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

There are many good answers here. I’ll just add here the general point to be made. You can summarise a distribution of values with a single number, the mean (or even two, say the standard deviation) but you always have to remember you are loosing information in doing so. That’s why you look at histograms before summarising.
So after looking at your histograms above, I would summarise the distributions in three different ways.

*

*In the first case: the distribution is multimodal, hence mode is a better metric than mean or median. Take both modes, assume a "cut" in the middle and report interquartile range for "left" and "right" distribution.

*In the second case: this may still be "normal-like" if the number of observations is small. You can report mean or median, and interquartile range around one or the other

*In the third case, the didstirbution is highly skewed. You could report median and interquartile range. Even better, look at the histogram using log-scale on the y axis first to see if there’s a second maximum - there seems to be one. If so, report median and interquartile range around first and second peak. You will need to choose and arbitrary value to say where the first distribution starts and the second ends

PS a better way for multimodal distributions would be to assume a parametric form for each component, and use maximum likelihood to estimate the relative components. But that’s likely an overkill in your case
