interquartile range

The NIST/SEMATECH e-Handbook of Statistical Methods sets it out in its account of the box plot. Order every measurement from lowest to highest. The lower quartile is the 25th percentile, the value a quarter of the way up that order, and the upper quartile is the 75th percentile, the value three quarters of the way up. The handbook calls the difference between the two the interquartile range, which it abbreviates IQ. Drawn as a box between the two quartiles, it represents the middle 50 percent of the data, what the handbook calls the body of the data, with the median marked inside it. So the number describes how tightly the middle half of a group clusters, and says nothing at all about the quarter at either end.

That is the point of it, and the reason it travels beside a median rather than a mean. A standard deviation is built from every value's distance from the mean, so a few extreme measurements pull it wider. An interquartile range only asks where the 25th and 75th ranked values fall, so those same extremes barely move it. The handbook puts that stability to work in the opposite direction, using the quartiles and the interquartile range to build the fence points of a box plot, beyond which a measurement counts as an outlier: an observation that lies an abnormal distance from other values in a random sample from a population.

One point of reading. Because the handbook's definition is a subtraction, upper quartile minus lower quartile, the interquartile range is strictly a single number, a width. Research papers commonly print it instead as the two quartile values themselves, so a follow-up time given as a median of 7.0 years with an interquartile range of 6.7 to 7.1 is naming the boundaries of the middle half, and the width is the 0.4 years between them. Read either way it answers the same question: how much did the middle of this group vary, and therefore how much weight does the median carry as a description of a typical case.