Thomas Schelling, in his fascinating book Micromotives and Macrobehavior (pp. 64-65 of the 2006 Norton edition) writes:

Ask people whether they consider themselves above or below average as drivers. Most people rank themselves above. When you tell them that, most of them smile sheepishly.

There are three possibilities. The average they have in mind is an arithmetic mean and if a minority drive badly enough a big majority can be “above average”. Or everybody ranks himself high in qualities he values: careful drivers give weight to care, skillfyl drivers give weight to skill, and those who think that, whatever sle they are not, at least they are polite, give weight to courtest, and come out high on their own scale. (Thsi is the way that every child has the best dog on the block.) Or some of us are kidding ourselves.

I’d long heard something similar in that “75 percent of students entering [insert elite college] think they’ll be in the top quarter of their class”; “top quarter” presumably means top quarter in GPA, so everybody is working on the same scale. But as Schelling points out, people probably judge driving on different scales. And I really don’t think people think intuitively in terms of means in the driving case; that requires doing arithmetic on numbers that don’t even exist.

So naturally I wondered how strong the effect is. Let’s assume that person $i$ has scores in two variables, $x_i$ and $y_i$, drawn from independent standard normal distributions. Let $z_i = (x_i + y_i)/\sqrt{2}$ be their “objective” score — weighting the two variables equally, and renomalizing to have variance 1.

Now consider someone with, say, $x_i = 1, y_i = 0.5$. Their objective score is $z_i = 1.5/\sqrt{2}$. Objective scores are standard normal; thus they are at to be at the $\Phi(1.5/\sqrt{2}) \times 100 = 85.56$ percentile of the distribution of objective scores.

But now say this person perceives the world around them using a subjective score of the form $(2x_i + y_i)/\sqrt{5}$ — since their $x$ is higher than their $y$, they naturally assume $x$ is the more important trait, following Schelling. Again we force the scores (from this person’s perspective) to be standard normal. Such scores have mean 0 and variance $\sqrt{5}$; this person’s score is $2(1) + 0.5 = 2.5$. So they are, according to their own perception, at the $\Phi(2.5/\sqrt{5}) \times 100 = 86.83$ percentile. The effect here is actually particularly weak, since this person is relatively well-balanced. If $x_i$ and $y_i$ are much different the effect is larger. For example if $x_i = 1$ and $y_i = -1$ then the individual in question is “objectively” exactly average, but they give themselves a subjective score of $1/\sqrt{5} = 0.45$ and therefore perceive themselves at the 67th percentile.

For a general person let $w_i = (2 \max(x_i, y_i) + \min(x_i, y_i))/sqrt{5}$ be their “subjective” score, weighting the variable in which they have the higher score double. Let $obj_i = \Phi(z_i)$ be the “objective” percentile rank, and let $subj_i = \Phi(w_i)$ be the ‘subjective” percentile rank. Then the mean $subj_i$ from a quick simulation is about 57 percent — people think they’re better than 57 percent of others, on average, when of course the “objective” truth is 50 percent.

Here’s a scatterplot of $z_i$ against $w_i-z_i$ — that is, the objective rank against the “perception gap”.

Here’s the distribution of people’s self-perceived ranks, from a simulation of ten thousand individuals. The histogram for objective ranks is nearly flat. The piling up near the right indicates that people are thinking more highly of themselves than would be objectively true. For a very quick measure, in this simulation of 10,000 individuals, 5,937 perceive themselves as above the median.

Here’s the distribution of “perception gaps”, the difference between subjective and objective ranks.

And somewhat surpisingly, about six percent of people have a lower subjective rank than objective rank. This is the “well-rounded” group that has both $x_i$ and $y_i$ positive, and $y_i/x_i$ between about $0.7$ and $1.4$.

One expects this effect to be stronger if there are more factors to be considered; I’ll save that for another post.