Zipf's law

**Zipf's law** is an empirical law formulated using mathematical statistics that refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution.

For example, Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table. Thus the most frequent word will occur approximately twice as often as the second most frequent word, three times as often as the third most frequent word, etc.: the rank-frequency distribution is an inverse relation.

For example, in the Brown Corpus of American English text, the word *the* is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's Law, the second-place word *of* accounts for slightly over 3.5% of words (36,411 occurrences), followed by *and* (28,852). Only 135 vocabulary items are needed to account for half the Brown Corpus.
The same relationship occurs in many other rankings unrelated to language, such as the population ranks of cities in various countries, corporation sizes, income rankings, ranks of number of people watching the same TV channel, and so on.

Zipf's law is one of a family of related discrete power law probability distributions. *Zipf distribution* is related to the zeta distribution, but is not identical.

Starting with an array of exactly 1,000,000 ones and picking 2 at random and combining them until only 100 are left results on average in the distribution above.

The results do NOT match Zipf's law (yellow line of dots). Some additional process must be invoked to explain Zipf's law.

Preferential attachment (intuitively, "the rich get richer" or "success breeds success") that results in the Yule–Simon distribution has been shown to fit word frequency versus rank in language and population versus city rank better than Zipf's law.

## Hapax legomenon

In corpus linguistics, a ***hapax legomenon*** is a word that occurs only once within a context, either in the written record of an entire language, in the works of an author, or in a single text. *Hapax legomenon* is a transliteration of Greek ἅπαξ λεγόμενον, meaning "(something) being said (only) once".

The related terms ***dis legomenon***, ***tris legomenon***, and ***tetrakis legomenon*** respectively refer to double, triple, or quadruple occurrences, but are far less commonly used.

*Hapax legomena* are quite common, as predicted by Zipf's law, which states that the frequency of any word in a corpus is inversely proportional to its rank in the frequency table. For large corpora, about 40% to 60% of the words are *hapax legomena*, and another 10% to 15% are *dis legomena*.

* Thus, in the Brown Corpus of American English, about half of the 50,000 distinct words are *hapax legomena* within that corpus.

*Hapax legomena* in ancient texts are usually difficult to decipher, since it is easier to infer meaning from multiple contexts than from just one. For example, many of the remaining undeciphered Mayan glyphs are *hapax legomena*, and Biblical (particularly Hebrew; see § Hebrew examples) *hapax legomena* sometimes pose problems in translation. *Hapax legomena* also pose challenges in natural language processing.

There are about 1,500 Hapax legomena in the Hebrew Bible; however, due to Hebrew roots, suffixes and prefixes, only 400 are "true" hapax legomena.