Outliers and the 1.5 IQR Rule

Eyeballing a dotplot or histogram for "the point way off to the side" works until two people disagree about which points count. Statistics needs a fixed rule so that outlier-calling doesn't depend on who's looking, what scale the graph uses, or how spread out the rest of the data happens to be. The 1.5 IQR rule gives a consistent, repeatable cutoff that anyone can apply to the same five-number summary and get the same answer.

Start with the interquartile range, IQR = Q3 - Q1, which measures the spread of the middle half of the data. Multiply it by 1.5 and use that distance to build two fences: a lower fence at Q1 - 1.5(IQR) and an upper fence at Q3 + 1.5(IQR). Any data value sitting outside either fence counts as an outlier; everything between the fences is typical. The fences stretch or shrink with the IQR, so a tightly clustered data set flags outliers close to the box, while a widely spread one allows more room before calling something unusual.

This rule connects directly to the boxplot: the fences are invisible lines just past the whiskers, and outliers show up as separate dots beyond them. It also sits next to measures of spread and shape from earlier in the unit — because the rule uses quartiles rather than the mean and standard deviation, it stays resistant to skew and to the very extreme values it's trying to catch, which is exactly why it's preferred over a standard-deviation-based cutoff for skewed or irregular distributions.