Can mode lie between mean and median?

The distribution is left-skewed if mean<median<mode.

The distribution is right-skewed if mean>median>mode.

Can mode lie between mean and median?

Solution 1:

First, left- or right-skewed tells you nothing about the mode.

For a case you are looking for try out $$0, 1, 1.5, 1.6, 2, 2, 12$$

Solution 2:

Paul's answer works, but it is also possible to have a distribution which is unimodal and either is continuous or is discrete with equal gaps in the support, such as the following distribution with median $2$, mode $3$ and mean $3.02$:

x   P(X=x)
0   0.16
1   0.17
2   0.18
3   0.19
4   0.06
5   0.055
6   0.05
7   0.045
8   0.04
9   0.03
10  0.02

It is possible to have the mean, median and mode in any order, though having the mode strictly between the other two may in a vague handwaving sense be less likely than other patterns, as I once tried to show