If the population variance (and so standard deviation) exist then:

The variance of the sample mean is where is the population variance and the sample size.

So the variance of the sample mean is smaller than the variance of the population and so the same goes for standard deviations.

Now if by spread you mean SD then the answer is yes.

However if the population variance does not exist, the sample mean also does not have a variance and so the answer is no.

CB