Quantiles, L-moments and modes: bringing order to descriptive statistics

Describing batches of data in terms of their order statistics or quantiles has long roots, but remains underrated in graphically-based exploration, data reduction and data reporting. Hosking in 1990 proposed L-moments based on quantiles as a unifying framework for summarizing distribution properties, but despite several advantages they still appear to be very little known outside their main application areas of hydrology and climatology. Similarly, the mode can be traced to the prehistory of statistics, but it is often neglected or disparaged despite its value as a simple descriptor and even as a robust estimator of location. This paper reviews and exemplifies these approaches with detailed reference to Stata implementations. Several graphical displays are discussed, some novel. Specific attention is given to the use of Mata for programming core calculations directly and rapidly.