How Well Does the Mean Describe a Multimodal Probability Distribution? When you have highly skewed, irregular or multimodal distributions:



In these instances, does it become more advantageous to use the median instead of the mean to infer properties of these distributions? Does it become less advantageous to use the mean in these examples?
Thanks

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*Your first plot shows a bimodal distribution that is close to symmetric, so it is quite likely that the mean would be close to or equal to the median. A mean or median is just a single number that summarizes some kind of information about the distribution. A single number will never tell you everything about the distribution, so it is hard to answer "how well"  it works because the answer depends on what is important for you. It does not tell you anything about the multimodality, but neither would a median.


*With your second plot, it is hard to say if the distribution is "irregular", or you just used the wrong parameters for the kernel density estimator. In kernel density estimation using a smaller bandwidth would always lead to curly shapes, while a high bandwidth would smooth such shapes more. The same applies to the histogram: with large bins it would be smoother, while with small bins. it would be a collection of peaks.


*The third plot shows a skewed distribution. Again, choosing between mean and median would depend on what kind of information you want to summarize. The If mean is so sensitive, why use it in the first place? thread discusses in detail why  we use means and what are the ideas behind that. TL;DR you actually may want the mean to be influenced by the extreme values, so it summarizes the "whole distribution" better.
