  "Wisdom of crowds" refers to the phenomenon that the average opinion of a
group of individuals on a given question can be very close to the true answer.
It requires a large group diversity of opinions, but the collective error, the
difference between the average opinion and the true value, has to be small. We
consider a stochastic opinion dynamics where individuals can change their
opinion based on the opinions of others (social influence $\alpha$), but to
some degree also stick to their initial opinion (individual conviction
$\beta$). We then derive analytic expressions for the dynamics of the
collective error and the group diversity. We analyze their long-term behavior
to determine the impact of the two parameters $(\alpha,\beta)$ and the initial
opinion distribution on the wisdom of crowds. This allows us to quantify the
ambiguous role of social influence: only if the initial collective error is
large, it helps to improve the wisdom of crowds, but in most cases it
deteriorates the outcome. In these cases, individual conviction still improves
the wisdom of crowds because it mitigates the impact of social influence.
