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I got a question today when talking about mean and median, IQR and variance. Is there a numerical measure of the robustness of a statistic?

I must confess that I had never thought about that before, so the question took me aback. I have not found anything yet and, to be honest, am not even sure if the question makes sense. Still, I keep thinking about it.

Is there some sort of score or number to say how much robust the median is than the mean? (for example)

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    $\begingroup$ From what I know, robustness is with reference to a particular assumption or set of assumptions, and given the variety of assumptions made in different analyses, I'd be surprised if there is any sort of all-encompassing measure of robustness. Perhaps others will suggest one and I'll learn something here. We can, however, certainly make statements such as statistic X is more robust than statistic Y with regards to assumption A, and that is something that could be quantified in a variety of creative ways based on your Monte Carlo simulations. $\endgroup$
    – ColorStatistics
    Commented Jan 4, 2021 at 20:04
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    $\begingroup$ There are two different terms that you should look into: "influence function" and "breakdown point". They both give numerical descriptions of different aspects of robustness but I can't do either of them justice in this space. (actually I couldn't do justice to influence function no matter how much space I had ). $\endgroup$
    – mlofton
    Commented Jan 4, 2021 at 20:27
  • $\begingroup$ @Luis: looks like mlofton made some good suggestions; I'll research those myself. I was thinking say the robustness of the t-statistic to the assumption of a random sample being from a normal distribution. How robust is it to the violation of normality? That could be creatively assessed with Monte Carlo studies and contrasted against the same violation for another statistic. That is what I was thinking. $\endgroup$
    – ColorStatistics
    Commented Jan 5, 2021 at 16:19
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    $\begingroup$ See stats.stackexchange.com/questions/511030/… from which you can use the breakdown point or the maximum of the influence function. And to answer your question, in a way the median is infinitely more robust than the mean. The mean is not robust. $\endgroup$
    – TMat
    Commented Mar 31, 2021 at 16:10

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