What Is Zipf's Law?

Zipf's law describes a striking regularity in language: if you rank all the words in a large text by how often they occur, each word's frequency is roughly proportional to 1 divided by its rank. The most common word appears about twice as often as the second, three times as often as the third, and so on. The result is a language with a tiny core of extremely frequent words and an enormous "long tail" of rare ones.

Zipf's law in real English

The classic Brown Corpus, about one million words of American English from 1961, shows the pattern clearly:

The fit is not exact, but the shape holds across corpora, genres and languages. In the Oxford English Corpus, Oxford's lexicographers reported that just 100 words make up about half of all the words used. Meanwhile, a large share of the distinct words in any corpus appear only once. These one-off words are called hapax legomena.

The formula

On a graph with logarithmic scales for both rank and frequency, the words fall close to a straight line with a slope of about −1. Later refinements, such as the Zipf-Mandelbrot law, adjust the curve for the very top and bottom ranks.

Where it comes from

George Kingsley Zipf, a Harvard linguist, set out the pattern in The Psycho-Biology of Language (1935) and explained it in Human Behavior and the Principle of Least Effort (1949). His idea was a trade-off: speakers save effort by reusing a few general words, while listeners need many specific words to avoid ambiguity. Whether that explanation is right is still debated; similar distributions appear in city sizes, incomes and website visits, and even some random processes produce Zipf-like curves. What is not debated is that the pattern is real and robust. A related observation, often called Zipf's meaning-frequency law, is that more frequent words tend to have more meanings, which fits the heavy polysemy of words like get, set and run.

Why Zipf's law matters for vocabulary learners

Fast start

Because a few words do so much work, the first 1,000–2,000 high-frequency words give beginners a large share of any text. That is why frequency-based study is so efficient early on.

Diminishing returns

The same curve explains why progress then feels slow. Each additional thousand words covers less text than the previous thousand. Going from 95 to 98 percent lexical coverage takes thousands more word families (Nation, 2006). The words you still need are individually rare, so you meet each one only occasionally.

Strategy change

The practical lesson: study frequency lists early, then switch to large amounts of reading and listening, where the rare-but-collectively-important words actually turn up. At that stage, a habit of saving unfamiliar words as you meet them, which is what Wordie is designed for, works better than any general list, because your own reading decides which tail words matter to you.

Text statistics

Zipf's law also explains why the type-token ratio falls as texts get longer: the most frequent words keep repeating while new types become rarer.

See it for yourself

1. Paste a long article into the word frequency counter.
2. Look at the top ten words; they will be mostly function words such as the, of, and and to.
3. Scroll to the bottom; most words appear once or twice.
4. Check how much of the text you can cover with the lexical coverage calculator.

Common misunderstandings

• "Rare words don't matter." Individually they are rare, but together they make up a substantial part of any text, and they often carry the key meaning.
• "It's an exact law." It is an approximate empirical pattern, not a rule words must obey.

Sources: Zipf, G. K. (1935), The Psycho-Biology of Language; Zipf, G. K. (1949), Human Behavior and the Principle of Least Effort; Kučera, H. & Francis, W. N. (1967), Computational Analysis of Present-Day American English (Brown Corpus); Nation, P. (2006), Canadian Modern Language Review.