Interquartile range

The interquartile range (IQR) is the difference between the third and first quartiles, spanning the middle half of a dataset. It measures spread while reducing the influence of extreme values.

Quartile: A quartile is one of three values that divide an ordered dataset into four equal parts. The first and third quartiles set the IQR’s lower and upper boundaries.
Box plot: A box plot displays a distribution using its quartiles, median, and whiskers or outlier markers. Its box length represents the IQR directly.
Range (statistics): The statistical range is the difference between the largest and smallest observations in a dataset. Unlike the IQR, it uses only the two extremes.
Empirical quantile: An empirical quantile is a quantile estimated from the ordered observations of a finite sample. Different empirical quantile rules can yield different sample IQRs.
First quartile: The first quartile is a value at or below which roughly 25% of ordered observations fall. It supplies the lower endpoint subtracted from the third quartile.
Tukey's fences: Tukey's fences are thresholds, commonly 1.5 IQRs beyond the quartiles, used to flag potential outliers. The IQR sets the scale for these conventional outlier thresholds.
Variance: Variance is the average squared deviation from a distribution's mean or a sample's mean. It uses all observations and is more sensitive to extreme values than the IQR.
Order statistics: Order statistics are the ranked observations in a sample arranged from smallest to largest. Interpolation or rank-selection choices determine which observations define sample quartiles.
Third quartile: The third quartile is a value at or below which roughly 75% of ordered observations fall. It supplies the upper endpoint of the middle-half span.
Robust statistics: Robust statistics are methods designed to remain useful when data contain outliers or departures from assumptions. The IQR is a robust alternative to spread measures sensitive to extremes.

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