In the Wikipedia article for mean-preserving spread, the following is claimed without citation:
If B is a mean-preserving spread of A, then B has a higher variance than A; but the converse is not in general true, because the variance is a complete ordering while ordering by mean-preserving spreads is only partial.
My intuition about the very meaning of "B is a mean-preserving spread of A" has been that the B distribution enjoys the same mean but a higher variance than A. I usually think of this as a risk-averse individual preferring the certainty of winning a dollar to a $.001$ chance of winning $1,000. In my mind, the individual would rank lotteries A and B as follows:
- Comparing the means of A and B
- Comparing the variances of A and B
And if 1 and 2 turn out to be equal, then the risk-averse individual is indifferent between A and B. But according to this article, that is false. What am I missing?