Case Study: Income Inequality — Why the Mean Lies

The Setup

In 2023, the Bureau of Labor Statistics reported that the average CEO of an S&P 500 company earned approximately $16.7 million per year. That same year, the U.S. Census Bureau reported that the *median* household income was about $75,000.

Now here's a thought experiment. Imagine 100 households in a room. Ninety-nine of them earn between $30,000 and $120,000 a year — a broad range, but all in the same general neighborhood. Then one CEO walks in earning $16.7 million.

What happens to the room's statistics?

The median barely moves. It's the income of the 50th person when everyone lines up from lowest to highest earner. The CEO goes to the end of the line, and the middle person stays roughly where they were.

The mean, though, explodes. That one $16.7 million salary gets added to the numerator. Divided by 100 people, it adds $167,000 to the average — all by itself. Suddenly the "average" household income jumps by an amount that exceeds what most households in the room actually earn.

This isn't a math trick. It's what happens to income data across the entire United States, every single day. And the consequences of choosing "mean" vs. "median" when discussing income ripple through policy debates, political campaigns, and how millions of people understand their own economic situation.

The Numbers: How Big Is the Gap?

Let's look at actual U.S. household income data (approximate, based on Census Bureau and Federal Reserve reports):

The mean is roughly 40% higher than the median. Think about what that means: if you read a news headline saying "American households earn an average of $105,000," you might feel reassured — or you might feel confused, because you know plenty of families earning far less.

The disconnect isn't your imagination. It's the math of skewed distributions.

Why Income Is Skewed Right

Income distributions are textbook examples of right skew. There's a natural floor (income can't go below zero for most measures), but there's no ceiling — some people earn millions or even billions. The result:

• A large cluster of households between $20,000 and $80,000
• A long tail stretching through $200,000, $500,000, $1 million, and beyond
• A handful of extreme values (CEO pay, hedge fund managers, tech founders) pulling the tail further and further right

Visual description (income distribution): A histogram of household incomes. The tallest bars are in the $20,000-$60,000 range, creating a prominent peak on the left side. The bars get progressively shorter as income increases, with a long tail extending to the right past $200,000, $500,000, and beyond. The distribution is strongly skewed right. A dashed red vertical line marks the mean (~$105,000) well to the right of a solid green line marking the median (~$75,000). Most of the histogram's area is to the left of the mean — meaning most households earn less than the "average."

The mean is pulled to the right by the long tail. The median sits in the middle of the data, not in the middle of the range. That's why they disagree — and why the median is more honest about what a "typical" household earns.

The Policy Consequences: When the Wrong Average Drives Decisions

The choice between mean and median isn't just academic. It shapes how we think about — and act on — economic reality.

Scenario 1: "The Economy Is Booming"

A politician announces: "Average household income has risen by 15% over the last decade!" The crowd cheers. But what if median household income rose only 3%? That would mean the top earners captured nearly all the gains — their rising incomes inflated the mean while most families barely saw a difference.

This actually happened in the United States during several periods. Between 2009 and 2019, the mean household income grew substantially, but the median grew much more slowly. The rising mean was driven largely by enormous gains at the top of the distribution. The "average" improved while the "typical" household's situation changed far less.

Scenario 2: CEO Pay vs. Worker Pay

In 1965, the ratio of CEO-to-worker compensation was approximately 21:1 — the average CEO earned about 21 times the average worker's salary. By 2022, that ratio was approximately 344:1.

But this comparison uses the mean CEO pay. If you used the median CEO pay (which is lower, because even among CEOs, pay is right-skewed — a handful of mega-CEOs pull the mean up), the ratio is less extreme. The choice of summary statistic literally changes the story about whether CEO pay has grown moderately or explosively.

Neither version is "wrong" — both are mathematically correct. But they serve different rhetorical purposes, and a critical consumer of statistics (which is what this course is training you to be) needs to ask: Which measure of center is being reported, and why?

Scenario 3: "The Typical Student Loan Balance"

News reports about student loan debt sometimes quote the mean balance (approximately $37,000) and sometimes the *median* balance (approximately $20,000). The gap is enormous — nearly double — and it's caused by the same skewness. A relatively small number of borrowers (medical students, law students, graduate students) carry very large balances ($100,000+) that inflate the mean.

A policy designed around "the average borrower carries $37,000" addresses a different problem than one designed around "the typical borrower carries $20,000." The first focuses resources on the tail; the second focuses on the center. The right policy depends on which number you believe is more relevant — and that decision rests on understanding the distribution's shape.

Case Analysis: Applying What You've Learned

Step 1: Recognize the Shape

Whenever someone reports an "average" for a distribution you suspect is skewed, your first question should be: Is this the mean or the median? For income, housing prices, company revenues, medical costs, and most financial data, the distribution is skewed right, meaning the mean will be higher than the median.

Step 2: Ask for Both Numbers

One number is never enough. Ask: What's the mean AND the median? The gap between them tells you about the shape of the distribution. A large gap signals strong skew. A small gap signals approximate symmetry.

Step 3: Consider the Spread

In income data, the standard deviation is enormous — often exceeding the mean itself. This means the "typical distance from the average" is larger than the average. That's a hallmark of an extremely skewed distribution and a signal that the mean is a poor summary of center.

The IQR is often more informative: it tells you the range that contains the middle 50% of earners, ignoring the extreme tails. For U.S. household income, the IQR is roughly $40,000 to $120,000 — a much more useful description of "where most people fall" than either the mean or the median alone.

Step 4: Look at the Whole Distribution

If you have access to the data (or a histogram), look at the shape. The five-number summary — minimum, Q1, median, Q3, maximum — tells you far more than any single number. A box plot of income data would show a compact box on the left with an enormously long right whisker and many outlier dots stretching toward millions of dollars.

The Deeper Lesson: Averages Can Hide Stories

This case study illustrates Theme 2 of this textbook: the human stories behind the data — who's affected, who's missing. When a politician reports the mean income, they're technically correct. But they're telling a story that makes most people's economic reality invisible. The mean income of $105,000 "represents" a country where the most common household income is about $35,000. The number is accurate. The story it tells is misleading.

As you develop your statistical skills, remember that every summary statistic is a compression — it takes an entire distribution and squeezes it into a single number. Some compressions lose more information than others. And the information lost is often the information that matters most: the shape of the distribution, the stories in the tails, and the lived experience of the typical observation.

Discussion Questions

1. A news article reports that "the average American household has $1,060,000 in wealth." The median household wealth is approximately $190,000. What does the ratio of mean to median (approximately 5.6:1) tell you about the wealth distribution? Is it more or less skewed than the income distribution?

2. If you were writing a factual article about the state of the American economy, would you report the mean or the median income? What if you were writing for a financial services company that wanted to attract wealthy clients? How might your choice differ — and is that ethical?

3. Some economists argue that neither the mean nor the median alone captures the full story of income inequality. They recommend measures like the Gini coefficient, which quantifies how evenly income is distributed across a population. Why might a single measure of center — even the "right" one — be insufficient for understanding inequality?

4. Professor Washington is studying whether bail amounts differ by race. If he reports only the mean bail amount for each group, what important information might he be hiding? What other statistics should he report?

Key Takeaways from This Case Study

• The mean is not resistant to outliers. In a skewed distribution like income, the mean is pulled toward the long tail and doesn't represent the "typical" value.
• The median is resistant and better represents the "typical" observation in skewed data.
• Always ask which average is being reported. "Average" is an ambiguous word — it could mean the mean, the median, or even the mode.
• The gap between mean and median tells you about the shape. A large gap indicates strong skew — and strong skew means the mean is misleading.
• Summary statistics serve rhetorical purposes. The choice of which number to report is never neutral. A statistically literate citizen asks: Why did they choose THIS number?