Do you ever get down on yourself?
You feel like you’re unpopular, like all your friends have way more friends than you?
Well look, kiddo, here’s the thing, bucko: everyone feels that way sometimes.
Now some people will tell you that it feels that way because of the false lens of social
media, or because you’re being too self-critical, but I’m here to tell you that you feel that
way because… it’s true.
Your friends are more popular than you.
Seriously.
I can prove it.
With math.
It’s called the friendship paradox, and it was first described by mathematical sociologist
and guy who should never be a middle school guidance counselor Scott L. Feld, in a 1991
paper in the American Journal of Sociology entitled “Why Your Friends Have More Friends
Than You Do.”
The simplest way to think of the friendship paradox is as a form of sampling bias.
So, if we were trying to figure out how many friends the average person has, we’d have
a list of all the people in the world, and how many friends they have—let’s call
that a friend count—and we’d just take the average of all the friend counts.
But, if we were trying to figure out how many friends an average person’s friend has,
we’d have to do something different.
We’d call up every person in the world and we’d say “please list the names of all
your friends.”
And they’d say “How did you get this number,” or actually more likely they’d say “你是怎么得到这个号码的”
and then we’d have to explain that we aren’t really calling them and that this is all part
of a hypothetical scenario.
That would calm them down, and eventually, every person in the world would give us a
list of their friends’ names.
Assuming that most people have more than one friend, we’d end up with a list of names
that’s way longer than the number of people in the world.
How?
Because there would be a bunch of repeated names on there.
Now, we have to ask ourselves, who will those repeats be?
Well, overwhelmingly, they’ll be the people with a lot of friends—after all, if you
have a lot of friends, then when people get asked who their friends are, your name will
come up a lot.
The effect is that when we average the friend counts of all the names on this extra-long
list of friends, the high friend counts of the people who are on the list a lot will
bring up the average, meaning that the number of friends held by the average friend will
always be higher than the number of friends held by the average person.
But according to YouTube statistics, only 23% of viewers will understand a concept after
just one example.
And that’s because people are dumb—for example, YouTube statistics show that 83%
of viewers will believe totally made up statistics.
So let’s do another example: let’s pretend like everyone in the world died in a robot
apocalypse except for three people who betrayed humanity to the robots in exchange for their
own survival.
And for no particular reason, let’s name those three people Bill, Jeff, and Elon.
Bill is friends with Jeff and Elon.
But Jeff and Elon are not friends with each other.
So, Bill has two friends, and Jeff and Elon each have one, so the average number of friends
a random person on the island has is 1.33.
But now let’s take the average number of friends someone’s friends have.
Hey Bill, who are your friends?
Jeff and Elon.
Okay, Jeff, who are your friends?
Bill.
And Elon?
Also just Bill.
So now we have a list of “people’s friends.”
Jeff, Elon, Bill, and Bill.
Look at that.
Bill shows up twice, just as we expected.
So when we now average how many friends the average friend has, we have to count Bill’s
high friend score twice, and find the average is 1.5—higher than the 1.33 of an average
person.
Essentially, unlike the number of friends an average person has, which is a simple average,
the number of friends an average friend has is a weighted average, and necessarily, popular
people will always be weighted more.
Now, the averages thing is a mathematical truth.
But as rapper Pitbull once said, “mathematical averages can often lead to misleading real-world
conclusions.
Mr Worldwide.”
If middle school taught me anything, it’s that popular people are more likely to have
popular friends, unpopular people are more likely to have unpopular friends, and people
with no friends tend to start YouTube channels.
The good news is that mathematicians and sociologists have worked on more complicated versions of
the friendship paradox and found that even when you take into account real-world social
dynamics, it’s still true that for most people, their friends have more friends than
they do.
The friendship paradox isn’t limited to friendships—in fact, researchers have a
number of other ways they can use it to embarrass us.
The same math also proves, for example, that most people’s sexual partners have more
sexual partners than them.
Also, data showing that more popular people tend to be rich and good looking means that
in addition to being more popular, most people’s friends are, on average, richer and better
looking than them.
And even the hot, rich, popular, promiscuous researchers weren’t immune: a 2014 study
found that on average, a researcher’s co-authors are more prominent and well-published than
they are.
But, it turns out that all this research wasn’t just to make us feel bad.
The friendship paradox actually has at least one useful application: studying infectious
disease.
A 2010 study tried to detect a flu outbreak by tracking two groups: one was random, but
the second was comprised of friends that the random group members recommended.
It turned out, the group of “friends” allowed researchers to detect a flu outbreak
2 weeks sooner than the random sample, because the friends tended to have more contacts.
So, don’t think of people with a lot of friends as popular: think of them as superspreaders.
What the friendship paradox really serves to demonstrate is the power of sampling bias:
the way in which you sample something can dramatically influence the result.
This is part of what makes statistics so complex: you typically have to take inherently flawed
data and process it in a way that draws meaningful conclusions.
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