There are many ways to state Zipf's Law but the simplest is procedural: Take all the words in a body of text, for example today's issue of the New York Times, and count the number of times each word appears. If the resulting histogram is sorted by rank, with the most frequently appearing word first, and so on ("a", "the", "for", "by", "and"...), then the shape of the curve is "Zipf curve" for that text. If the Zipf curve is plotted on a log-log scale, it appears as a straight line with a slope of -1.
The Zipf curve is a characteristic of human languages, and many other natural and human phenomena as well. Zipf noticed that the populations of cities followed a similar distribution. There are a few very large cities, a larger number of medium-sized ones, and a large number of small cities. If the cities, or the words of natural language, were randomly distributed, then the Zipf curve would be a flat horizontal line.
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