Reading today about illusory superiority to improve the Wikipedia article, I came across something tangential but intellectually delightful.

**Most people have fewer friends than their friends (on average) have.**

When I first read it, it sounded impossible, but it’s a practically inevitable fact.

It’s not specifically about friendship, but a mathematical fact about any relation which is symmetrical and which varies across a population. Consider a set of people such as the students in a college. There are some odd cases we have to ignore, for example:

- nobody knows anybody else (all have 0 friends)
- everybody is part of an insular couple (all have 1 friend each)
- everybody knows everybody else (all have the same large number of friends)

Instead we’re considering realistic situations: each person in the group knows some others but not necessarily all the others. People differ in popularity: some have more friends than average, some fewer. Oh, and it’s essential that friendship is a two-way thing: if A is B’s friend, then B is A’s friend.

In these circumstances, you can imagine counting how many friends someone has, and comparing it to the average of how many friends their friends have. Most of the time, that second number will be bigger.

I’m not sure I follow the rigorous mathematical proof, but you can get a grasp of it intuitively but drawing networks and counting the number of connections each point in the network has. The paper that made me aware of this includes one such diagram (see below).

Another intuition pump: imagine one student comes along who is amazingly popular and instantly makes friends with everybody. That new person knows lots of people who are less popular than him, but on the other hand, everybody else now knows someone more popular than they are. This might not be good for everyone else’s happiness if social comparison is important.

Another way to get the point is to imagine that you’re an averagely popular person. Consider the least popular people: the total loners or the insular couples. They have fewer friends than you. The set of your friends isn’t a representative sample of the population, because it doesn’t include these people: it’s biased towards people who have friends. So even though your popularity is the same as the population average, the popularity of your friends, on average, is higher than that.

The irony is that when you actually do the surveys and get people to evaluate their own popularity, they (on the whole) describe themselves as having more friends than their friends have. The friendship paradox shows this is mathematically impossible, and so demonstrates that there’s a self-enhancement effect (an illusory superiority) in people’s estimates of how popular they are. It’s as though we find it hard to handle the idea of being less popular than our friends, and the biases kick in to prevent us from acknowledging that.

**Reference:** Zuckerman, Ezra W.; John T. Jost (2001). “What Makes You Think You’re So Popular? Self Evaluation Maintenance and the Subjective Side of the “Friendship Paradox”“. *Social Psychology Quarterly* **64** (3): 207–223. doi:10.2307/3090112.