why is it possible that the set of data with the greater median has some values that are lower than values in the other set of data?
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this the answer of your question
Answer:
Think of a data set with three items in it. 9, 10, and 11. In this case the mean and the median are both 10. If you start increasing the highest number, 11, the mean jumps ahead of the median. Let's say you have 9,10, 1000. This is basically the "Bill Gates walks into a bar and make the average person a millionaire" affect. Now 10, and 11 constant and let's lower the 9, so let's say we have 0, 10, 11. Now the median is still 10 but the mean is going down. So in general (though not necessarily), if the median is lower then the mean, you have major outliers in the high end of the distribution, and if the mean is lower then median you have major outliers in the low end.
In the case of income distribution for example, I don't think there's any country where the mean income is lower then the median, because the income distribution always looks like a very large amount of poor and middle class people with a very small super-rich minority, hence major outliers on the high end.
A society where the median is higher then the mean would consist of a very large middle class, no extreme wealth, and a small and extremely poor minority.