Zipf’s Law Explained

Zipf’s law is a mysterious relationship that describes the frequencies of words or the populations of cities to inverse relationships. The law was coined by an American linguist named George Kingsley Zipf. He discovered this law by looking at the relative frequencies of words in a large text, like the famous book Moby Dick.

So named after his name, this law states that, in many classical books, the repetition of a given word is inversely proportional to its position in the frequency table. So the law can be expressed as Pn ∝ 1/n^a. Zipf’s law is also known as the “Power Law”.  A rule which is explaining the relation between the frequency of a set of objects or events and their size.

Check out the Archimedes Principle!

Here in his law, he was talking about the most common or most often used words in the English language. So if you take a book and you calculate the frequency each word appears, then you will find that the word “the” will appear more frequently than any other word out there. So we can easily verify the most common word in the English language in terms of its usage right? If we have to express it more mathematically then, usually it occurs about 7% of the time. So what is that mean? It means that 7% of words in a typical text are the word “the”. Below you can find all the commonly used words in English –


If we use Zipf’s law (Pn ∝ 1/n^a) then we can easily notice that the second most frequent word in the English language which is ‘of’ happens about half as frequently as the most frequent word ‘the’. Likewise, the third most frequent word ‘to’ happens about a third as frequently as ‘the’. And so on. The seventh most frequent word ‘for’ happens about one-seventh as frequently as ‘the’.

So if we try to draw a graph according to their frequencies then the graph will look like this –

So right now, you may have a question like, “Is the Zipf’s law only applies to the English language?” The answer is “No” It has been observed that Zipf’s law applies to all languages out there. It is not all about a language, This law even applicable to population, birth, death, online internet usage, largest cities, the income of the richest countries and many more can be described in terms of Zipf’s Law. All these things make it a powerful yet mysterious law of all time. That’s all about Zipf’s Law.