The fact that the mean and variance were sufficient statistics for a univariate normal distribution truncated at a fixed point was known to Fisher by 1931 [2]. Hotelling [3] has recently observed the corresponding fact for the truncated multivariate normal distribution. It is the aim of this note to point out that these are special cases of a general result, namely: If a family of distributions admits a set of sufficient statistics, then the family obtained by truncation to a fixed set, or by fixed selection, also admits the SAME set of sufficient statistics.