Can anyone explain this mathematically?

[Decrement]

I have a question that i have wondered ever since my undergraduate days. Is it possible to have a unimodal distribution such that the mode is in-between the mean and the median? Ie. Median<Mode<Mean, or Mean<Mode<Median.

I have experimented with various asymmetrical distributions such as the Chi-Square, Lognormal, Pareto, Weibull etc and I’ve found using different parameter values that this is not possible.

Does anyone have any idea how this could be proved generally, or even provide some sort of explanation based on some relation to the cumulate distributive function/maxima of the distribution?

I'm hoping someone here can come up with something, most of my tutors said they’d "try" to find an answer but obviously had no clue.

[omega]

Very interesting question indeed. Did you try any discrete distributions? Like this one below:

0.0
1.0
2.0
3.0
3.0
3.1
4.0
5.0
3.2
3.3


Descriptive Statistics: C1

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum
C1 10 0 2.760 0.454 1.436 0.000 1.750 3.050 3.475 5.000

And the mode of 3 is between the mean of 2.76 and the median of 3.05.

[Decrement]

Originally Posted by omega

Very interesting question indeed. Did you try any discrete distributions? Like this one below:

0.0
1.0
2.0
3.0
3.0
3.1
4.0
5.0
3.2
3.3


Descriptive Statistics: C1

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum
C1 10 0 2.760 0.454 1.436 0.000 1.750 3.050 3.475 5.000

And the mode of 3 is between the mean of 2.76 and the median of 3.05.

I didn't consider discrete distributions, but yes i would love to know if it is true for all continuous distributions, it would be easyest if a similar counterexample could be found to put this conjecture to rest!

[omega]

We can make a continuous distribution that is very similar to the discrete distribution above, with almost the same probabilities at 0, 1, 2, etc., and very small probability in between. The descriptive statistics would be almost the same. So there are continuous distributions that satisfy the inequality.