In this article we introduce a new class of test statistics designed to detect the occurrence of abnormal observations. It derives from the joint distribution of moment- and quantile- based estimators of power variation σ^r, under the assumption of a normal distribution for the underlying data. Our novel tests can be applied to test for jumps and are found to be generally more powerful than widely used alternatives. An extensive empirical illustration for high-frequency equity data suggests that jumps can be more prevalent than inferred from existing tests on the second or third moment of the data. (Co-author: Pawel Janus.)