Skewed Distributions: Choose the Summary for the Question

"A mean does not lie, and a median is not automatically honest. Trouble starts when a valid summary is used to answer the wrong question."

What Skew Means

A distribution is skewed when its shape is asymmetric. Right-skewed data have a longer or heavier right tail; left-skewed data have a longer or heavier left tail. Income, house prices, and response times are often right-skewed in a specified population, while scores near a ceiling can be left-skewed.

In many simple unimodal cases, right skew goes with mean > median and left skew with mean < median. That ordering is a useful heuristic, not a theorem for every possible distribution.

Mean and Median Answer Different Questions

The mean uses every magnitude and is sensitive to extreme values. It is the balance point, connects directly to totals and expected values, and is often the right target for budgets or resource allocation. The median depends on ranks and is more robust to a few extreme magnitudes; it is often better for describing a typical position.

For a right-skewed income distribution, the mean can be much higher than the median. Calling the mean "what a typical person earns" can mislead, but the mean remains meaningful for total income divided by population. Report the purpose, shape, and spread instead of declaring one statistic universally honest.

Operational Principle

When a summary is called "average," ask which average and what decision it serves. Mean, median, quantiles, and totals can all be valid. A skewed distribution usually deserves more than one number.

Wealth as an Example

Household wealth data are typically strongly right-skewed, so published mean wealth can exceed median wealth substantially. The exact gap and upper-tail share depend on the country, year, unit of analysis, valuation method, and data source. A power law may be a candidate model for part of the upper tail, but skewness alone does not prove a pure 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 headline that reports an average for a skewed variable. Locate the mean, median, population, and at least one quantile. Rewrite the headline for two targets: the typical case and the aggregate resource question.

Hint

There is no reliable universal ratio between a mean and median in right-skewed data. Use the actual distribution or reported quantiles.

What Skew Means

Mean and Median Answer Different Questions

Operational Principle

Wealth as an Example

Code

External links

Exercise

Progress

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