Power laws and concentration

The upper tail of incomes follows a power law, consistent across the countries he studied. The "80/20" wording is not his.

The original evidence that value piles up in a heavy tail.

Juran named the "Pareto principle" and later wrote that "I had mistakenly applied the wrong name to the principle" and that it "as a universal was not original with Pareto." He dates "vital few, trivial many" to a 1954 paper and later preferred "useful many."

80/20 is a rule of thumb about concentration, not a law with a fixed ratio.

Rank–size regularities (Zipf's law) across word frequencies, city sizes and other human data. Zipf's law is a form of power law (see Newman, 2005).

Ranked customer or keyword revenue often has this shape.

The same skewed shape shows up in word frequencies, papers per scientist, city sizes, incomes and species per genus. Simon traces it to "a similarity in the structure of the underlying probability mechanisms": growth in which the big get bigger.

A mechanism for why accounts and channels that are already large keep pulling ahead.

A "Cumulative Advantage Distribution … which models statistically the situation in which success breeds success."

The formal version of compounding: referrals, expansion, word of mouth.

Networks become power-law ("scale-free") because they keep growing and "new vertices attach preferentially to sites that are already well connected."

In partner, affiliate and referral networks, most of the value goes to the best-connected.

The "law of proportionate effect": when growth is proportional to current size, the result is lognormal.

Proportional growth produces a lognormal distribution: heavy-tailed, but not always a true power law (see Mitzenmacher, 2004).

A readable review of the evidence (city sizes, earthquakes, personal fortunes and more) and the theories that produce power laws.

The standard primer on the math.

The common method (a straight-line fit on a log-log chart) "can produce substantially inaccurate estimates." Testing 24 data sets properly, the power law is supported "in some cases" and "ruled out" in others.

Don't claim a strict power law from a chart. The business point only needs a heavy tail, and that's easy to check: split the book at its average, then split each half again.

"lognormal and power law distributions connect quite naturally." It links Gibrat to Simon.

Check whether customer value is lognormal before calling it a power law.

"Mediocristan," where no single observation moves the total much, versus "Extremistan," where one observation can dominate it.

Customer revenue lives in Extremistan.

"The sample mean is rarely in line with the population mean" under fat tails, and inequality measures such as top-share statistics "produce wrong results."

Average LTV, average deal size and top-20% shares are unstable in heavy-tailed books.

Across "5 studies involving 198 samples including 633,263 researchers, entertainers, politicians, and amateur and professional athletes," performance "is not normally distributed—instead, it follows a Paretian (power law) distribution."

Rep output is likely heavy-tailed too, so bell-curve quotas and evenly shared lead flow mislead.

Changes in work have produced stars, "a few individuals who contribute a disproportionate amount of output."

Hiring, keeping and routing work to top sellers.

Concentration statistics are easy to compute and hard to interpret. The paper models "the true level of relevant concentration" against the apparent level in a given window.

Your top-20% share depends on the time window. Read value over time, not one month.

Six years of Nielsen panel data (more than 100,000 households) give a mean brand-level Pareto ratio of 0.73; "the Pareto principle generally holds" across packaged-goods categories. It is higher for niche brands and lower for big brands.

Even in groceries, the top fifth of buyers carries most of the volume.

"A brand's heaviest 20% of buyers generally contribute not much more than half of a brand's sales": 59% on average, from 68% (dog food) to 44% (hair conditioner).

The counterpoint: concentration varies by industry. Packaged goods are flatter; B2B and recurring-revenue books are usually far more concentrated. Measure your own book.

Plot cumulative profit against customers ranked from most to least profitable and the curve rises above 100%, then falls. As the IMA's statement citing this paper puts it: "Typically, about 20% of customers generate anywhere from 150% to 300% of company profits," most break even, and the bottom "10% … destroy anywhere from 50% to 200%."

The profit version of the power law. The tail isn't just small: part of it costs you money.

Children of unusually tall or short parents are, on average, closer to the mean ("two-thirds of the height-deviate of its mid-parentage"). His quincunx, the Galton board, shows why "mediocrity is so common": many small independent nudges add up to a bell curve.

Two lessons. Last quarter's extremes tend to regress next quarter. And the bell curve comes from adding many small effects, which revenue doesn't do: it compounds.