Arbitrary, possibly randomized estimators of a one-dimensional parameter are considered. It is shown that suitable averages of their distribution functions are more spread out than particular distribution functions, which are defined in terms of the weight functions by which the averages are taken over the parameter space and in terms of the family of distributions for the random quantity on which the estimators are based. In this way bounds are provided for the performance of arbitrary estimators of the parameter. As consequences of this nonasymptotic spread inequality, which will be proved under mild regularity conditions, a local asymptotic minimax inequality and a generalization of the classical results on superefficiency can be derived, thus showing the strength of our spread inequality.