Power-law Distribution

Definition

A power-law distribution is one where how often a value occurs falls as a fixed power of its size, so a few items are enormous and a long tail of items are tiny. Averages mislead for such data, and size against rank forms a straight line on log-log axes.

In plain terms

Many real quantities behave this way: word frequencies, city populations, citations, and sales per product. The mean is dragged up by the biggest items while the median sits far below it, so “the average product sells X” is true of almost no product. Plotting size against rank on logarithmic axes gives a roughly straight line.

A word of caution: many other distributions look straight over a limited range, so claiming a true power law needs careful fitting and testing, not just a good-looking plot.

Example

A synthetic marketplace where the item at rank k sells 1,000 / k units a day (the simplest power law, often called Zipf’s law):

Across the top 1,000 items the total is about 7,485 units a day. The top 10 items (1% of the items) account for about 2,929, or 39% of all sales, while the median item sells well under 2 a day.

Common misunderstandings

• “The average is a fine summary.” For heavy-tailed data the mean and the median can differ by orders of magnitude.
• “80/20 is a law of nature.” It is an illustration; the real split depends on the data.
• “A straight line on a log-log plot proves a power law.” Other shapes can mimic it over a limited range.