Turn 30 Into 300 and the Interquartile Range Does Not Move

Quartiles have no single definition

The value 7.25 for the first quartile comes from interpolating between the second and third observations, and it is one of several defensible answers. Different statistical packages implement different conventions — some interpolate, some take the nearest observation, some exclude the median from each half before splitting — and the same ten numbers can produce first quartiles that differ by a whole unit. The disagreement shrinks as the dataset grows and never quite disappears, so an interquartile range quoted between two systems should name the method, and a difference of a few per cent between two tools is a convention rather than a bug.

Where the 1.5 in the outlier rule comes from

The convention that flags anything more than one and a half interquartile ranges beyond a quartile is a choice, not a theorem. It was proposed as a rule of thumb that catches genuinely unusual points without flagging too many from a well-behaved distribution, and it does that job reasonably for data shaped roughly like a bell. On strongly skewed data — incomes, waiting times, file sizes — it flags a large share of perfectly ordinary observations at the long end, because the distribution simply is long there. Treat a flag as an instruction to look at the value rather than as permission to remove it.

Frequently asked questions

When should I use the IQR instead of the standard deviation?

Whenever the tails are either untrustworthy or not the subject. Response times, incomes and file sizes all have long right tails in which a handful of values dominate the standard deviation, and reporting the median with the interquartile range describes the typical experience far better. Use the standard deviation when the distribution is roughly symmetric and when the extremes are the point — in quality control or in risk work, the rare large deviation is precisely what is being measured, and a robust statistic would hide it.

Does the IQR work on a small sample?

Poorly below about a dozen values, because each quartile then rests on one or two observations and moves whenever either of them does. The robustness that makes the measure attractive comes from having enough points that the middle half is genuinely a crowd; with eight numbers, deleting one can shift a quartile by a visible amount. Below that size the honest presentation is the sorted list itself — ten numbers fit on a line and tell the reader everything a summary would have hidden.