The aggregation of information from a finite number of document rankings into a representative ranking is an indispensable step in many research areas involving information retrieval. However, aggregation methods such as the median method face the difficulty that document rankings are considered categorical data; and the ordering structure is not necessarily a total ordering. Examples of such ordering structures are obtained when they are derived from desirable properties of effectiveness metrics. The geometric median is an aggregation operator of central tendency, which extends the concept of the median from total to partial orders. Based on lattice theory, this work formally explores when it is possible to compute the medians on orderings derived from properties of effectiveness metrics. This is found to be possible for most orderings, except when document swapping is considered. The median is computed with the median procedure, and some recommendations are made regarding the algorithm to make data processing feasible.