Scalar and vector partitions of the probability score (PS) in N-state (N > 2) situations are described and compared. In N-state, as well as in two-state (N = 2), situations these partitions provide similar, but not equivalent (i.e., linearly related), measures of the reliability and resolution of probability forecasts. Specifically, the vector partition, when compared to the scalar partition, decreases the reliability and increases the resolution of the forecasts. A sample collection of forecasts is used to illustrate the differences between these partitions in N-state situations.
Several questions related to the use of scalar and vector partitions of the PS in N-state situations are discussed, including the relative merits of these partitions and the effect upon sample size when forecasts are considered to be vectors rather than scalars. The discussions indicate that the vector partition appears to be more appropriate, in general, than the scalar partition, and that when the forecasts in a collection of forecasts are considered to be vectors rather than scalars the sample size of the collection may be substantially reduced.