the distribution of the mean is normal. sample variance follows chi-sq distribution hence, sample SD should follow chi distribution since the squared of a chi distributed rv follows chi-sq distribution. For the sample range, it is a lot more complicated. But it can be understood in the following way. Fisher–Tippett theorem says that the maximum follows gev distribution (generalised extreme distribution) n hence the minimum should also follow gev distribution since min(x)= -max(-x). Now the sample range is the difference between these two. If the shape parameters for these two distributions (the max n min) are zero (which is rarely the case) then the difference between them (the range) follows logistic distribution.