The Mean of Factors

§ 01Why the ordinary average fails here

Start with something simple. A sum of money grows by 50 percent in one year and falls by 50 percent the next. The arithmetic mean of those two returns is zero — add them, divide by two, and you get nothing. But the money itself is not back where it started. A hundred pounds becomes a hundred and fifty, then that hundred and fifty is halved to seventy-five. The arithmetic mean said no change; the actual result is a loss of 25 percent. The average lied, not through carelessness but because it was the wrong tool for the job.

The arithmetic mean is built for addition. You have five bags of apples; you want to know the average number per bag; you add the contents and divide by five. That works because the bags are independent — what is in one does not alter what ends up in another. Compound growth does not work that way. Each period's result is applied to the result of the period before it. The factors multiply. An average of multiplied factors cannot, therefore, be a sum divided by a count. It has to be a product divided differently — which is exactly what the geometric mean is.

§ 02How the geometric mean is built

Take a sequence of growth factors rather than percentages. A 50 percent gain is a factor of 1.5; a 50 percent loss is a factor of 0.5. Multiply them together: 1.5 × 0.5 = 0.75. Now take the root that corresponds to the number of periods — two periods, so the square root of 0.75, which is roughly 0.866. That is the geometric mean of the two factors: a number below one, confirming that something was lost each year on average, which matches the actual result. The arithmetic mean of the same factors — (1.5 + 0.5) ÷ 2 = 1.0 — said nothing was gained or lost. It was, in the compound sense, wrong.

The geometric mean is always less than or equal to the arithmetic mean. They are equal only when every factor in the sequence is identical — no variation at all. The moment any variation enters, the geometric mean falls below the arithmetic one. This is not a curiosity; it is a theorem, and it follows directly from the mathematics of convexity. Dispersion itself has a cost: two sequences with the same arithmetic average but different volatility will compound to different endpoints, and the more volatile one ends up lower. The smoother path accumulates more.

The arithmetic mean of those two returns is zero — add them, divide by two, and you get nothing.

This relationship between dispersion and the gap between the two means is why a fall needs a bigger rise than the arithmetic framing suggests. A 50 percent decline requires a 100 percent recovery just to return to the start — the multiplicative nature of the sequence is already embedded in that asymmetry, and the geometric mean is the average that captures it honestly.

§ 03What it measures, and where it appears

The geometric mean appears wherever things grow by successive multiplication: compound interest calculations, the measurement of population growth over time, the construction of index price series. When analysts describe the annualised return of a market over several decades, a precise reading of that figure should be the geometric mean of the annual factors — the rate that, compounded uniformly over the same number of years, would reproduce the actual endpoint. The arithmetic mean of the same annual figures would give a higher number that no investor ever collected, because no investor received averages; they received the actual sequence, product and all.

The distinction is not semantic. A long sequence of factors with moderate variance can show an arithmetic average comfortably above zero while the geometric mean sits closer to, or even below, the starting value. The arithmetic figure describes the centre of a distribution of single-period outcomes; the geometric figure describes what happened to the accumulated whole. They are answers to different questions, and using the first to answer the second is a category error dressed up in arithmetic.