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?
Can mode lie between mean and median
statistics
Related Solutions
The answer is "No," as the article linked to by leonbloy indicates. Here's an example of the kind of situation in which you would get mean < mode < median. The black rectangle below contains the mean, the blue rectangle the mode, and the red rectangle the median.
The black rectangle is $5 \times 1/5$, the blue $1/4 \times 4$, and the red $1 \times 3$. The median is in the red rectangle because the areas of the three rectangles are 1, 1, and 3. The blue rectangle clearly contains the mode. Placing the origin at the lower left corner of the blue rectangle, we see that the mean is at $(-2.5)(1) + (1/8)(1) + (3/4)(3) = -1/8$, and so the black rectangle contains the mean.
Of course, this can be scaled to produce an actual probability density function or smoothed to obtain a continuous pdf without changing mean < mode < median.
Added: One of our own users, Henry, has written a detailed article on the relationship between the mean, median, mode, and standard deviation in a unimodal distribution. It's definitely worth a look.
I guess that you are making a confusion between the values of the random variable (let X) and the frequencies.
We are not talking about the average height of the bars in the plot (i.e. the average frequency), we are talking about the average value of X, which is weighted by the frequency. The formula is the sum of X times the frequency of X over the sum of frequencies.
The mode is the value of X that achieves the highest frequency (i.e. the highest bar); it is not the frequency that occurs most often (often all frequencies are different).
The median is the value of X that reaches the half of the number of samples (you must accumulate the bars from left to right until you reach 50% of the total bar height); it is not "the bar in the middle".
The value of X which is the midpoint between the extremes is called the midrange.
In the case of a symmetric histogram, all these values are equal.
Best Answer
First, left- or right-skewed tells you nothing about the mode.
For a case you are looking for try out $$0, 1, 1.5, 1.6, 2, 2, 12$$