The 1.5 x IQR rule
The standard fence method needs three numbers from the data: the first quartile Q1, the third quartile Q3, and their difference, the interquartile range. Build two fences: a lower fence at Q1 − 1.5 × IQR and an upper fence at Q3 + 1.5 × IQR. Any value outside the fences is flagged an outlier. The method is resistant by construction: the quartiles themselves barely move when an extreme value arrives, so the fences stay put while the suspect is judged against them.
Delivery times in minutes: 30, 31, 32, 32, 33, 33, 34, 35, 95. Quartiles: Q1 = 32, Q3 = 34, so IQR = 2. Fences: 32 − 3 = 29 and 34 + 3 = 37. The 95-minute delivery sits far outside: flagged. Notice what the mean would have done: it rises to 39.4 under the outlier's pull, above every genuine delivery, while the median stays 33. Judging spread with fences rather than the mean is the whole point.
Flagged is not the same as wrong
An outlier flag opens an investigation; it does not close one. The legitimate fates of a flagged value: a recording error (fix or remove, and log it), a genuinely different case (report it separately), or a real extreme from the same process (keep it, and prefer resistant summaries or methods). Deleting points because they are inconvenient bends every estimate that follows; with samples of a few hundred, roughly 1 in 100 normal values crosses the fences by design, so flags alone never justify deletion.
Testing for outliers with z-scores computed from the mean and SD of the same data. The extreme value inflates the SD and drags the mean toward itself, hiding its own tracks (masking). The fence method and median-based z-scores resist this; plain z-scores on small samples largely cannot see the problem they are meant to find.
Frequently asked questions
Why 1.5 times the IQR?
Tukey's convention. For normal data the fences sit about 2.7 SD from the centre, flagging roughly 0.7% of good values. A stricter 3 x IQR fence marks 'far out' points, about 1 in 500,000 for normal data.
Should I delete outliers?
Only documented errors, with the deletion reported. Real extremes carry information; remove them and your spread estimates describe a world that does not exist.
What if half my data gets flagged?
The method is being applied to data it does not fit: heavily skewed or zero-inflated sets flag their whole upper tail. Transform first (logs are the usual move) or use a method designed for skew.