Zipf's Law Explained

Main Ideas / Concepts

• Zipf’s Law (Zipf’s Law Explained) describes a mysterious relationship connecting:
  • Word frequency in texts, and
  • City population (and other “sizes” in general) via an inverse relationship.
• The law was coined/discovered by American linguist George Kingsley Zipf.
• Zipf discovered it by analyzing relative frequencies of words in large texts, using Moby-Dick as an example.

What the Law States (as Presented)

• In “many classical books,” the repetition frequency of a word is inversely proportional to its position in a frequency ranking/table.
• The subtitles provide a mathematical form:
  • (p_n) is directly proportional to (1 / n^a)
    • i.e., (p_n \propto \frac{1}{n^a}), described as “(1) by (n) raised to the power (a)”.
• It is also called a power law: a rule explaining the relationship between the frequency of objects/events and their size/rank.

Example Using English Word Frequencies

• The subtitles focus on the most common words in English:
  • In a typical text, “the” appears about ~7% of the time.
  • The 2nd most frequent word (“of”) appears about half as often as “the”.
  • The 3rd most frequent word (“to”) appears about a third as often as “the”.
  • The 7th most frequent word (“for”) appears about one-seventh as often as “the”.
• A graph of word rank vs. frequency would follow the decreasing pattern implied by Zipf’s Law.

Scope / Where Else It Applies (According to the Subtitles)

Zipf’s Law is stated to apply beyond English:

• All languages (as claimed).
• It is also claimed to apply to many non-language domains, such as:
  • Population
  • Birth and death
  • Online/internet usage
  • City sizes / largest cities
  • Income of the richest countries
  • More examples (unspecified)

Sources / Speakers Featured

• George Kingsley Zipf — American linguist credited with coining/discovering the law
• Moby-Dick — the example text analyzed
• No other specific speakers are identified in the subtitles (the narrator is not explicitly named).