科学与数学thestar.com编辑精选
当美国空军发现“平均数的缺陷”
When the U.S. air force discovered the flaw of averages
2016 — 2020，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这篇文章讲述美国空军在二战后为设计驾驶舱而测量数千名飞行员、寻找“平均身材”的故事：统计学家 Daniels 发现，4063 名飞行员中竟没有一人同时符合全部 10 项尺寸的平均范围。这一发现推翻了“按平均值设计就能适合大多数人”的假设，催生了可调节的座舱设计。
为什么经久不衰：它用一个经典案例揭示了高维空间中“平均人”根本不存在的数学事实，提醒我们任何按平均值设计的系统都会系统性地排除许多人，这一教训至今适用于产品设计、教育和数据科学。
HN 评论精选
把偏离平均当作缺陷的思维方式，正在导致对人类自然差异的过度医学化。
让我触动的是，我们今天在谈论人的心智时仍在重复“诺玛式”的错误——把神话中的平均人当作标准，把差异当作问题。我很想知道还有多少自然差异被“医学化”了：有朋友因睡不满平均的 8 小时就被当作失眠症治疗；我丧母悲伤时有人建议我吃抗抑郁药；我担心许多学童的差异被医学化，只是因为他们给使用工业时代教育模式的、不堪重负的老师带来了不便。我没有解决方案，但这确实令人不安。
wpietri查看英文原文
The thing that strikes me here is how much we repeat the mistake of Norma (taking the mythical average body and assuming that differences mean problems) today in talking about people's minds. I started using computers before it was fashionable, before my fellow nerds started being worth billions and ending up on magazine covers. It's hard to describe now how much our difference was seen as wrong, as the sign of a problem. I have to wonder how much other natural differences gets medicalized. I know I have friends who take drugs for "insomnia" even though the only problem they experience is that they don't always get the "average" night of eight hours uninterrupted sleep. (Which anyhow is a modern invention. [1]) When I was in mourning after my mom died, a few buttinskys suggested I talk to my doctor about antidepressants, even though actual experts thought I was doing fine. And I worry about the number of schoolkids who have their differences medicalized because, in effect, they are inconvenient for overburdened teachers using industrial-age models of education. I have no solution here, but I definitely find it troubling. [1] For more, see this podcast, especially the "Til Morning is Nigh" segment: http://backstoryradio.org/shows/on-the-clock-4/
“没有平均人”本质是维度灾难：维度越多，同时接近各维度平均的概率指数级下降。
注意，这并非人类不符合数学理想导致的意外，而是把对一、二、三维的直觉套用到高维空间的结果。单个维度落在中间 50% 的概率是 50%，两个维度同时是 25%，三个维度约 12%，到十个维度就只剩约 0.01%——一千个人里平均只有一个。即便是空军后来采用的方案（9 个维度、每个维度覆盖 90% 的人），也意味着总体只有 38% 的人被覆盖。
lalaithion查看英文原文
Note that this is not some sort of unexpected effect due to humans not behaving mathematically ideally; this is what happens when your intuitions about 1, 2 and 3 dimensions are applied to higher dimensional spaces. Consider the goal of being in the middle 50% of a random value on one dimension; you have a 50% chance. But if you have two dimensions, and you want to be in the middle 50% for both of them, it's a 25% chance. And if you have 3, it's a ~12% chance. This already is counterintuitive, but when you ramp it up to 10 dimensions, it's a ~0.01% chance. That means that if you have a thousand people, only one of them (on average) will be in the middle 50% of all of them. Even the praised end state of the air force, with 9 dimensions and a required support of 90% of individuals of each dimension, implies that only 38% of individuals will be supported overall. (Granted, human dimensions aren't wholly uncorrelated.)
用简单的概率计算就能预见结果，问题在于人们错误假设了各维度高度相关。
虽然事后诸葛亮，但有些本该更早被察觉。Daniels 把“平均”宽松地定义为落在中间 30% 区间，而只要取 10 个维度中的 3 个，就只有不到 3.5% 的飞行员在三个维度上都平均。30% 的三次方就是 2.7%，基本概率而已。我猜大家当时都假设了身体尺寸会高度聚集，而不是近似独立的变量。
keenerd查看英文原文
While hindsight is 20:20, parts of this should have been more obvious. > Daniels generously defined as someone whose measurements were within the middle 30 per cent of the range of values > Daniels discovered that if you picked out just three of the ten dimensions of size ... less than 3.5 per cent of pilots would be average sized on all three dimensions. 30% raised to the third power is 2.7%. Basic probability. I guess everyone assumed there would be heavy clustering instead of largely independent variables?
按“平均用户”设计的表单和系统至今仍在排除大量真实用户。
HN 上早就有人指出：普通的网页表单排除了很多人。许多人填不了姓名/用户名/密码/社保号/街道地址/银行账户这些字段——而且不只是偏远地区的人。我的商业伙伴家里没有可投递的街道地址，只有邮政信箱，收亚马逊快递简直是噩梦。总之，不存在“平均人”。
JoeAltmaier查看英文原文
Been noted here on HN: the average web form excludes lots of people. Lots of folks cant fill in: First/middle/last name; unique username/password; social security number; street address; bank account. And not just folks in Micronesia or whatever. Heck my business partner hasn't got a deliverable street address - no mailbox at his house. He has a PO box. Is a nightmare getting Amazon deliveries that aren't lost, or left in the bushes, or returned undeliverable. Anyway, yes, there is no average person.
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