the Zipf distribution

The Zipf distribution is the law of 'a few things dominate'. Rank items by how often they occur — words by frequency, cities by population, websites by traffic, wealth by holder — and Zipf's pattern says the frequency of the item ranked k is roughly proportional to 1/k^s for some exponent s near 1. The most common word ('the') is about twice as frequent as the second, three times the third, and so on. The top of the list towers over a long, slowly fading tail of rare items.

Made precise as a discrete distribution over ranks k = 1, 2, ..., N, the pmf is P(X = k) = (1/k^s) / H, where H is the normalizing sum H = (1/1^s + 1/2^s + ... + 1/N^s), called a generalized harmonic number, included so the probabilities add to 1. The single shape parameter s controls how steeply frequency drops with rank: s = 0 flattens to the discrete uniform, while larger s makes the head ever more dominant. Zipf is the discrete sibling of the continuous Pareto distribution and a close relative of the power law; all three describe heavy-tailed, scale-free phenomena where the average can be a poor summary because a handful of giants distort it.

Two honest caveats are worth flagging. First, real data often follow Zipf only over a middle range of ranks, deviating at the very top and the very bottom, so 'Zipf's law' is an empirical regularity, not an exact decree. Second, when the exponent s is at or below 1 and N is allowed to grow without bound, the harmonic sum diverges and the mean can fail to exist — a hallmark of heavy tails that makes ordinary averaging and the usual central limit reasoning treacherous.