Variance changes proof

Question

I have, say a vector of data, A of size(N*1). How do I prove or disprove: if some of the data in A, say A(1:N/2), their variance becomes large, the whole vector A's variance becomes larger? That is to ask, if a subset of population's variance becomes larger, does it mean the whole population's variance increases too? Note here the mean of population could change too. I am looking for a strict mathematical proof. Thank you!

Answer 1

I'll interpret this to mean that you have $N$ data points, and that you throw out the first $N/2$ and replace them with $N/2$ new data points in such a way that the variance of the original first $N/2$ data points was less than that of the new first $N/2$ data points.

It's fairly straightforward to see that this can decrease the variance of the set. Suppose, for instance, that to begin with your data set was {1, 1, 19, 20}. The variance of {1,1} is zero. Replacing those with {19, 20}, which has variance greater than zero, we get {19, 20, 19, 20}, which has variance far lower than the original data set.