The Tyranny of the Average

what a 1733 piece of mathematics taught me about building companies

In Joseph Wright of Derby’s An Iron Forge, the drama is not the machine, but the moment raw iron is pulled from the common mass and forced, under fire and pressure, into another form. It is a painting about technology before productivity, transformation before output — and the uncomfortable truth that not everything placed near the flame is remade.

In 1733, Abraham de Moivre noticed something that should unsettle anyone trying to be exceptional.

Flip a fair coin enough times, tally the heads, and the counts arrange themselves into a curve — the bell. A century later Laplace generalized it; in 1920 George Pólya gave it a name that has been quietly governing your expectations ever since: the central limit theorem. Its claim, stated carefully, is this. Take many independent things, each contributing a little, none of them dominant, and add them up. Almost regardless of what those individual things look like, their sum collapses into the same smooth, symmetric bell. And by its companion law — the law of large numbers — the average of those things concentrates, tighter and tighter, on the mean.

That is not a metaphor yet. It is arithmetic. The variance of an average of n independent efforts shrinks like 1/n. Pile up more independent, ordinary inputs and you don’t get a wilder result — you get a more certain one, pressed ever harder against the middle. The mathematics has a recipe for producing the average, and the recipe is: many small, independent, interchangeable contributions, no single one allowed to dominate.

Here is the trap. That recipe is exactly the one most of us are taught to follow with a life.

A sensible career: a bit of this skill, a bit of that, a diversified spread of reasonable bets, none too risky, each independent of the others, advice taken from many sources, edges sanded off. It is the most rational-sounding way to live — and to the precise extent that you build yourself this way, out of many small averaged inputs, the math has all but written your expected result. You converge on the mean. Not because you failed, but because you followed, with unusual discipline, the exact instructions for manufacturing an average.

I want to be careful here, because this is where these arguments usually cheat. The theorem is about sums of independent random variables; a life is not literally such a sum, and I won’t pretend the equation proves anything about you. The bell isn’t even a life’s default shape — most things that matter (skill, reputation, wealth) compound, and compounding is multiplicative, which bends a distribution into a long tail on its own, for everyone, with no heroics required. So the honest claim is narrower and more useful than “math says you’ll be average”: the bell describes the parts of your life you deliberately average — the hedged portfolio, the balanced schedule, the attention sprayed evenly across ten priorities. To the degree those parts dominate, you inherit their mathematics. And their mathematics is the mean.

So the real question is not “how do I do the normal things better?” Doing the normal things better just buys a tighter, more confident seat inside the same bell. The question is: which of your bets are you quietly putting into the averaging machine — and which could you pull out of it?

Three ways out, and the price on each

The theorem, read honestly, hands you the exits. The bell is a consequence of three assumptions — independence, finite variance, and no single dominant term. Outliers live exactly where one of those breaks. There are three doors, and — the part the motivational version omits — each has a price, because the same move that opens the upside opens the downside.

Door one: compounding. The bell needs your contributions to be independent and interchangeable, each a fresh small draw. So stop drawing fresh. Pour effort into one thing long enough that each year multiplies the last instead of starting over — skill on skill, reputation on reputation, the second product riding the distribution of the first. This is not “introduce statistical correlation”; it is the older and stronger thing, multiplicative growth, and multiplicative growth is precisely what bends a thin Gaussian into a long tail. It is the mathematics under the oldest founder cliché: do one thing, obsessively, until it compounds. The price is concentration risk — compounding multiplies whatever it is fed, including a mistake — so the honest version of “go all in” is not faith. It is to concentrate on the one bet whose downside you can survive and whose upside is uncapped, to make the early moves cheap and reversible, gather signal, and only then pour in. Founders who win this way rarely bet the company on conviction; they sequence small, high-information bets and feed the one that answers.

Door two: fat tails. The bell needs finite variance. Some games don’t have it. In a power-law game a single outcome can be larger in magnitude than all the others combined, and the tidy bell never forms — the sum still settles, but onto a heavy-tailed law, never the Gaussian. The mathematician’s parable is the Cauchy distribution: average a thousand draws from a Cauchy and you get back — a Cauchy. The same wild, unconcentrated thing you started with. Averaging buys you nothing. Venture capital is this distribution wearing a suit: a handful of investments return the fund and the median one is a rounding error — which is why spraying small uncommitted bets across everything is futile. You cannot diversify your way into the tail; you have to own enough of the one that hits. Alice Bentinck, who has spent twelve years at Entrepreneur First watching people build companies from nothing, says it without the math: “Being a founder is taking on a power law. Very few individuals succeed, but when they do succeed, they succeed to an insane degree.” But a fat tail is fat on the left as well: the same distribution that holds the fund-returner holds the wipeout. Choosing this game means choosing a bet whose expected value may be lower than the salary you gave up — chosen for the shape of the prize, with both ends of the shape in view.

Door three: refuse to be a random draw. Bentinck’s larger finding is that her most successful founders share a disposition she calls Personal Exceptionalism — from the investor Michael Dearing, who described people who see their work as “snowflake-special” and “operate way outside the bounds of normal for their peer group.” It is, she’s careful to say, not arrogance: “a quiet belief that even if the world is falling around you, you will be the one to be ok.” This is the one door that is psychological, not statistical, and I won’t dress it up as math: believing you’re the exception does not change the distribution. What it changes is your willingness to take the insane goal, break the rule written “for other people,” and absorb the rejection that sends ordinary samples back toward the mean. It is a behavioral edge, and a real one — but treat the source honestly. An accelerator that selects hard for exceptional people and then reports that its winners felt exceptional has the selection baked in. The honest claim is not “believe and you’ll win.” It is: the people who took the fat-tailed bet and survived it tended to share this disposition — which tells you nothing about the identically-convinced people it flattened. EF’s founders, drawn together as strangers, have built more than $10 billion of companies. We do not have the number for the ones who didn’t.

The tail is selected, not summoned

It is tempting to conclude that outliers are manufactured — that the right environment simply produces them. The evidence says something narrower: the tail is selected and amplified, not conjured from nothing.

Roger Bannister ran the first sub-four-minute mile on the 6th of May, 1954. The myth is that a dam broke and a flood of runners poured through; the reality is quieter. John Landy followed in forty-six days — but only about ten men had broken four minutes two and a half years later. The barrier was partly real, and what dissolved it was not new physiology but a demonstration that the tail was reachable, plus the slow training that reached it.

Or take Iten, a town in Kenya’s Rift Valley, where the Irish missionary Colm O’Connell has coached athletes who became roughly twenty-five world champions. That is not luck and not one man’s magic — but it is not proof that anyone can be made a champion either. Iten sits on top of an altitude-adapted, biomechanically economical, Kalenjin-concentrated talent pool; what the place does is find and amplify what is already there — above all the runner whose ability whispers rather than shouts. And the science is precise about the limit. Deliberate practice carries you from beginner to expert; there it dominates. But among people who have already paid that price, it barely separates them — in elite-versus-elite samples it explains on the order of one percent of the difference. Past the ante, the residual is something practice cannot supply. Environments like Iten reject the mean on your behalf, ruthlessly; they cannot exempt you from the odds.

And the odds are the honest part. Every name in this essay is drawn from the right tail; the people who showed identical conviction and identical obsession and simply lost do not write essays, and we never count them. The fat-tailed bet is fat in both directions. Personal Exceptionalism without a growth mindset, Bentinck warns, makes someone hard to work with and slow to take feedback — and, at the limit, simply deluded. The tail is real, and reachable, and most who sprint at it miss.

Why the AI age sharpens the trap

You would think cheap, abundant creation softens all this. It does the opposite.

In 1900 the electric dynamo had been commercially viable for two decades, and you could see it everywhere except in the productivity statistics. Factories had bought electric motors and bolted them where the steam engine used to sit, and gotten almost nothing for it. The economist Paul David told this story in 1990 to explain why the computer, then equally invisible in the numbers, would take its time. Electric motors were under 5% of American factory power in 1899; they passed 55% only around 1919; the productivity surge waited until the 1920s. Roughly forty years passed between the technology and the payoff.

The lag had a cause, and the cause is the whole lesson. The gain did not come from the motor. It came from reorganizing the entire factory around it — tearing out the central steam engine and its overhead forest of shafts and belts, giving every machine its own “unit drive,” and, freed from the geometry of the driveshaft, laying the plant out by the flow of the work. Notice the unit of analysis: David is describing firms restructuring capital over decades — which is exactly why the lesson is not “try harder.” The factory didn’t get more productive by running the steam engine faster; it got more productive by rebuilding what it was around the new primitive. The individual translation isn’t do more. It’s restructure what you do.

This is the shape of the AI moment, and it is not comforting. When a machine produces the competent, median version of almost anything — the draft, the design, the analysis — the floor of competence rises and flattens. Everyone now holds the same tool and reaches for the same average output; the bell gets tighter and more crowded than it has ever been. Adopting AI is bolting the motor where the steam engine sat: a sensible, universally available input that the averaging will wash straight into the mean, precisely because everyone has it. Abundance of competence does not lower the bar to matter. It makes distinctiveness scarcer, and therefore dearer. The payoff was never in the tool. It is in doing the expensive, disorganizing, four-decade thing — rebuilding the whole system around the new primitive — while everyone else is still admiring their new motor.

The honest conclusion

So here is the argument with the comfortable parts removed, and the uncomfortable ones kept in the same breath.

The central limit theorem is a description of strategies, not of souls. It says nothing about which life is worth living, and a balanced, competent, present life in the broad middle is not a failure — by the measures that actually survive scrutiny, it is where most genuinely good lives are lived. The math is about outputs and odds, never about worth, and anyone who sells you the percentile as the meaning is selling you something. The tail is also positional: it has room for almost no one, most who reach it pay a price you can see and many pay one you can’t, and “stop averaging” is a wager, not a commandment — addressed only to those who have already decided to play.

But if you have decided — eyes open, both ends of the distribution in view — then the theorem is a map. The bell is the default for whatever you average; you leave it only by doing what the math calls non-average and the world calls unreasonable. Compound one thing until it multiplies instead of resets. Choose the fat-tailed games, and own enough of the bet to catch the tail — the one whose downside you can survive. Refuse to model yourself as one more interchangeable draw, while staying honest that the refusal is a bet, not a birthright. And when a tool arrives that everyone can hold, don’t bolt it to the old machine and call it progress — rebuild the machine.

Most people do the same things. That is not their failure; it is a theorem. Choosing, with open eyes, to break it is the only bet that was ever worth the premium.

Sources & notes

The mathematics The central limit theorem (de Moivre’s 1733 normal approximation to the binomial; generalized by Laplace, 1810–1812; named by George Pólya, 1920) concerns the standardized sum of many independent, finite-variance terms; the concentration of the average on the mean (variance ∝ 1/n) is its companion, the law of large numbers. The bell requires independence, finite variance, and no single dominant term. It fails when any breaks: under infinite variance the average of i.i.d. Cauchy draws is itself Cauchy — no concentration (a standard textbook result); fat-tailed sums converge instead to heavy-tailed α-stable laws. Multiplicative compounding of independent shocks produces a right-skewed lognormal, heavy-tailed without any change in strategy. The theorem describes distributions of outcomes; it carries no claim about any individual’s worth — its use here as a lens on strategy is deliberate analogy, flagged as such.

Sources Power-law founding & “Personal Exceptionalism”: Alice Bentinck, “Personal Exceptionalism” (Entrepreneur First, 2026), quoting Michael Dearing, “The Cognitive Distortions of Founders” (Harrison Metal); EF cumulative company value “$10bn” per the essay. VC return concentration: the Babe-Ruth-effect / power-law literature (e.g., Dixon; Evans on Horsley Bridge data). Sub-four-minute mile: Bannister, 6 May 1954; Landy +46 days; ~10 men within ~2.5 years (de-mythologized vs. the “flood” legend). Iten / Brother Colm O’Connell: ~25 world champions coached over ~40 years, atop the documented altitude + biomechanical + Kalenjin substrate of East-African distance running (Wilber & Pitsiladis). Deliberate practice: Macnamara, Hambrick & Oswald, Psychological Science (2014) for the cross-domain variance-explained figures; Macnamara, Moreau & Hambrick, Perspectives on Psychological Science 11(3) (2016) for the ~1% elite-versus-elite residual in sport; the “10,000-hour rule” is Gladwell’s popularization, disavowed by Ericsson. Productivity paradox: Paul A. David, “The Dynamo and the Computer,” AER P&P 80(2) (1990) — electric motors <5% of US factory power in 1899 → ~55% by 1919, the surge arriving in the 1920s via “unit drive” and factory reorganization; echoing Solow’s 1987 “you can see the computer age everywhere but in the productivity statistics.”