How to use box plots to detect outliers?

Question

Suppose for simplicity that we have Gaussian distributed data with some outliers, whose typical characteristic is getting values that are far from the mean. Suppose my sample size is N.

I was thinking that box plots could handle such a situation, but then I thought that, even without outliers, the region Q1 − 1.5IQR and Q3 + 1.5IQR should contain around 99.3% of observations, i.e. we would have N * 0.7 samples declared as outliers even in the absence of outliers (and even under the hypothetical condition of perfect normality...).

Is there a way to circumvent this behavior? For example, we could check the difference between the observed number of samples outside the whiskers and the expected one N * 0.7. Then declare that there is an outlier outside the whiskers only if this difference is statistically relevant. This way we would have the opportunity to declare that no outliers are present, maybe...

By varying the default threshold of 1.5, maybe we could then also localize the outlier.

Would it make sense such a technique for detecting outliers? Are there techniques for implementing this rough idea?

Answer 1

I think all your computations are correct, so with that criteria you expect outliers at a given frequency. I think this makes sense, even by chance you can get extreme values, right? The point you raised seems to be somehow related to the question "how likely is to observe this specific value,assuming a certain underlying distribution?", which in turn can be addressed using quantiles. Not all outliers would be equal at that point, and maybe you can decide how extreme a point needs to be in order to label it as an anomaly.