为何朋友的好友比你多？ Why your friends seem to have more friends than you do

你是不是会经常觉得自己的朋友所拥有的朋友比自己多呢？别苦恼，这绝对不是你的问题！不相信的话看看下面的调查吧。

You spend your time tweeting, friending, liking, poking, and in the few minutes left, cultivating friends in the flesh. Yet sadly, despite all your efforts, you probably have fewer friends than most of your friends have. But don’t despair — the same is true for almost all of us. Our friends are typically more popular than we are.
你花时间写微博、交朋友、挖八卦，还用剩下的时间来亲自建立友谊。但悲哀的是，尽管做了这么多努力，你的朋友可能还是比你大多数朋友少。但不要绝望，我们大多数人都是这样。一般来说我们的朋友比我们受欢迎。

Don’t believe it? Consider these results from a colossal recent study of Facebook by Johan Ugander, Brian Karrer, Lars Backstrom and Cameron Marlow. (Disclosure: Ugander is a student at Cornell, and I’m on his doctoral committee.) They examined all of Facebook’s active users, which at the time included 721 million people — about 10 percent of the world’s population — with 69 billion friendships among them. First, the researchers looked at how users stacked up against their circle of friends. They found that a user’s friend count was less than the average friend count of his or her friends, 93 percent of the time. Next, they measured averages across Facebook as a whole, and found that users had an average of 190 friends, while their friends averaged 635 friends of their own.
不相信？想想最近Johan Ugander、Brian Karrer、Lars Backstrom 、Cameron Marlow最近写的有关Facebook的论文吧。（信息披露：Ugander是康奈尔的学生，我是他博士论文答辩委员会成员。）他们调查了Facebook上所有的活跃用户（当时为7.21亿人，约为世界人口的10%），其中共有690亿对好友关系。研究人员研究了用户们的数量与其朋友的朋友数量，发现93%的用户的朋友数量低于他朋友的朋友数量的平均值。之后，他们从整体上对Facebook进行了研究，发现平均一个用户有190个朋友，而他们的朋友平均有635个朋友。

Studies of offline social networks show the same trend. It has nothing to do with personalities; it follows from basic arithmetic. For any network where some people have more friends than others, it’s a theorem that the average number of friends of friends is always greater than the average number of friends of individuals.
对线下社交网络的研究也发现了同样的趋势。这与每个人的性格无关，只是一个基本的算数问题。一个人朋友的朋友平均数量总是大于这个人朋友的数量。只要社交网络有人朋友多有人朋友少，这条定理就必然存在。

This phenomenon has been called the friendship paradox. Its explanation hinges on a numerical pattern — a particular kind of “weighted average” — that comes up in many other situations. Understanding that pattern will help you feel better about some of life’s little annoyances.
这一现象被称之为好友悖论，其根本原因是一种在很多其他情况下都会出现数学模型（一种特别的“加权平均数“模型”）。了解了这种模型能帮你化解一些生活中的小烦恼。

For example, imagine going to the gym. When you look around, does it seem that just about everybody there is in better shape than you are? Well, you’re probably right. But that’s inevitable and nothing to feel ashamed of. If you’re an average gym member, that’s exactly what you should expect to see, because the people sweating and grunting around you are not average. They’re the types who spend time at the gym, which is why you’re seeing them there in the first place. The couch potatoes are snoozing at home where you can’t count them. In other words, your sample of the gym’s membership is not representative. It’s biased toward gym rats.
比如说你现在走进了一家健身房。看看周围，是不是觉得好像每个人的身材都比你好？恩，你也许是对的，但这种情况无法避免而且完全不需要因此觉得惭愧。如果你只是一个普通的健身房会员，根据预期你就应该看到这样的场景，因为那些大汗淋漓喘着粗气的人不代表平均值。他们是那种会花时间在健身房的类型，所以你才会第一眼就看到他们。那些沙发土豆正窝在家里你没法把他们算在内。换句话说，你所取的健身馆会员这个样本不具有代表性，它其中健身迷的比例偏多。

This is also why people experience airplanes, restaurants, parks and beaches to be more crowded