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.

This is 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."

Treat 80/20 as a rule of thumb about concentration. There's no law that says the ratio is fixed.

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).

Rank your customers or keywords by revenue and you'll see 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.

Why do your biggest accounts and channels keep pulling ahead? This gives you a mechanism for it.

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

It's the formal version of the compounding you get from referrals, expansion and 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 your partner, affiliate and referral networks, most of the value goes to whoever is best connected.

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

When growth is proportional, you get a lognormal distribution. That's heavy-tailed, but it isn't 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.

Start here for the math. It's the standard primer.

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 off a chart. Their method fits the tail and tests the fit, which a business rarely needs. For the business point you only need to see how much of the book sits above the average: split it at the average, split each half again, and compare each part's share of the accounts with its share of the revenue.

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

Before you call your customer value a power law, check whether it's lognormal.

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

Your customer revenue lives in Extremistan, where one account can dominate the total.

"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."

In a heavy-tailed book, your average LTV, average deal size and top-20% share are all unstable.

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."

Your reps' output is heavy-tailed too. So bell-curve quotas and splitting lead flow evenly will mislead you.

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

It matters for how you hire, keep and route work to your 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, so read value over time instead of one month.

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

Even at the grocery store, 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).

This is the counterpoint. Concentration varies, and it isn't always 80/20. In the same paper's appendix, the top 20% of buyers carried 67% of sales at listed companies on average, and 58% at subscription firms. Measure your own book before you plan around a ratio.

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%."

This is the power law in profit terms. Part of your tail is small, and 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.

There are two lessons here. Last quarter's extremes tend to drift back toward average next quarter. And the bell curve comes from adding many small effects, while revenue compounds.