Revision sheet: Understanding Income Distributions

📋 Course Outline

1. Income histograms and log scale
2. Density plots of income
3. ECDFs and country comparison

📖 1. Income histograms and log scale

🔑 Key Concepts & Definitions

• Income histogram : An income histogram is a bar chart that groups income values into intervals and shows how often each interval occurs as frequency.
• Log income : Log income is an income variable transformed by taking its logarithm so income changes are displayed on a multiplicative scale.

📝 Essential Points

• The histograms in the material use Income on the horizontal axis and Frequency on the vertical axis for Europe incomes.
• One histogram switches to Log income on the horizontal axis while keeping Frequency on the vertical axis for Europe incomes.

💡 Memory Hook

Log scale turns “multiplying income” into “adding distance” on the x-axis, so spread at high incomes looks less compressed.

📖 2. Density plots of income

🔑 Key Concepts & Definitions

• Income density : Income density is a distribution summary that describes how concentrated income values are across possible incomes.
• Density plot of income : A density plot of income is a smooth curve replacing histogram bars, showing the estimated income density across the income axis.

📝 Essential Points

• The density plots label Income on the horizontal axis and Income Density on the vertical axis for Europe incomes.
• The material also provides density plots for the UK with Income on the horizontal axis and Income Density on the vertical axis.

💡 Memory Hook

Density plots are “histograms with no bins”: smoother curve, same idea of distribution shape.

📖 3. ECDFs and country comparison

🔑 Key Concepts & Definitions

• ECDF : An ECDF is an empirical cumulative distribution function that gives the proportion of observations at or below each income level.
• Empirical cumulative distribution of income : The empirical cumulative distribution of income is the step-by-step cumulative fraction computed from observed incomes, displayed as ECDF versus income.

📝 Essential Points

• The ECDF figure uses Income on the horizontal axis and ECDF (empirical cumulative distribution) on the vertical axis for the UK.
• A country comparison overlays ECDF curves for IT and UK on the same axes to compare cumulative income patterns.

💡 Memory Hook

ECDF answers: “What share of people earn ≤ this income?” and the curve climbs toward 1.

⚠️ Common Pitfalls & Confusions

1. Switching from Income to Log income changes the meaning of the x-axis, so the same numerical tick marks do not represent the same income values.
2. Confusing Frequency (histogram counts) with Income Density (density curve height) leads to incorrect interpretations of which distribution places more mass.
3. Thinking the ECDF is a probability density instead of a cumulative proportion makes you read the y-axis incorrectly.
4. Comparing countries with different ECDF shapes requires using the same income scale and axes; otherwise the visual comparison is misleading.
5. Interpreting the ECDF curve as a smooth line instead of a cumulative step-by-step function can distort how you read intermediate values.

✅ Exam Checklist

1. Identify that income histograms plot Income versus Frequency with grouped intervals.
2. State that the log-scale histogram replaces Income by Log income while still plotting Frequency on the vertical axis.
3. Recognize that a density plot shows a smooth curve with Income on the x-axis and Income Density on the y-axis.
4. Distinguish Income Density plots from histogram frequency counts when comparing distribution shapes.
5. Define ECDF as an empirical cumulative distribution function giving the share of observations at or below each income level.
6. Read ECDF graphs with Income on the x-axis and ECDF on the y-axis.
7. Explain that the UK ECDF and the IT/UK ECDF overlay are used to compare cumulative income patterns across countries.
8. Use the axes labels (Income, Frequency, Income Density, ECDF) to choose the correct interpretation for each plot type.