I recent survey from Greenberg, Quinian, and Rosner found that 80% of Americans believe they will be in top half of the population in income in the next 5 years. Of course, it is impossible for 80% of the people to be in the top half, right? Is this an optimism bias where people foolishly think their prospects are better than they really are?
Maybe not. If we look at the income distribution, it is severely skewed, but still has a big mass near the mode. So if we take the mode as what the average American sees as the average, and use a +/- 20% as the precision that most people attribute to their perceptions of other peoples' income, the belief may not be incorrect. Think of the simple example where:
10% of Americans make less than $10k/year
10% make between $10k and $20k
60% make between $20k and $50k
10% make between $50k and $100k
10% make over $100k
If the typical person does not have the precision to discriminate within the $20-50 group, then anyone making more that $20k is already in the top "half." So 80% (10+10+60) are in the top half. I just made up these numbers, but you can see that any distribution with a big center would have this same effect. So maybe the average American is not as irrationally optimistic as we think (not that optimism is bad thing, of course).