Distribution of a mean values

Suppose we've got n observations, where n is around 100 000. We don't know anything about their distribution. In the next step, we divide the whole sample in k groups of the same size and in each of the groups we calculate the average value. Question is - what can we say about the distributions of the averages that we get this way?

If the original sample is random with finite mean and variance, then each of the k subgroups will have the same mean with a variance reduced by dividing by k.