29. My Friends Are More Popular than I Am:

The Friendship Paradox

The Question:

Have you noticed that, in general, your friends have more friends on Facebook than you have? That they have more followers on Twitter than you have? That your boyfriend/girlfriend had more romantic partners than you had? In short, that you are less popular, on average, than your friends?

Depressing!

The Paradox:

No, don’t be miserable. It’s simply a fluke of statistics.

Nevertheless, the fact is, however, that the phenomenon is real. In 1961, in the study The Adolescent Society, the sociologist James Coleman showed that most pupils in the twelve high schools that he analyzed had fewer friends than their friends had.

In the digital age, you, dear reader, can easily verify the phenomenon yourself with data from the social networks. Most probably you will be able to confirm that the mean number of friends of your friends, or followers of your followers, is greater than the number of your own friends or followers. In 2012, the average Facebook user had 245 friends. But the average friend on Facebook had 359 friends. Only those who had over 780 friends had friends who on average had smaller networks than their own.

The phenomenon can be observed in different contexts too: people disproportionately experience restaurants, beaches, airports and highways as more crowded than they actually are on average. Students experience the average class size as being larger than it actually is.

If you are a scientist in the publish-or-perish business, you’ll discover that your co-authors have more co-authors than you and, what’s even more disheartening, they also have more publications than you.

Background:

Individuals tend to use the number of friends that their friends have as a basis to determine whether they themselves have an adequate number of friends. Scientists compare the number of their own publications to their peers’ publication lists. Unfortunately, when using these as measures of one’s own social or academic competency most people will feel relatively inadequate. While not all individuals have fewer friends than the mean of her or his own friends, most people do.

Scott Feld, also a sociologist, discovered the mathematical explanation for the paradox while investigating causes and consequences of patterns in social networks. He presented it at the Sunbelt Social Network Conference in Santa Barbara, California, in 1986 and his paper, Why Your Friends Have More Friends Than You Do, was published in the American Journal of Sociology in 1991, three decades afer Coleman’s study.

The explanation is surprisingly simple. (See below.)

Dénouement:

First, people with lots of friends are more likely to be among your circle of friends. Second, when they are, they significantly raise the average number of friends that your friends have.

To illustrate, consider the popular people, i.e., those with many friends or many followers. They show up in many groups. Wallflowers, on the other hand, with few friends or followers, show up in only few groups. Hence, the popular few will be encountered by many more individuals than the many introverts.

In Feld’s words, “If there are some people with many friendship ties and others with few, those with many ties show up disproportionately in sets of friends. For example, those with 40 friends show up in each of 40 individual friendship networks and thus can make 40 people feel relatively deprived, while those with only one friend show up in only one friendship network and can make only that one person feel relatively advantaged.” (That was in pre-digital times, when a group of 40 friends was considered as very large.)

In the same manner, students experience the average class size as being larger than it is according to the colleges’ data because, by definition, many students attend the popular classes, while few students attend the sparsely attended classes. Thus, they experience a higher average class size than exists for the college because many students are in the large classes, while few students experience the small classes.

To give a numerical example, in a class of 150 students 150 students will state that their class has 150 students. In a class of 10 students, 10 students will state that it has 10. Therefore, while the true average class size is 80 (= [150 + 10]/2), students will, on average, claim a class size of 141 (= [150x150] + [10x10] / 160). The underlysing reason is that when computing the weighted average of the class size, the numbers 150 and 10 show up twice: once as the numbers to be averaged and once more as their weights.

Additional info:

By the way, unless you are a fitness freak or body builder, most people you see at the gym are in better shape than you are. Again, no reason to be depressed. These fitter people spend hours each day at the gym, which is the reason why you encounter them in the first place. Whenever you show up, they are likely to be there. The lazybones, those people who most probably are less fit than you, are absent and you rarely get to see them. In other words, the people you encounter at the gym are not representative of the general population.

//// the same holds, for sexual partners/////

Corrections, comments, observations: