Plan for the average and you'll be wrong every time

A statistician drowned crossing a river of average depth four feet. Everyone tells that story about uncertainty. It isn't about uncertainty at all — and you can switch the whole effect off from here without removing a single drop of it.

What you're seeing

The curve is the rule of your situation: how things turn out along the bottom, what you get up the side. The amber band along the bottom is everything the input might actually be — not one number, a range. Every dot in that band gets pushed up through the curve and stacked on the left. That stack is what really happens.

Two lines cut across it. The dashed one is the plan: take the average input, run it through the curve once, done. That is what a spreadsheet does. The solid one is reality: the average of all the outcomes that actually occurred. They are not the same line, and the distance between them is the mistake — a mistake you make even though every number you typed in was correct.

Now find out what is causing it. Drag the spread to its widest. The gap opens up, and the obvious conclusion is that uncertainty is to blame. So test that: leave the spread exactly where it is and press straighten the curve. The uncertainty has not moved. Every dot is still there, still scattered just as widely. And the gap is gone — not small, exactly zero.

That is the whole exhibit. Averaging is only safe on a straight line. On a straight line the outcomes that land above the average and the ones that land below cancel perfectly, however wildly they scatter. Put any bend in the curve and they stop cancelling: one side gets stretched more than the other side gets squashed, and the leftover is the gap. Uncertainty supplies the raw material. Curvature is what converts it into an error.

The direction is worth knowing too. Bend the curve upward and reality comes out above the plan — surprises help you. Bend it downward and reality lands below — surprises cost you, every time. Press the shop's ceiling for the version that ruins real forecasts: a shop can sell less than it expected but never more than it stocks, so its curve bends down, and a plan built on average demand always predicts profits the shop will not see. Deadlines, capacity, budgets, road networks: the bends in the world mostly point downward, which is why plans built on averages so reliably disappoint.

And the river. Its average depth was four feet, and the statistician was six feet tall, so the plan said dry. But drowning is not a smooth curve — it is the sharpest downward bend there is. Survival does not improve when the water is shallower than average; it just ends when the water is deeper. All the shallow parts in the world cannot average away one deep one.

The rule, exactly. The curve is f(x) = 5 + (x−5) + c(x−5)² and the input X is spread evenly over 5 ± a, so its variance: the average squared distance from the mean is a²/3. The plan is f(5). Reality is the average of f across the whole band. Because the curve is exactly quadratic, their difference is not an approximation but an identity: reality − plan = c · Var(X) = c a²/3 Read it off. The slope is absent — tilting the curve changes nothing. Set c = 0 and the gap is exactly zero for every spread. And the spread enters squared: doubling your uncertainty does not double the error, it quadruples it. This is Jensen's inequality: for a curve that bends one way everywhere, the average of the outputs always sits on one definite side of the output of the average, with the bend's sign fixing which side. Verified in node (improve/verify/96-flaw-of-averages.js). The closed form is checked against deterministic Simpson quadrature of the true expectation across 42 curvature-and-spread combinations, agreeing to 6×10⁻¹⁴, and the page's own sampler is shown to converge to it. Four negative controls. A straight curve must return exactly zero at every spread including the widest — so the exhibit's central claim is tested, not asserted. The intuitive law “the error grows in proportion to the spread” is fitted to the narrow case and then shown to under-predict the wide case by 75%, refuting it. A downward bend must produce a negative gap, refuting the common misreading that the flaw of averages always flatters the plan. And a deliberately sign-flipped implementation must be caught by the quadrature comparison. The gate also caught a real bug while this exhibit was being written: the check's first random-number generator used a textbook multiplier that silently overflows JavaScript's exact-integer range, so its draws were not uniform and the sampled average converged to the wrong number — caught as a non-converging test before a line of the page had been drawn.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 96.