Skewed Distributions: When the Average Lies

"The mean is the most weaponized number in statistics. Skewed distributions are where it does the most damage."

What Skew Means

A distribution is skewed when it isn't symmetric. The two flavors:

Right-skewed (positive skew): the long tail is on the right. A few large values stretch the right side. Mean > median. Examples: income, house prices, time-to-complete-task, file sizes.
Left-skewed (negative skew): the long tail is on the left. A few small values stretch the left side. Mean < median. Examples: human lifespan (everyone dies young or old, the long tail is short lives), exam pass rates near the ceiling.

Right-skew is by far the more common in everyday data, because so many real quantities are bounded below by zero but unbounded above (you can't have negative income or negative time).

The Mean Drifts Toward the Long Tail

The median is the value that splits the distribution into two equal halves; it doesn't care how extreme the tail values are. The mean is the balance point; it does care, and it drifts toward whichever side has the long tail.

In a right-skewed distribution like income, the mean is dragged upward by the few high earners. The 'average' income in a country is almost always larger than what a typical citizen earns. The median tells you what the middle person makes; the mean tells you a number that nobody actually makes.

The Canonical Citizen Disaster

When someone hands you a mean from a skewed distribution and calls it 'typical,' they have already lied — sometimes on purpose, more often by reflex. The median is the honest summary for skewed data. The mean is the honest summary for symmetric data. Asking 'which shape produced this average?' is the citizen's defense.

Dad's Wealth-Distribution Example

Dad once analyzed the Korean wealth distribution and found what you already suspect: it is severely right-skewed. The top 0.1% holds a disproportionate share, the upper tail stretches far to the right, and the 'average wealth' figure that headlines love is meaningfully larger than the median household actually holds. Anyone who quotes the mean as 'how rich a typical Korean is' has skipped the question of what shape the distribution has — and that skip is the whole mistake. (Track 05 will show that this isn't even skew; it's the next animal in the zoo, the power law.)

Code

Mean vs median on lognormal 'income'·python

import numpy as np
rng = np.random.default_rng(11)

# Synthetic right-skewed 'income' data — lognormal is the textbook model.
N = 50_000
income = rng.lognormal(mean=10.5, sigma=0.9, size=N)   # in your country's currency

print(f"Median income: {np.median(income):>12,.0f}")
print(f"Mean income:   {np.mean(income):>12,.0f}")
print(f"Top  1% threshold: {np.quantile(income, 0.99):>12,.0f}")
print(f"Top 0.1% threshold: {np.quantile(income, 0.999):>12,.0f}")
print(f"Maximum:       {np.max(income):>12,.0f}")

# The mean is comfortably larger than the median.
# The top 0.1% threshold is comically larger than the median.
# A headline that says 'average income is X' is technically true and
# practically misleading. The median is the citizen-honest number.

Exercise

Find a news headline that quotes an 'average' for something likely right-skewed (income, house price, donation, salary, charitable giving). Estimate by how much the median would differ from the quoted mean, and whether the headline's argument would still hold if the median were used instead. Most arguments built on a skewed-distribution mean dissolve quietly when the median replaces it.

Hint

A rule of thumb: in many right-skewed distributions in everyday life, the mean is 1.3x–2x the median. Substituting changes 'typical' to a much more honest number.

What Skew Means

The Mean Drifts Toward the Long Tail

The Canonical Citizen Disaster

Dad's Wealth-Distribution Example

Code

External links

Exercise

Progress

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