  Even though the forecasting literature agrees that aggregating multiple
predictions of some future outcome typically outperforms the individual
predictions, there is no general consensus about the right way to do this. Most
common aggregators are means, defined loosely as aggregators that always remain
between the smallest and largest predictions. Examples include the arithmetic
mean, trimmed means, median, mid-range, and many other measures of central
tendency. If the forecasters use different information, the aggregator ideally
combines their information into a consensus without losing or distorting any of
it. An aggregator that achieves this is considered efficient. Unfortunately,
our results show that if the forecasters use their information accurately, an
aggregator that always remains strictly between the smallest and largest
predictions is never efficient in practice. A similar result holds even if the
ideal predictions are distorted with random error that is centered at zero. If
these noisy predictions are aggregated with a similar notion of centrality,
then, under some mild conditions, the aggregator is asymptotically inefficient.
