I'm looking at this picture on wikipedia, comparing the median and mean of an arbitrary distribution. But what does it mean exactly? From the figure, it looks like the mean is the center of mass of the distribution (in the sense that you could balance the distribution on that point). However, since the median has $50\%$ to its left and right, it seems a more reasonable candidate for the center of mass. Which one is it? And what is the geometric interpretation of the other metric?
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The median is the point that splits a set into two equal subsets. The mean is the average of all elements in the set. Both are measures of center, but the mean is more sensitive to the extreme values (i.e., outliers).
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The image is not clear. The prob dist has 50% of its values by quantity to the left of the median, and 50% to the right. Each value is put in ascending order and is counted once in computing the median, no matter its value. The dist has 50% of its mass/"area under the curve" to the left of the mean. This is a quick computation using the formula for expected value (technically the discrete and continuous cases are slightly different but the idea is the same).
The two sets $\{1,2,3\}$ and $\{1,2,100\}$ have the same median. But the second set's center of mass is at $34$.