Each member decides correctly on a two-option question with the same independent probability. The group then votes by majority. Watch how group accuracy depends on individual competence and on group size.
With each member correct 60% of the time, a majority of 9 is correct 73.3% of the time — better than the 60% individual rate.
The theorem is double-edged. Above 50% competence, enlarging the group drives majority accuracy toward certainty; below it, the same mechanism drives it toward zero. Independence of the votes is assumed throughout.
Twelve people independently estimate an ox’s weight (true value 1198 lb, the dashed gold line). The crowd average (navy) sits close to the truth. Now let each person see and drift toward the first public guess, and watch the average decay.
Crowd-mean error: 10.5 lb · typical individual error: 129.2 lb. With independent judgments the crowd beats the average member by roughly 12×.
Averaging cancels large, uncorrelated errors; that is the whole mechanism. Correlating the judgments destroys it, which is why collecting opinions independently, before any discussion, protects a crowd’s accuracy.
Three analysts start from different estimates and each trusts the others by a fixed set of weights. Every round, each opinion becomes the trust-weighted average of all three. Step the clock and watch them meet.
Round 0: opinions 70.0, 40.0, 90.0 (spread 50.0). The limit is the influence-weighted mean 63.81, using the stationary weights 5/21, 9/21, 7/21 — not the plain average 66.67.
Whoever is trusted most, not whoever is most confident or most correct, pulls the consensus furthest. Analyst 2 carries weight 9/21 and so dominates the agreed value on the default settings.