Why Averages Can Lie: Mean, Median, Mode, and Distribution
Ten people sit in a bar, each earning $50,000 a year. The average income is $50,000. Then a billionaire walks in. The average income in the room is now over $90 million, but nobody already sitting there got a cent richer.
The mean didn't make a calculation error. It did exactly what it's defined to do. The problem is that "average" hides the most important thing about data: its shape.
Nobody Is Average
In 1950, the U.S. Air Force had a problem: pilots kept losing control of their planes. Cockpits had been designed in 1926 around the average body measurements of pilots.
Lieutenant Gilbert S. Daniels measured 4,063 pilots on 10 body dimensions, including height, chest, sleeve length and thigh circumference. He defined "average" generously: within the middle 30% of the range for each dimension.
How many pilots were average on all 10 dimensions? Zero. Not a single one. Even on just three dimensions, fewer than 3.5% qualified.
The Air Force responded by requiring adjustable seats, pedals and helmet straps. Designing for the average meant designing for nobody. Todd Rose retells the story in his 2016 book The End of Average.
Three Averages, Three Different Answers
"Average" can mean three different things:
- Mean: add all values and divide by how many there are
- Median: the middle value when sorted
- Mode: the value that appears most often
Take these nine house prices (in thousands of dollars):
210, 225, 240, 240, 255, 270, 290, 310, 1,900
| Measure | Value |
|---|---|
| Mean | $437,778 |
| Median | $255,000 |
| Mode | $240,000 |
One mansion drags the mean up by more than $180,000. The median barely notices it. That's why housing prices and household incomes are usually reported as medians.
When to Use Which
| Situation | Best measure | Why |
|---|---|---|
| Symmetric data, no outliers | Mean | Uses every value |
| Skewed data (income, house prices) | Median | Resistant to extremes |
| Categories (shoe sizes, favorite color) | Mode | Mean is meaningless |
| Rates of growth | Geometric mean | Growth multiplies |
A useful rule of thumb: if the mean is well above the median, the data is right-skewed, with a long tail of large values. Income, wealth, city sizes and website traffic all look like this.
Same Average, Completely Different Data
In 1973, statistician Francis Anscombe published four small datasets now known as Anscombe's quartet. All four have:
- The same mean of x (9) and y (about 7.50)
- The same variance
- The same correlation (about 0.816)
- The same best-fit line (y ≈ 3 + 0.5x)
Graph them, though, and they look nothing alike. One is a clean linear trend, one is a curve, one is a straight line with one outlier, and one is a vertical stack with a single distant point. His point: always plot your data. Summary statistics alone can describe very different realities.
Spread Matters as Much as Center
Two cities both have an average daily high of 20°C. In one, it's 18–22°C every day. In the other, it swings from 0°C to 40°C. Same mean, very different lives.
The standard deviation measures that spread:
σ = √( Σ(x − μ)² / N )
A small σ means values cluster near the mean. A large σ means they're scattered. The z-score calculator uses both numbers to tell you how unusual a value is: z = (x − μ) / σ.
Simpson's Paradox: When Averages Reverse
In 1973, UC Berkeley's graduate admissions looked biased: about 44% of male applicants were admitted versus about 35% of female applicants.
But when researchers (Bickel, Hammel and O'Connell, in a 1975 Science paper) broke the numbers down by department, most departments admitted women at an equal or higher rate. Women had mostly applied to highly competitive departments with low admission rates for everyone. The overall average reversed the trend in the parts.
That's Simpson's paradox, and it's a reminder that an average of groups can point the opposite way from the groups themselves.
Two Concepts Worth Knowing
Distribution
A distribution describes how often each value occurs. The famous bell-shaped normal distribution is symmetric, so its mean, median and mode are identical. Real data often isn't normal. See the standard normal table for the ideal case.
Outliers
An outlier is a value far from the rest. The interquartile range (IQR) method flags any value more than 1.5 × IQR below the first quartile or above the third. Whether to remove an outlier depends on whether it's an error or a real, important value.
Quick Answer: What Is the Difference Between Mean, Median and Mode?
The mean is the sum divided by the count, the median is the middle value when the data is sorted, and the mode is the most common value. The mean is pulled by outliers. The median resists them, so it's preferred for skewed data like incomes and house prices.
Try Them Yourself
- Statistics Formulas: mean, median, variance and standard deviation
- Z-Score Calculator: measure how far a value is from the mean
- Standard Normal Table: the symmetric ideal distribution
- Z Tables: probabilities for any z-score
- Random Number Generator: generate data and compare its mean and median
- Unlocking Insights With Regression Analysis: go beyond averages
Next time you see "the average," ask two questions: which average, and what does the distribution look like? Those two questions will protect you from most statistical spin.