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I did not say that the list that I provided is the exhaustive list. I just wanted add more to it. So if the Q is "is mean=median" What constitute sufficiency other than arithmetic progression and/or evenly spaced sets? Anything about the relationship that can be expressed in terms of range, relationship between the smallest or largest number, or the relationship between median and other numbers etc.?
Perhaps the easiest way to understand when mean = median is this one reqiurement:
1) When the distribution above the mean is symmetrical to the distribution below the mean.
So think of the middle point. Whatever pattern of data you have above the middle point is mirrored on the other side (below) the middle point.
When this happens, the mean will effectively be the same as the median (the middle number by rank).
This is satisfied when data is evenly spaced out. It's also true in other mirror-like cases.
Sure the distribution does not necessarily have to be exactly mirrored on both sides. But as long as they roughly cancel each other out, then you'll have a situation where the mean = median.