom 80% Share (%)

cum_80 = ((v1 + v2 + v3 + v4) / total) * 100

Top 20% Share (%)

top_20_share = (v5 / total) * 100

Variables

VariableDescriptionDefault
v1Income 1 (lowest)10
v2Income 220
v3Income 330
v4Income 440
v5Income 5 (highest)100
totalDerived value= v1 + v2 + v3 + v4 + v5calculated

How It Works

How to Calculate the Lorenz Curve

Method

1. Sort incomes from lowest to highest 2. Calculate cumulative population share: 20%, 40%, 60%, 80%, 100% 3. Calculate cumulative income share at each point 4. Plot cumulative income share (y) vs. cumulative population share (x)

The 45-degree diagonal represents perfect equality. The Lorenz curve bows below; the area between the diagonal and the curve (relative to the total area below the diagonal) is the Gini coefficient.

Worked Example

Five people earn 10, 20, 30, 40, 100.

v1 = 10v2 = 20v3 = 30v4 = 40v5 = 100
  1. 01Total income = 200
  2. 02Bottom 20%: 10/200 = 5%
  3. 03Bottom 40%: 30/200 = 15%
  4. 04Bottom 60%: 60/200 = 30%
  5. 05Bottom 80%: 100/200 = 50%
  6. 06Top 20% earns 100/200 = 50% of total income

Frequently Asked Questions

What does the Lorenz curve show?

It visualizes the distribution of income (or any quantity) across a population. Each point shows what cumulative share of income is held by the bottom x% of the population. A deeper bow below the diagonal means more inequality.

How is the Gini coefficient related to the Lorenz curve?

The Gini coefficient = A / (A + B), where A is the area between the equality line and the Lorenz curve, and A + B is the total area under the equality line (0.5). So Gini = 2A.

Can two distributions have the same Gini but different Lorenz curves?

Yes. If two Lorenz curves cross, they can enclose the same total area with the diagonal but have different shapes. This is why the full Lorenz curve is more informative than the single Gini number.

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