On Assessing the Importance of the Mode: Generic Measures of Distributional Peakedness, Mean-Modal Differences and Modally Focused Kurtosis

More often than not the mean is used to represent a population’s outcomes, but as the most frequently observed outcome, there are good reasons for using the mode in that role. Its relevance for that purpose hinges on the extent to which it differs from the other centrality statistics employed to represent a populations central location and its prominence or “Peakedness†in the population’s distributional density profile. Peakedness is frequently measured using Pearsons Kurtosis measure however, that measure has been shown to be more about the fatness of a distributions tails rather than its Peakedness. Here, generic probabilisticly based measures of the extent to which location points differ and the extent to which distributions are peaked, suitable for any discrete or continuous potentially multidimensional data environment are proposed together with a modally focused analogue of Pearsons Kurtosis measure, appropriate for skewed distributions. Applications in five very different distributional environments demonstrate the usefulness and general applicability of these new measures.