Replacing outlier removal from IQR to MAD A common outlier removal formula is Q3 + IQR * 1.5 and Q1 - IQR * 1.5
Outliers can also be removed using Mean Absolute Deviation and Median Absolute Deviation.
Is anyone aware of any rules of thumb around these two latter methods? Similar to how IQR is scaled to 1.5 and tends to work well on many distributions?
 A: Boxplot outliers (according to your 1.5 IQR rule) are characteristic of
many common distributions. For example, among normal samples of size 100,
there is on average about one boxplot outlier per sample, and over half
of such samples have at least one outlier.
nr.out=replicate(10^6,  length(boxplot.stats(rnorm(100))$out))
mean(nr.out)
[1] 0.924752
mean(nr.out > 0)
[1] 0.521987

Here are boxplots for 20 normal samples of size 100.

Same for samples of size 100 from an exponential population.

A: Under some distributional assumption you could find a rule that would highlight as unusual a similar fraction of the distribution as that box plot rule does - at least in large samples.
E.g. For a normal distribution asymptotically 0.7% of the distribution is extreme by the box plot rule.
If you did median absolute deviation from the median 'MAD' as your spread and median as location, median + mad would be similar to Q3 and median - mad would be similar to Q1 (with a symmetric distribution), so median ± 4 mad should be like the boxplot bounds. The behaviour would be less similar with skewed distributions but may still be reasonable.
If you wanted some rule in terms of say mean and mean deviation or median and mean deviation you could set it so that in very large samples it also cut off about 0.7% of the distribution - but these would be more affected by outliers and less similar in behavior than the other two would be.
A simple calculation indicates that the large-sample equivalent cutoffs (at the normal) to the boxplot rule is about 3.38 mean deviations from the center (whether mean or median).

The limits calculated each way there are very similar. However, if we choose a heavier tailed symmetric distribution, the one based on mean deviation from the mean will be wider. If we choose a skewed distribution, all three will tend to be somewhat different.
