## Math proves that you aren’t crazy!

If you think about your friend group, chances are that it might seem like your average friend has more friends than you. Oddly enough, that statement is true for most people. It might seem like a contradiction that everyone could have more friends than everybody else, but if you look at the math, it actually makes sense.

Let’s take a sample set of people:

I’ve drawn lines between them to represent friendship. Let’s assume that friendship goes both ways despite that not always being the case in real life.

| Number of Friends | Their Friends’ Numbers of Friends | Their Friends’ Average Number of Friends |

Alice | 2 | 3, 5 | 4 |

Bob | 2 | 5, 5 | 5 |

Claire | 3 | 2, 3, 5 | 3.3 |

David | 5 | 2, 2, 3, 3, 4 | 2.8 |

Emma | 5 | 2, 2, 3, 3, 4 | 2.8 |

Fred | 3 | 3, 5, 5 | 4.3 |

George | 2 | 4, 5 | 4.5 |

Hannah | 3 | 1, 4, 5 | 3.3 |

Ian | 4 | 2, 3, 5, 5 | 3.8 |

Julie | 1 | 3 | 3 |

Average | 3 | | 3.7 |

I’ve bolded the number of people here who have more friends than their average friend and as you can see, that isn’t the case for most people. How is that possible? People with lots and lots of friends — like David, Emma and Ian — skew the results.

People with many friends will turn up in the third column more often since they are more people’s friends — this increases the average number of connections most people’s friends have. People like Julie who only have one friend don’t turn up very often — only for Hannah — so Julie’s number doesn’t decrease the average number of connections most people’s friends have.

“Your friends are not a typical sample. People with a lot of friends will be over-represented among them, because there’s a greater chance that you are one of their friends. People with few friends will be under-represented,” explains Ian Stewart in “Professor Stewart’s Casebook of Mathematical Mysteries.”

If you look at the average in column 2, you see that a typical person has three friends and that exactly half of the group has more friends than everybody else. The average in column 4 is skewed by the misrepresentative samples. In other words, David, Emma and Ian are making everyone else feel lonely — or maybe people are just happy to be their friends!