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# Why You're Probably Less Popular Than Your Friends

Where averages and individual perspectives diverge

Image: David Malan Getty Images

Are your friends more popular than you are? There doesn’t seem to be any obvious reason to suppose this is true, but it probably is. We are all more likely to become friends with someone who has a lot of friends than we are to befriend someone with few friends. It’s not that we avoid those with few friends; rather it’s more probable that we will be among a popular person’s friends simply because he or she has a larger number of them.

This simple realization is relevant not only to real-life friends but also to social media. In Twitter, for example, it gives rise to what might be called the follower paradox: most people have fewer followers than their followers do. Before you resolve to become more scintillating, remember that most people are in similar, sparsely populated boats.

The number of friends we have is typical of many situations in which the average deviates from individuals’ experience. Another is class size. Let’s imagine a small department offering three courses for the semester. One is a survey course with 80 students, one an upper-level course with 15 students, and one a seminar with five students. Now what is the average class size? Clearly, it is (80 + 15 + 5)/3, or 33.3 students. This is the number the department is likely to publicize.

But once again, let’s adopt the perspective of the average person and reexamine these numbers. Eighty of the 100 students find themselves in a class with 80 students, 15 find themselves in a class of 15 students, and five in a class of five students. Thus, the average student’s class size is (80 × 80 + 15 × 15 + 5 × 5)/­100, or 66.5 students. This number is less likely to be publicized by the department.

Of course, the argument applies to many situations. Consider population density. The average number of human beings per square mile of the earth’s land surface is low. Looked at from the perspective of the average human being, how­ev­er, the density is much higher because most humans reside in cities. Thus, we can conclude that despite being more crowded together than average, most of us are less popular than average.

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Paulos teaches mathematics at Temple University.

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1. 1. musicis4worship 02:18 PM 1/20/11

I find the idea at the end intriguing. Have heard the population growth stats put in this way, which I think are inaccurate to be sure: "By 2050 there will be 2 feet of space for every person on the planet." They calculate this based on supposedly the current trend of doubling our population every 50 years. They claim this rate is unsustainable at our PRESENT rate of growing food etc. What they underestimate is our ability to adapt. So... panic is not an option. Massive grave digging isn't either. China's idea of one child per family didn't work out too well, now that they have so few women in ratio to men. It's just a mess. Geo-engineering is a mess. Forecasting fails to account for the ingenuity of our species, and forces we cannot predict or control. One invention can change everything. One catastrophe. One disease. People make their plans based on faulty assumptions and mess things up that would have been fine had they left them alone. That's my soap box. Hope the Bilderberg's are reading this :)

2. 2. carlofab 11:01 PM 1/20/11

Everything in the article is about the present.

Still it seems to me legitimate to say that the present rate of population growth is unsustainable. It is not a prediction to state what the population will be in fifty years at the present rate of growth. That is simple mathematics, and could appear as a problem on a math test.

Don't know how many believe the population will actually reach such numbers, but I for one doubt it. Malthus was right, and we seem no more able to control our population than other animals. That the leaves natural controls which are extremely unpleasant. We are safe from predators that cull animal herds, but not starvation, disease, or other horrors.

The technology that supports our present numbers is extremely fragile. Cities produce no food and little water. These must be imported 24/7 by air, rail, or trucks. Any disruption, such as an earthquake, can cause a food crisis in days. So would a nuclear exchange. The number of people killed would be insignificant compared those who perish due to disruption of the infrastructure that distributes food and other essentials.

On a larger scale we have become dangerously dependent on an electronic infrastructure that is extremely fragile. Nuclear weapons detonated in the stratosphere would create electronic pulses capable to frying electronic grids over wide areas. So could a massive sun storm, or other natural source. Or a modest little virus.

A simple economic collapse might also do it. During the recent credit crunch, banks had little to loan because their assets were in unsalable real estate loans. This froze world shipping. The example I read was of a shipper of toys in China to a buyer in San Francisco. The buyer must send a letter of credit from a bank to assure the shipper he will be paid. Banks had no capitol for such purposes. So the ship doesn't sale, the buyer has no toys to sell, and toy factories in China close down.

A large scale economic collapse would bring on such a crisis so quickly there would be little time to react. So would a widespread blow to electronic infrastructure.
These are the things that make our present population level possible, and it would suffer a precipitous
decline without them.

What seems most likely to me is a new super virus.

None of the above bring about human extinction. But they would be fantastic catastrophes. At least those who survive, after perhaps a generation, would enjoy a better lifestyle. Labor would be in great demand.

3. 3. chkrvrty in reply to musicis4worship 01:23 PM 1/21/11

I agree that the 2 feet of space per person is an unnecessarily panic-inducing hypothesis. Clearly that's much much more dense than is sustainable, and population studies have shown that populations will not exceed the sustainable number that drastically.

However, I feel that some of your alternate "didn't work" examples are red herrings. The Chinese population control method has resulted in gender imbalance not because of the policy itself, but because they didn't understand the social benefit of men in that society and that, given the restriction of one child, many families would do whatever possible to ensure that one is a child. Geo-engineering has not (thus far) worked very well, but it was never meant to be a SOLUTION to the climate or population problems, rather a preventative measure. And it's also a work in progress, as is China's social engineering.

I think you're forgetting that humans' ability to adapt is BECAUSE of their ability to think outside the box and come up with unique solutions to problems. Was the concept of irrigation also a mess? At some point, even that concept may have been seen as "playing God by manipulating rain". Now we take such advances for granted. But they have allowed society to progress (or digress, based on your perspective) to where we are today.

In summary, I agree completely about humans' ability to adapt. However, I disagree that adaptation involves doing nothing. By definition, in order to adapt you have to change.

4. 4. bhallmar in reply to musicis4worship 02:15 PM 1/21/11

I disagree about the one child policy "not working out too well." It is significantly more complicated than that. While perhaps unfortunate, the gender imbalance in China actually has a beneficial demographic upside - it will reduce the population more. IOW think of it as surplus male labor that won't reproduce. Terrible at the human level, but not so terrible at the demographic level. I'm not arguing for it, I'm just saying that it's not so simple.

5. 5. chkrvrty in reply to chkrvrty 02:47 PM 1/21/11

"The Chinese population control method has resulted in gender imbalance not because of the policy itself, but because they didn't understand the social benefit of men in that society and that, given the restriction of one child, many families would do whatever possible to ensure that one is a MALE child."

6. 6. carlnz 11:48 PM 1/21/11

This logic also explains another dispiriting fact that I have noticed - you are probably in worse shape than the people around you at the gym.

Statistically, a gym member who spends 20 hours a week working out is more likely to end up standing next to you at the squat rack than another member who makes it in for half an hour a week. Therefore, when you go in to the gym, you are likely to be surrounded by unusually frequent trainers. This perhaps explains why many people feel a bit intimidated when they go to the gym.

However, on the upside, if you end up in hospital (perhaps because of your growing beer gut), your stay is most likely to be shorter than that of most of the other patients there.

7. 7. thejustkat 01:35 PM 1/22/11

I have lost all respect for SA. This article demonstrates a fundamental failure to understand the word "average". The word the author was looking for was "median". It is impossible, by the definition of the word "average" for "most" people to be either above or below average. It's the f*cking average. Seriously pathetic. The author should shave their head in shame and never pretend to know anything about math or science again.

8. 8. StatHack in reply to thejustkat 09:01 PM 1/22/11

Sadly, it is you who are mistaken. You should be ashamed to speak so harshly of someone's alleged misuse of statistical terms while making such an obvious error yourself. By definition it is impossible for the majority to be above or below the *median*. However, in many distributions the mean (average) and median are not equal, and therefore, by definition, more than half (the majority) are either above or below average.

9. 9. zstansfi 10:47 PM 1/22/11

To sum up, this article attempts to illustrate a very basic statistical concept: namely, that in a skewed distribution the majority of individual values will lie above or below the mean. Of course, I'm not sure why this is being discussed by Scientific American, as I'm pretty sure all of this magazine's readers have moved beyond grade school math.

Even more vexing is this question:

"Are your friends more popular than you are?"

What the hell does that mean?

Clearly the average person should be expected to have some friends who are less popular and some who are more popular than him or herself. So obviously this question cannot be stating that the average person is the LEAST popular of his/her friends. As a result, it is clear that this question is implying that the average person is not the MOST popular of his/her friends.

...Wow, a real eye-opener right there.

10. 10. Raghuvanshi1 03:02 AM 1/25/11

I think your definition of friend is vague.You are mixing up friend and acquainted. Movie star is may acquainted thousand fan but there are not friend.same way popular hero of any field have many acquainted, those who is not popular may have few friend.I don't understand why you raise this kind of sundry question?

11. 11. pajamasam in reply to musicis4worship 10:10 PM 1/26/11

Actually China's population control program worked quite well (ignoring the associated social and human rights issues). China's population growth rate is now one of the lowest in the world, in part due to the low proportion of women. Of course it would be great if China's population were actually decreasing, but the US is not a position to preach on this issue.

12. 12. asociologist 10:34 AM 1/27/11

This idea was much more profound, and much more thoroughly explained when it was published the first time 20 years ago. See "Why your friends have more friends than you do." By Scott Feld in The American Journal of Sociology, 1991.

13. 13. fsmaura 12:56 PM 1/29/11

I am not a mathematician. I understand the formula for obtaining the average: A+b+c/3.
I would appreciate explanation of the second formula:
the numbers are squared, added, and divided by 100.
Thank you.

14. 14. pedromgf 12:58 PM 1/31/11

This thesis leads to a paradox: many people would have no chosen friends, since they are more popular than every person they know.

15. 15. rm8877 12:21 PM 2/1/11

The average class size from the perspective of the student is correct only if each student only takes one class in the department. A more reasonable assumption may be that students take 1, 2, or 3 courses, and that all students are required to take the survey course. In this case the average class size would range from 70.99 (where 5 students take 3 courses) to 71.56 (where 20 students take 2 courses).

16. 16. bucketofsquid 01:14 PM 2/1/11

I find the blatant inaccuracy of the fundamental assertions in the "free" part of this article to be contemptable. I stress the free part because as a non-paying reader I was only able to read the teaser. I'm therefore a little confused by the discussion about population growth rates because that has nothing to do with friend ratios.

First of all, there is a big different between aquaintence and friend. A friend is someone that I choose to spend time with. An aquaintence is someone that I know.

Then there is the assertion that my friends have more friends than I do. Taken as a group the total number of friends of friends is obviously larger because there are more base points to work from. If we go on an individual basis then any given friend is equally likely to have more or less friends than I do thus showing that the primary assertion is useless in the real world. Had the article stated that one or more of my friends are more popular than I am, then I would have no issue with it because I can easily prove that to be true. By clustering all of my friends into a group and applying an invalid mathematical assertion to the group, all they do is demonstrate the basic dishonesty of advertisers.

As far as social networks go, I turn down most friend requests I recieve. I find it somewhat disturbing when teenaged girls I've never met want to be friends with a crusty old coot like me. Are they totally unaware of predators? Why aren't their parents paying attention?

17. 17. jstreet 01:43 AM 2/2/11

This sounds a little like the Groucho Marx paradox/joke, stated as "I would never join a club that would accept people like me as members."

The relation "popular" is ill defined.

It is not symmetric.

If we mean "x is popular with y" to mean "x likes y", then as Shakespeare has demonstrated, popular is not symmetric.

It's not reflexive either: "Joe is popular with himself" isn't always true as suicide proves.

Transitivity fails also. "Einstein is popular with his wife." "Einstein's wife is popular with her mother." It doesn't follow that "Einstein is popular with his mother-in-law."

There is, however a new logic of relations which fail to be reflexive, symmetric and transitive.

18. 18. berries@blueyonder.co.uk 10:51 AM 2/6/11

Re-Why you're probably less popular than your friends:

If I am less popular than my friends, what about those who regard ME as a friend? Are the two sets necessarily different?

Ian Berry

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