Lorenz curves and the Gini coefficient

Interactive Lorenz curve and Gini coefficient explorer with Idaho and US Census data.

Line up every household from poorest to richest. A Lorenz curve asks: what share of all income goes to the bottom 20%? The bottom 40%? If income were shared equally, the curve would follow the diagonal. The Gini coefficient measures the gap between the diagonal and the curve. It runs from 0 (everyone has the same income) to 1 (one household has everything).

census = ({
  "Idaho": {shares: [3.89, 9.90, 15.70, 23.04, 47.48], gini: 0.4345},
  "United States": {shares: [3.00, 8.55, 14.39, 22.59, 51.47], gini: 0.4809}
})
lorenz = shares => {
  const total = d3.sum(shares)
  let cum = 0
  return [{x: 0, y: 0}, ...shares.map((s, i) => ({x: (i + 1) / 5, y: (cum += s / total)}))]
}
// Trapezoid rule on five quintile points
giniOf = pts => 1 - d3.sum(pts.slice(1), (p, i) => (p.x - pts[i].x) * (p.y + pts[i].y))

viewof place = Inputs.radio(["Idaho", "United States"], {value: "Idaho", label: "Compare with"})

Build your own economy. Here is the average household income in each fifth, in thousands of dollars:

viewof q1 = Inputs.range([0, 400], {step: 5, value: 15, label: "Poorest 20%"})
viewof q2 = Inputs.range([0, 400], {step: 5, value: 40, label: "Next 20%"})
viewof q3 = Inputs.range([0, 400], {step: 5, value: 65, label: "Middle 20%"})
viewof q4 = Inputs.range([0, 400], {step: 5, value: 100, label: "Next 20%"})
viewof q5 = Inputs.range([0, 400], {step: 5, value: 230, label: "Richest 20%"})

mine = lorenz([q1, q2, q3, q4, q5])
theirs = lorenz(census[place].shares)
myGini = d3.sum([q1, q2, q3, q4, q5]) > 0 ? giniOf(mine) : NaN
theirGini = giniOf(theirs)
Plot.plot({
  width: 560,
  height: 520,
  marginLeft: 56,
  marginBottom: 46,
  style: {fontSize: "13px", background: "transparent"},
  x: {domain: [0, 1], label: "Share of households, poorest to richest →", tickFormat: "%"},
  y: {domain: [0, 1], label: "↑ Share of all income", tickFormat: "%", grid: true},
  marks: [
    Plot.areaY(theirs, {x: "x", y1: "y", y2: "x", fill: "#176b87", fillOpacity: 0.12}),
    Plot.line([{x: 0, y: 0}, {x: 1, y: 1}], {x: "x", y: "y", stroke: "#597287", strokeDasharray: "5 5"}),
    Plot.line(theirs, {x: "x", y: "y", stroke: "#176b87", strokeWidth: 3}),
    Plot.dot(theirs, {x: "x", y: "y", fill: "#176b87", r: 4}),
    Plot.line(mine, {x: "x", y: "y", stroke: "#2f7d5b", strokeWidth: 3, strokeDasharray: "8 4"}),
    Plot.dot(mine, {x: "x", y: "y", fill: "#2f7d5b", r: 4}),
    Plot.text([{x: 0.55, y: 0.6}], {x: "x", y: "y", text: () => "Perfect equality", rotate: -45, fill: "#597287", dy: -8}),
    Plot.text([{x: 0.8, y: theirs[4].y}], {x: "x", y: "y", text: () => place, dx: 8, dy: 14, textAnchor: "start", fill: "#176b87", fontWeight: 700}),
    Plot.text([{x: 0.6, y: mine[3].y}], {x: "x", y: "y", text: () => "Your economy", dx: 8, dy: 14, textAnchor: "start", fill: "#2f7d5b", fontWeight: 700})
  ]
})

html`<div class="stat-row">
  <div class="stat" style="border-left-color:#2f7d5b"><div class="k">Your economy · Gini</div><div class="v">${isNaN(myGini) ? "—" : myGini.toFixed(3)}</div></div>
  <div class="stat"><div class="k">${place} · Gini from these 5 points</div><div class="v">${theirGini.toFixed(3)}</div></div>
  <div class="stat"><div class="k">${place} · official Census Gini</div><div class="v">${census[place].gini.toFixed(3)}</div></div>
  <div class="stat" style="border-left-color:#e5ad3d"><div class="k">Richest 20% / poorest 20% (yours)</div><div class="v">${q1 > 0 ? (q5 / q1).toFixed(1) + "×" : "—"}</div></div>
</div>`

Try this

Set all five sliders to the same income. What does the curve look like, and what is the Gini?
Now double every household’s income. Does the Gini change? What does that tell you about what the Gini measures, and what it doesn’t?
Take $20k from the richest fifth and give it to the poorest fifth. How far does the Gini fall? Now move $20k from the middle fifth to the next fifth. Which transfer matters more?
Try to match Idaho’s curve. Then try to match the United States. What separates the two?
Why is the official Census Gini higher than the one computed from just five points? (Hint: what’s hidden inside each fifth?)

Data: U.S. Census Bureau, American Community Survey 2024 1-year estimates, tables B19082 and B19083. Margins of error on the Gini (90% confidence): Idaho ±0.007, United States ±0.001.