The Math That Explains Why Bell Curves Are Everywhere

The central limit theorem is what makes bell-shaped distributions so ubiquitous.

中心极限定理使得钟形分布如此普遍。

No matter where you look, a bell curve is close by.

无论你往哪里看，钟形曲线都在附近。

Place a measuring cup in your backyard every time it rains and note the height of the water when it stops: Your data will conform to a bell curve.

每次下雨时，在后院放置一个量杯，并记录停雨时的水位高度：您的数据将符合钟形曲线。

Record 100 people’s guesses at the number of jelly beans in a jar, and they’ll follow a bell curve.

记录 100 个人对罐子里软心豆粒糖数量的猜测，他们会遵循钟形曲线。

Measure enough women’s heights, men’s weights, SAT scores, marathon times — you’ll always get the same smooth, rounded hump that tapers at the edges.

测量足够多的女性身高、男性体重、SAT 成绩、马拉松时间——您总会得到边缘逐渐变细的光滑、圆形的驼峰。

Why does the bell curve pop up in so many datasets?

为什么钟形曲线会出现在这么多数据集中？

The answer boils down to the central limit theorem, a mathematical truth so powerful that it often strikes newcomers as impossible, like a magic trick of nature.

答案归结为中心极限定理，这是一个如此强大的数学真理，常常让新手觉得不可能，就像大自然的魔术一样。

“The central limit theorem is pretty amazing because it is so unintuitive and surprising,” said Daniela Witten , a biostatistician at the University of Washington.

华盛顿大学生物统计学家丹妮拉·威滕 (Daniela Witten) 表示：“中心极限定理非常令人惊奇，因为它是如此不直观且令人惊讶。”

Through it, the most random, unimaginable chaos can lead to striking predictability.

通过它，最随机、最难以想象的混乱也能带来惊人的可预测性。

It’s now a pillar on which much of modern empirical science rests.

它现在已成为现代实证科学的支柱。

Almost every time a scientist uses measurements to infer something about the world, the central limit theorem is buried somewhere in the methods.

几乎每次科学家使用测量来推断有关世界的某些事情时，中心极限定理都隐藏在方法中的某个地方。

Without it, it would be hard for science to say anything, with any confidence, about anything.

没有它，科学就很难有信心地谈论任何事情。

“I don’t think the field of statistics would exist without the central limit theorem,” said Larry Wasserman , a statistician at Carnegie Mellon University.

卡内基梅隆大学统计学家 Larry Wasserman 表示：“我认为如果没有中心极限定理，统计学领域就不会存在。”

“It’s everything.”

“这就是一切。”

Perhaps it shouldn’t come as a surprise that the push to find regularity in randomness came from the study of gambling.

寻找随机性中的规律性的推动力来自于赌博研究，也许这并不令人意外。

In the coffeehouses of early-18th-century London, Abraham de Moivre’s mathematical talents were obvious.

在 18 世纪初伦敦的咖啡馆里，亚伯拉罕·德·莫弗尔 (Abraham de Moivre) 的数学天赋显而易见。

Many of his contemporaries, including Isaac Newton and Edmond Halley, recognized his brilliance.

他的许多同时代人，包括艾萨克·牛顿和埃德蒙·哈雷，都认识到他的才华。

De Moivre was a fellow of the Royal Society, but he was also a refugee, a Frenchman who had fled his home country as a young man in the face of anti-Protestant persecution.

德莫弗是英国皇家学会会员，但他也是一名难民，一名法国人，年轻时因面临反新教迫害而逃离祖国。

As a foreigner, he couldn’t secure the kind of steady academic post that would befit his talent.

作为一个外国人，他无法获得适合他才能的稳定学术职位。

So to help pay his bills, he became a consultant to gamblers who sought a mathematical edge.

因此，为了支付账单，他成为了寻求数学优势的赌徒的顾问。

Abraham de Moivre made early mathematical investigations into games of chance.

亚伯拉罕·德莫弗尔对机会游戏进行了早期的数学研究。

Joseph Highmore (1736)/Public Domain

约瑟夫·海默 (1736)/公共领域

Flipping a coin, rolling a die, and drawing a card from a deck are random actions, with every outcome equally likely.

抛硬币、掷骰子以及从一副牌中抽一张牌都是随机动作，每种结果的可能性均等。

What de Moivre realized is that when you combine many random actions, the result follows a reliable pattern.

德莫弗意识到，当你结合许多随机动作时，结果会遵循一个可靠的模式。

Flip a coin 100 times and count how often it comes up heads.

抛一枚硬币 100 次，然后数数正面朝上的次数。

It’ll be somewhere around 50, but not very precisely.

大概在 50 左右，但不是很精确。

Play this game 10 times, and you may get 10 different counts.

玩这个游戏 10 次，您可能会得到 10 种不同的计数。

Now imagine playing the game 1 million times.

现在想象一下玩这个游戏 100 万次。

The bulk of the outcomes will be close to 50.

大部分结果将接近 50 个。

You’ll almost never get under 10 heads or over 90.

你几乎永远不会遇到低于 10 个头或超过 90 个头的情况。

If you make a graph of how many times you see each number between zero and 100, you’ll see that classic bell shape, with 50 at the center.

如果你制作一个图表，记录你看到 0 到 100 之间每个数字的次数，你会看到那个经典的钟形，中心是 50。

The more times you play the game, the smoother and clearer the bell will become.

玩游戏的次数越多，铃声就会变得越流畅、越清晰。

De Moivre figured out the exact shape of this bell, which came to be called the normal distribution.

德莫弗尔弄清楚了这个钟的确切形状，这被称为正态分布。

It told him, without his having to actually play the game, how likely different outcomes were.

它告诉他，无需他实际玩游戏，不同结果的可能性有多大。

For instance, the probability of getting between 45 and 55 heads is about 68%.

例如，出现 45 到 55 次正面的概率约为 68%。

De Moivre marveled with religious devotion at the “steadfast order of the universe” that eventually overcame any and all deviations from the bell.

德莫弗尔对“宇宙坚定的秩序”充满宗教虔诚而惊叹不已，这种秩序最终克服了与钟声的任何偏差。

“In process of time,” he wrote, “these irregularities will bear no proportion to the recurrency of that order which naturally results from original design.”

“随着时间的推移，”他写道，“这些不规则现象将与原始设计自然产生的秩序的重复出现不成比例。”

The Doctrine of Chances , first published in 1718, was the first textbook on probability theory.

《机会论》首次出版于 1718 年，是第一本概率论教科书。

A second edition, published in 1738, featured the work that would lead to the central limit theorem.

第二版于 1738 年出版，其中介绍了导致中心极限定理的工作。

He used these insights to sustain a meager life in London, writing a book called The Doctrine of Chances that became a gambler’s bible, and holding informal office hours at the famed Old Slaughter’s Coffee House.

他利用这些见解在伦敦维持着微薄的生活，写了一本名为《机会论》的书，该书成为赌徒的圣经，并在著名的老屠夫咖啡馆举行非正式的办公时间。

But even de Moivre didn’t realize the full scope of his discovery.

但即使是德莫弗尔也没有意识到他的发现的全部范围。

Only when Pierre-Simon Laplace ran with the idea in 1810, decades after de Moivre’s death, was its full reach uncovered.

直到 1810 年，也就是德莫弗去世几十年后，皮埃尔-西蒙·拉普拉斯 (Pierre-Simon Laplace) 提出这个想法时，它的影响力才被全面揭示。

Let’s take an example slightly more complex than coin flips: dice rolls.

让我们举一个比抛硬币稍微复杂一些的例子：掷骰子。

Every roll of a die has six equally likely outcomes.

每掷一次骰子都有六个同样可能的结果。

If you repeatedly roll the die and tally the results, you’ll get a chart that looks flat — you’re bound to see about as many rolls of 1 as you do 2 or 4 or 6.

如果您反复掷骰子并计算结果，您将得到一个看起来平坦的图表 - 您一定会看到与 2、4 或 6 一样多的掷骰子。

Now roll that die 10 times and take the average.

现在将骰子掷 10 次并取平均值。

You’re likely to get somewhere around 3.5.

您可能会得到 3.5 左右的值。

Repeat the experiment many more times and graph up all the results.

多次重复实验并绘制所有结果。

You’ll get a bell curve that peaks at 3.5, with a precisely defined structure on either side.

您将得到一条峰值为 3.5 的钟形曲线，两侧都有精确定义的结构。

That’s the magic of the central limit theorem.

这就是中心极限定理的魔力。

You started with a distribution of possible outcomes that has no structure at all — equal chances of rolling 1 through 6.

你一开始就对可能的结果进行了分布，这种分布完全没有结构——从 1 到 6 的概率相等。

But by taking an average of multiple measurements, then repeating that process over and over, you get a precise, predictable, mathematical structure: the bell curve.

但通过多次测量的平均值，然后一遍又一遍地重复这个过程，你就会得到一个精确的、可预测的数学结构：钟形曲线。

Laplace distilled this structure into a simple formula, the one that would later be known as the central limit theorem.

拉普拉斯将这一结构提炼成一个简单的公式，这个公式后来被称为中心极限定理。

No matter how irregular a random process is, even if it’s impossible to model, the average of many outcomes has the distribution that it describes.

无论随机过程多么不规则，即使不可能建模，许多结果的平均值也具有它所描述的分布。

“It’s really powerful, because it means we don’t need to actually care what is the distribution of the things that got averaged,” Witten said.

“它真的很强大，因为这意味着我们不需要真正关心被平均的东西的分布情况，”威滕说。

“All that matters is that the average itself is going to follow a normal distribution.”

“重要的是平均值本身将遵循正态分布。”

Averaging might seem like something it takes a human to do, but the central limit theorem applies invisibly to all kinds of things we can observe in the world, like human heights.

平均似乎需要人类才能完成，但中心极限定理无形地适用于我们在世界上可以观察到的各种事物，例如人类的身高。

“Somebody’s height might depend on their dad’s height, and their mom’s height, and their genetics, and their nutrition, and all these little effects that add up,” said Jeffrey Rosenthal , a statistician at the University of Toronto.

多伦多大学统计学家杰弗里·罗森塔尔说：“一个人的身高可能取决于他们父亲的身高、母亲的身高、他们的遗传、他们的营养，以及所有这些加起来的小影响。”

Those effects are unrelated to each other (generally, your dad’s height has nothing to do with the food you eat).

这些影响彼此无关（通常，你父亲的身高与你吃的食物无关）。

“It’s kind of like averaging a bunch of little effects,” said Rosenthal, which is why height approximately follows a normal distribution.

罗森塔尔说：“这有点像对一堆小影响进行平均。”这就是为什么身高大致遵循正态分布。

Daniela Witten, a biostatistician at the University of Washington.

丹妮拉·威滕 (Daniela Witten)，华盛顿大学生物统计学家。

This is why all kinds of datasets seem to conform to this beautiful shape spontaneously.

这就是为什么各种数据集似乎自发地符合这个美丽的形状。

“Anywhere that there’s an average under the hood, if it’s an average over enough things, then you’re going to end up with a normal distribution,” Witten said.

威滕说：“只要有一个平均值，如果它是足够多事物的平均值，那么最终就会得到正态分布。”

The theorem also gives statisticians the power to tell when something fishy is happening.

该定理还赋予统计学家判断可疑情况何时发生的能力。

Say you’re sipping coffee at Old Slaughter’s when a patron hands you a coin and bets that you can’t get 45 heads in 100 flips.

假设您正在 Old Slaughter’s 喝咖啡，这时一位顾客递给您一枚硬币，并打赌您在 100 次抛掷中不能得到 45 个正面。

You try, and only get 20.

你尝试一下，但只得到 20 个。

How can you tell whether he gave you a trick coin and the process is not as random as it ought to be?

你如何判断他是否给了你一枚特技硬币，而且这个过程并不像应有的那样随机？

Thanks to the central limit theorem, you know that the numbers up to 20 only cover 0.15% of the bell, so there’s only a 0.15% chance a fair coin would give such a bad result.

由于中心极限定理，您知道 20 以内的数字仅覆盖钟的 0.15%，因此一枚公平的硬币产生如此糟糕结果的可能性只有 0.15%。

You’re almost certainly being had.

几乎肯定你已经被骗了。

That’s the true power of Laplace’s formula.

这就是拉普拉斯公式的真正威力。

He knew that averaging over any process gives you a bell curve, which lets you say something about that process, without knowing anything deeper about how it works.

他知道对任何流程进行平均都会产生一条钟形曲线，这让您可以对这个流程进行一些描述，而无需更深入地了解它是如何工作的。

Despite its centrality to modern science, the central limit theorem has limits of its own.

尽管中心极限定理对现代科学至关重要，但它也有其自身的局限性。

It only works when you’re combining many samples, and those samples need to be independent.

它仅在您组合许多样本时才有效，并且这些样本必须是独立的。

If they’re not — for example, if you only run a national presidential poll in a single small town in Maine — repeating the experiment won’t get you closer to the expected bell curve.

如果不是——例如，如果你只在缅因州的一个小镇进行全国总统大选——重复实验并不会让你更接近预期的钟形曲线。

And sometimes in science, the outliers can be more important than the average.

有时在科学中，异常值可能比平均值更重要。

“The ‘hundred-year flood’ is suddenly happening more often,” said Richard D.

“‘百年一遇的洪水’突然变得更加频繁，”理查德·D·

De Veaux , an applied statistician at Williams College.

De Veaux 是威廉姆斯学院的应用统计学家。

“These days, modeling extreme events is probably as important as modeling the mean.”

“如今，对极端事件进行建模可能与对平均值进行建模一样重要。”

Fortunately, the idea behind the central limit theorem — the power and reliability of averages — has been used far and wide to extend the power of statistics.

幸运的是，中心极限定理背后的思想——平均值的力量和可靠性——已被广泛使用，以扩展统计的力量。

Statisticians often formulate a version of the central limit theorem for whatever specific problem they’re working on.

统计学家经常为他们正在研究的任何具体问题制定一个版本的中心极限定理。

“There’s so many more complicated things where if you’re clever you can write it as a sample mean plus some error,” Wasserman said.

“有很多更复杂的事情，如果你聪明的话，你可以把它写成样本均值加上一些误差，”沃瑟曼说。

In those cases, you can use a variant of the theorem to simplify the problem.

在这些情况下，您可以使用定理的变体来简化问题。

The central limit theorem is a pillar of modern science, ultimately, because it’s a pillar of the world around us.

归根结底，中心极限定理是现代科学的支柱，因为它是我们周围世界的支柱。

When we combine lots of independent measurements, we get clusters.

当我们结合大量独立的测量结果时，我们得到了聚类。

And if we’re clever enough, we can use those clusters to find out something interesting about the processes that made them.

如果我们足够聪明，我们可以利用这些集群来找出有关它们的制造过程的一些有趣的东西。

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获取发送到您的电子邮件收件箱的最重要新闻的摘要

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《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。

Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

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版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。