Interquartile Range
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset — the span covered by the middle 50% of observations, with the most extreme 25% on each end deliberately excluded from the calculation. It is a measure of spread, answering the same basic question the range and standard deviation answer — how spread out is this data — but doing so in a way that's specifically built to ignore the influence of extreme values at either tail, which is the property that gives it its primary practical advantage over the full range.
Because the IQR is computed entirely from the middle half of the data, a single extreme outlier anywhere beyond Q1 or Q3 has zero effect on its value — Q1 and Q3 themselves don't move at all in response to how extreme the most extreme observations happen to be, only in response to how many observations fall below or above a given point. This is a fundamentally different behavior from the range (maximum minus minimum), which is defined entirely by the two most extreme points and can be dramatically inflated by a single unusual value, or the standard deviation, which incorporates every value in its calculation and is therefore pulled, though less dramatically than the mean, by extreme observations as well. The IQR's complete insulation from tail behavior is a deliberate design tradeoff: it sacrifices information about the extremes specifically in exchange for a robust, outlier-resistant measure of the data's typical spread.
The IQR's most consequential practical application is as the basis of the standard outlier-detection rule formalized by statistician John Tukey: any observation falling below Q1 minus 1.5 times the IQR, or above Q3 plus 1.5 times the IQR, is conventionally flagged as a potential outlier, a threshold that's become a widely used default across statistical software despite being, fundamentally, a convention rather than a law derived from first principles — the choice of 1.5 as the multiplier is a reasonable, well-established rule of thumb, not a universally correct cutoff, and some fields and specific analyses use different multipliers (2 or 3, for a stricter or looser threshold) depending on how conservative the flagging needs to be. This 1.5×IQR rule is exactly what determines where a box plot's whiskers end: rather than extending all the way to the dataset's absolute minimum and maximum, a standard box plot's whiskers extend only to the most extreme value that falls within this threshold, with anything beyond plotted as individual points specifically to visually highlight them as flagged outliers.
The IQR plays the same outlier-resistant role for spread that the median plays for central tendency, and the two are frequently reported together specifically because they share this robustness property: median and IQR form a natural pairing for skewed or outlier-heavy data, the way mean and standard deviation form the natural pairing for roughly symmetric, outlier-free data, and choosing the wrong pairing — reporting a mean and standard deviation on heavily skewed data, for instance — risks presenting a summary that's technically correct but meaningfully distorted by exactly the extreme values the median-and-IQR pairing was built to be resistant to.
Read the interquartile range as the spread measure built specifically to ignore what happens at the tails: its complete insensitivity to extreme values is a deliberate tradeoff, not a limitation, that makes it the right choice for characterizing spread on skewed or outlier-heavy data where the range or standard deviation would be unduly influenced by a small number of extreme observations, and its role as the basis for Tukey's 1.5×IQR outlier-flagging rule is what directly determines how a box plot's whiskers, and the points flagged beyond them, actually get drawn.