Data envelopment analysis (DEA) is a method for measuring the relative efficiencies of a set of decision-making units (DMUs) that use multiple inputs to produce multiple outputs. In this paper, we study the measurement of DMU performances in DEA in situations where input and/or output values are given as imprecise data. By imprecise data we mean situations where we only know that the actual values lie in certain intervals, or cases in which data are given only as ordinal relationships. In this paper, we present two distinct approaches obtaining the upper and lower bounds of efficiency which the DMU under evaluation can have with imprecise data. The optimistic approach seeks the best score among the various values of the efficiency score, while the pessimistic approach seeks the worst score. The main idea of the paper is illustrated using an example. Also, two real-world cases are presented to demonstrate how the efficiency interval is interpreted. The efficiency interval not only describes the actual situation in more detail, but also relieves the psychological pressure on all the evaluated DMUs and the decision-maker.