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

The mean means what it means
Whenever you compute a single real value that describes some aspect of a distribution ---whether this is the mean, mode, standard deviation, kurtosis, a particular quantile, or whatever--- that quantity measures what it measures and not what it doesn't measure.  So the mean always measures the mean, irrespective of whether the distribution is unimodal, bimodal, trimodal, etc.  Now, you ask whether the mean is good to "infer properties of these distributions".  This begs the natural question, which properties?  If the property of interest to you is the "centre" of the distribution, then obviously the mean will represent that property extremely well.  On the other hand, if the property of interest to you is something else (e.g., the mode) then the mean might represent that very poorly.
All of this is just another way of saying that real quantities computed from distributions generally represent only one aspect of the distribution, and there is a loss of information when transitioning from the distribution to a descriptive quantity.  So if you want to use descriptive quantities to represent properties of the distribution, you need to be specific about what properties are of interest to you.  There is no single quantity (other than the distribution itself) that will give you "the properties" of the distribution.
