Median and Geometric Mean as Crowd Aggregators

mechanism

A crowd’s signal depends not only on its guesses but on the rule used to combine them. The median resists extreme estimates; the geometric mean better represents quantities spread across multiplicative scales.

Faced with 800 guesses of an ox’s weight, Galton did not need to identify the smartest judge. He took the median—turning a noisy pile of opinions into one resistant central estimate.

The combining rule shapes the answer

Aggregation works when individual errors differ enough to cancel, the core promise of wisdom-of-the-crowds. But the function decides which errors get leverage. A median cares about rank, so a wildly high or low guess cannot drag the result far. A geometric mean combines values multiplicatively, reducing the dominance of very large numbers when estimates span wide ranges. Neither rule creates wisdom: each extracts a different kind of centre from the same crowd.

Where it shows up

The ox-weight median

With 800 ox-weight estimates, the median made extreme guesses largely irrelevant. It obtained robustness without requiring someone to decide precisely which guesses deserved removal.

Pune to Paris

The Pune–Paris experiment first discarded outliers, then used the geometric mean. That pairing separated two choices often blurred together: which observations to retain and how to aggregate those that remain.

No formula repairs a synchronized mistake

A robust centre protects against scattered extremes, not correlated-error. If participants copy one another or share the same mistaken premise, the median and geometric mean can both return a confidently wrong consensus. Nor are they interchangeable: the median discards information about distance, while the geometric mean suits multiplicative scales rather than every quantity.

Choose the centre before seeing the guesses

Before collecting estimates, write down the aggregation rule. Use the median when a few extreme values are plausible; use the geometric mean when values are positive and differ by ratios across a wide range. Precommitting prevents you from selecting the formula—or the outliers—that produces the answer you wanted.

Episodes that teach this

The Wisdom of the Crowd Effect: Theory, Experiment, Examples and Explanation 2,169 views
Galton took a median of 800 ox-weight guesses; the Pune-Paris experiment used the geometric mean after discarding outliers.