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I was given this question by my friend who said he got it from his teacher (which meant he couldn't clarify the terms in this question).
The question is: What is a better measure of central tendency? The mean or the median?

I'm not sure what 'central tendency' means but I was thinking it means "the middle". If so, I thought the median by definition will be the best, since its definition is that it's the middle value.
I can't think of any case when the mean is better. In a sample with outliers, the median beats the mean.
In censored observations, median also beats the mean.

When does the mean beat the median?

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    $\begingroup$ It's a meaningless question. The mean and the median are different features of the 'central tendency' with different uses; they can be better at something but they can't be better without qualification. $\endgroup$
    – Mees de Vries
    Jun 26, 2019 at 10:48
  • $\begingroup$ There is no 'better measure' unless you specify a situation. $\endgroup$
    – StubbornAtom
    Jun 26, 2019 at 12:14

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Consider a sample of the Marron and Wand model number 16, a mixture of normal density functions. The mean is going to be near the $0$ value, and that is what is the central tendency for us in that case. On the other hand, the median is going to be, probably, on one of the two "mountains" of density. Marron and Wand model number 16

Now, if we simulate a sample of this distribution with size $1000$, we obtain a mean of $0.0679332$ which is the center of the density for us, while we obtain a median of $2.176665$, been in this case the mean so much better. This happens because the simulated sample has the following density plot:

Simulated sample density

And as you can see the values near of the $0$ practically never happen, so the median is going to be biased to one of the two "mountains".

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