How can it be shown that the variance of combined predictions is 1/n I was reading this article.
In it, when referring to Ensemble Learning, this was stated:

It can be theoretically shown that the variance of the combined
  predictions are reduced to 1/n (n: number of classifiers) of the
  original variance, under some assumptions.

Assuming that we have n classifiers, and we combine their predictions, how can it be shown that the combined predictions have variance 1/n?
 A: The key here is the fact that you are combining the performance of difference classifiers, and thus the effect of one bad classifier is "diminished" because the other $n-1$ may not be so bad. The article writes:

The predictions of all the classifiers are combined using a mean, median or mode value depending on the problem at hand

Suppose the classifiers $C_1, C_2, \dots, C_n$ each yield a final prediction. This prediction is random because the original training data is assumed to be random, i.e., following a distribution. Say each classifier independently defines a distribution centered around the true value $\mu$ and has a variance $\sigma^2$.
Then since combining is done by say, taking the mean of all the classifiers, due to independence the variance of the final prediction is
$$\text{Var}\left(\dfrac{C_1 + C_2 + \dots + C_n}{n} \right) = \dfrac{Var(C_1) +  Var(C_2) + \dots Var(C_n)}{n^2} = \dfrac{\sigma^2}{n} \,. $$
(Of course a lot of this is based on assumptions of equal variance of classifiers and independence of the classifiers, both of which may not be true. But the general idea is what is above).
