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The maths of friendship

Did you know that the number of friends you have is correlated to the size of certain parts of your brain? That you are far more likely to share genes and neural patterns with friends than non-friends? Or that social networks in the modern world closely resemble those in Norman England?

All this comes from new research by Robin Dunbar, the world-renowned evolutionary psychologist who famously discovered Dunbar’s number: how our capacity for friendship is limited to around 150 people.

Dunbar discussed his latest findings on a recent ‘Invest like the Best’ podcast, and here are some of the fascinating takeaways, along with context from his book, Friends.

“Our entire modern world is built around this number, 150”

Dunbar stumbled across Dunbar’s number due to his studies on whether the size of an animal’s brain is linked to the size of group they live in.

“Hunter-gatherers like us live in multilayered social systems. So I was looking for a layering within their social systems that was about a 150 in size rather than, say, 2,000 or 25. And sure enough there is one. It's generally called a clan or a regional grouping or something like that. It's basically a unit that shares hunting territory as much as anything… After that, we just kept finding this number everywhere… our entire modern world is built around this number, 150.”

When I first heard Dunbar talking about seeing clusters of 150 people everywhere, I thought he was simply seeing what he wanted to see. Like spotting an elephant shaped cloud or seeing Jesus on a bit of toast. Then I read Dunbar’s book. In a lot of cases, it seems that collectives of around 150 people really have popped up an awful lot – from village sizes recorded by William the Conqueror’s England to agricultural cooperatives in Italy, off-grid sects and factory sizes.

Obviously, very few of us live in villages or are part of agricultural collectives today. But Dunbar still sees complex human systems clustering around this number 150 – from the way tax is collected in Scandinavia, to the amount of people in offices and the size of trailer park neighborhoods in Germany. Why? Because, according to Dunbar, 150 is the maximum number of meaningful relationships a person can maintain at any time.

Dunbar’s layers of friendship

Your personal social network, all the family, extended family and friendships you have, looks pretty much like the ripples on a pond where you throw a stone. You have these series of layers going out from you, which increase in size but the quality of the relationship and the frequencies that you contact them gets smaller and smaller, smaller, lower and lower.”

Here's how Dunbar defines the different layers:

  • 1500 faces you can recognize (including people you don’t know personally)
  • 500 acquaintances
  • 150 meaningful relationships – your wedding or funeral attendees
  • 50 friends – your wedding or funeral attendees during Covid restrictions?
  • 15 people in your sympathy group, whose death would truly upset you
  • 5 ‘cry on the shoulder’ close friends

The number of people in a further out layer also includes the people in all the closer layers. So, the 1500 faces you can recognise includes the 500 people who are at least an acquaintance, plus 1000 others. Your sympathy group of 15 includes your 5 closest friends plus ten others, and so on.

Do Dunbar’s layers add up?

For the closest layers, Dunbar’s numbers seem roughly right. For a bit of context, I moved to this part of the world from Scotland about 18 months ago. Because I live 17,000 kilometres from most of my family and friends, I decided to use WhatsApp messages instead of real-life interactions to test Dunbar’s theories.

Except for my partner, there are three or four people I have messaged every few days over a long period. There are a few more people I’ll message once or twice a month, and a bigger group I’ll message every few months – usually around their birthday or if something funny happens to their football team.

I also thought about how I share big news. I’ve only told a few people about my new job, and I told them in different ways. I told my partner as soon as I got the offer and I told my brother, parents and one friend the next time we spoke. I guess this reveals my inner circle, which adds perfectly to Dunbar’s layer of five. Those are the people I felt would want to know and wouldn’t just think I was bragging.

I struggle to see how Dunbar put a number on the further out layers. But I find it fascinating that modern research – including studies of over a million Facebook accounts and email exchanges by 30,000+ college students – tend to suggest that personal networks remain similarly sized to how they were in Norman times.

Dunbar’s most important layer

Even though Dunbar christened 150 as his number, he sees this inner layer of 5 as the most important for human health.

“The best predictor of your mental health and physical health well-being, and even how long you're going to live into the future from today is predicted by the number and quality of friendships you have in that layer. The five is an average. So if you only have three don't panic yet, because introverts tend to prefer to have fewer people but have stronger friendships.”

Again, I was skeptical about this. I turned to Dunbar’s book for more information. As it turns out, the whole first chapter is dedicated to showing several studies that suggest the quantity and quality of social connections you have is reflected by your health. Not just your mental health, as might be expected, but your chances of avoiding and surviving physical ailments too.

Although it was interesting, I couldn’t help but wonder if it’s correlation rather than causation. For example, playing a sport is the best way I know to meet people as an adult. If a lot of other people find that too, wouldn’t their better health be at least partly due to playing more sport?

Dunbar, though, goes all in on friendship:

“You’ll certainly do yourself a favour by eating better, taking more exercise and popping the pills they give you, but you’ll do considerably better just by having some friends”.

The 1.5 closest friends you can have

Dunbar’s ‘five loved ones’ is usually shown as his tightest layer of friendship, but he takes it further and says that the closest possible level of relationship are the 1.5 people found in your ‘intimates’ layer. Why 1.5? Because, according to Dunbar, men and women behave differently.

His and apparently several other studies suggest that most women end up with two people in the inner sanctum - their romantic partner and a ‘BFF’, their best friend forever. Men only have room for one. So, when they find a romantic partner, their best friend gets relegated to ‘just’ a close friend – hence the average of 1.5.

This rings true for me. Meeting my partner didn’t stop me from talking to my closest friends on an almost daily basis. But I certainly played golf less often and no longer needed a drinking buddy – Dunbar’s definition of a best friend – on Tuesday nights anymore.

Why friendships fade

If you don’t meet up once in a while face-to-face, nothing on the face of earth or Facebook is going to stop that friendship gradually becoming an acquaintanceship.

Studies suggest that humans spend about 20% of their waking day in social situations. Those 3.5 hours a day need to be split between the 150 friends and family in your network. And, of course, we don’t split the time evenly.

According to Dunbar, about 40% goes to closest five, 20% goes to the additional ten people who make up your ‘sympathy group’, and the remaining 40% is spread between the 135 other people you know.

Keeping someone at a closer friendship layer requires a bigger investment of time. If it becomes harder to make this happen – due to distance or something else – they will drift to a more distant layer. A study Dunbar mentioned on students leaving high school suggests that this can happen quite quickly for anyone who isn’t family. Just six months without contact can see somebody drift out a layer, with the rate of decline slowing as time goes on.

The family immunity aspect seems right. If anything, I keep in more regular contact with my Mum, Dad and brother than before I moved Down Under, even if it isn’t ‘quality’ time of the in-person variety. I also wonder if there’s a bit of Nassim Taleb’s Lindy effect going on.

The Lindy effect suggests that something’s remaining lifespan is best predicted by its lifespan to date. When I look at the only friend from home still in my ‘inner five’, it’s the one I’ve been friends with for longest. Unfortunately for my liver, it looks like my friend Bruce will be around for at least another 20 years – especially as he recently made the move from Scotland to Australia too.


Joseph Taylor is an Associate Investment Specialist, Morningstar Australia and Firstlinks.


June 10, 2024

"Can you squeeze Pi and Schrodinger’s cat in there somewhere too?":

By your command:
e^(i * pi) + 1 = 0

Is the square of a number = -1 if it is never observed?

May 27, 2024

On the fun side of Dunbar hitting his eponymous number of 150, humans can rest easy that it is more than sufficient to dabble in so many friends. Imagine the cost of birth day and festival cards, and time wasted in gossiping about the friends and non-friends!

As for the magic number 150, what can one do as we do not get to choose our brain sizes? It is a non-brainer!

The famous evolutionary biologist Richard Dawkins has revealed that the Pope might be your and my long-lost half sibling, thanks to the dispersal of genes. That should prove redemptive enough…

May 23, 2024

Eagerly awaiting a Dudley formula for this...

May 23, 2024

Appears Dunbar numbers have significant error ranges, probably due to being averages of a noisy observations. I suggest the number 2 at play:

Layer Dunbar Dudley Formula
1 5.000 4.645 = SQRT(2) * EXP((2 ^ (1 / 4)) * 1)
2 15.000 15.256 = SQRT(2) * EXP((2 ^ (1 / 4)) * 2)
3 35.000 50.109 = SQRT(2) * EXP((2 ^ (1 / 4)) * 3)
4 150.000 164.581 = SQRT(2) * EXP((2 ^ (1 / 4)) * 4)
5 500.000 540.562 = SQRT(2) * EXP((2 ^ (1 / 4)) * 5)
6 1,500.000 1,775.463 = SQRT(2) * EXP((2 ^ (1 / 4)) * 6)

May 26, 2024

?? Dudley's not just a pretty number

Peter Vann
May 28, 2024


exponential growth at play with all constants in the formula the number 2 (don’t forget 4= 2^2)

Brendan O’Brien
June 09, 2024

I love it when the maths gurus get a whiff of predictability.
Can you squeeze Pi and Schrodinger’s cat in there somewhere too?


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