How does a boxplot's 1.5 x IQR rule decide that a data point is an outlier?

## The construction A boxplot is built from five numbers. Sort the data, take the first quartile `Q1` (the value below which a quarter of the data falls), the median, and the third quartile `Q3`. The box spans `Q1` to `Q3` and is crossed by the median. The interquartile range is `IQR = Q3 - Q1`, the width of that box. The fences are then - lower fence: `Q1 - 1.5 * IQR` - upper fence: `Q3 + 1.5 * IQR` Every observation strictly outside a fence is drawn as its own marker. Everything inside is summarised by the whiskers. ## The detail people get wrong The whiskers are not drawn at the fences. Each whisker extends from the box to the most extreme observation that is still inside the fence on that side. If your largest non-flagged value sits well short of the upper fence, the upper whisker is short. This is why two boxplots with identical boxes can have very different whisker lengths, and why you cannot read the fence position off the plot directly — you can only read where real data stopped. ## Why 1.5 The multiplier is a convention introduced by John Tukey, chosen so that the rule flags few points on well-behaved data while remaining simple enough to apply by hand. It has a clean interpretation under a normal distribution: quartiles sit at about `0.6745` standard deviations either side of the centre, so `IQR` is about `1.349` standard deviations, and the fences land at about `0.6745 + 2.0235 = 2.698` standard deviations from the centre. The probability of a normal observation falling beyond that is roughly 0.7% in total across both tails. In a clean normal sample of 1,000 points you should therefore expect about seven flagged points and be entirely unsurprised by them. Tukey also defined outer fences at `Q1 - 3 * IQR` and `Q3 + 3 * IQR`, sometimes drawn with a different marker; points beyond those he called far out. Nothing about either multiplier is a hypothesis test — there is no null distribution, no p-value and no error-rate guarantee attached to the rule. ## Why it is robust The fences depend only on `Q1` and `Q3`. Both are quantiles, so both are functions of rank rather than magnitude, and both can absorb a large amount of contamination before they move. That is the rule's key advantage over a mean-and-standard-deviation cutoff: the extreme values you are hunting for do not participate in setting their own threshold. A cutoff built from the mean and standard deviation is contaminated by exactly the points it is meant to catch. ## Where it misleads: asymmetric data The rule is symmetric about the box, but many real distributions are not. Response times, incomes, session lengths and claim sizes have a long right tail by nature. On such data the upper fence sits close to the bulk in relative terms while the tail extends far past it, so the rule flags a fistful of points — sometimes dozens — none of which are errors. A clean, right-tailed sample can easily produce ten or twenty flagged points, and the correct reading is that the distribution is asymmetric, not that the data is dirty. The practical consequences: - The count of flagged points is a statement about distribution shape at least as much as about data quality. - Applying the rule inside groups can be very different from applying it to a pooled column, because pooling several differently-centred groups widens `IQR` and hides genuine within-group extremes. - On tiny samples the quartiles themselves are unstable, so the fences jump around; with fewer than about ten points the rule tells you very little. ## What to do with a flagged point The flag is a prompt to look, not a licence to delete. A flagged point deserves a provenance check — where did the value come from, is it physically possible, does an independent source corroborate it — before anyone decides it is contamination. Deleting on the strength of the fence rule alone systematically shaves real tail mass off asymmetric data, and tail behaviour is often exactly what the analysis was about. ## What interviewers listen for The formula stated correctly with quartiles rather than the mean, the whisker detail, an awareness that 1.5 is a convention with roughly 0.7% coverage under normality, and the judgment to say that skewed data will trigger the rule without anything being wrong.