# Why John Tukey set 1.5 IQR to detect outliers instead of 1 or 2?

To define outliers, why we cannot use: Lower Limit: Q1-1xIQR Upper Limit: Q3+1xIQR

OR

Lower Limit: Q1-2xIQR Upper Limit: Q3+2xIQR

By definition, 50% of all measurements are within $\pm0.5IQR$ of the median. Compare this - heuristically - with a normal distributions where 68% are within $\pm\sigma$, so in that case IQR would be slightly less than $\sigma$. Cutting at $\pm 1.5IQR$ is therefore somewhat comparable to cutting slightly below $\pm3\sigma$, which would declare about 1% of measurements outliers. This matches quite well with the habit of using "$3\sigma$" as a bound in many simple statistical tests. On the other hand, cutting at $\pm1IQR$ would be like cutting near $\pm 2\sigma$, making about 5% outliers - too many; and cutting at $\pm2IQR$ would be like cutting at $\pm4\sigma$, thus turning even many quite extreme measurements into non-outliers. So $\pm 1.5IQR$ is also what Goldilocks would choose.
• The phrasing $\pm 1.5 IQR$ seems to imply that something is an outlier if it's more than $1.5 IQR$ away from the median, which is not correct. – Richard Rast Jul 9 '18 at 13:11