So much so that people, organizations, and whole countries

以致很多人、機構甚至整個國家

base some of their most important decisions on organized data.

將已整理的數據 作為他們一些最重要決定的依據

But there's a problem with that.

但這做法有一個問題

Any set of statistics might have something lurking inside it,

任何一組統計數據 都有可能潛伏一些因素

something that can turn the results completely upside down.

這些因素有時可能完全改變結論

For example, imagine you need to choose between two hospitals

例如，想像你需要 從兩間醫院中選擇一間

for an elderly relative's surgery.

適合年老的親人來做手術

Out of each hospital's last 1000 patient's,

在各自醫院最近收治的 1000 個病人中

900 survived at Hospital A,

醫院 A 有 900 人存活

while only 800 survived at Hospital B.

而醫院 B 只有 800 人存活

So it looks like Hospital A is the better choice.

所以看起來醫院 A 是比較好的選擇

But before you make your decision,

但在你作決定前

remember that not all patients arrive at the hospital

要記得並不是所有病人入院時

with the same level of health.

都有相同的健康情況

And if we divide each hospital's last 1000 patients

若我們把各間醫院最近收治的 1000 個病人

into those who arrived in good health and those who arrived in poor health,

分成入院時健康良好和欠佳這兩組

the picture starts to look very different.

情況變得截然不同

Hospital A had only 100 patients who arrived in poor health,

醫院 A 只有 100 人入院時健康欠佳， 而當中 30 人存活

of which 30 survived.

但醫院 B 則有 400 人， 而他們能保住 210 人的性命

But Hospital B had 400, and they were able to save 210.

所以對於入院時健康欠佳的病人， 醫院 B 是較好的選擇

So Hospital B is the better choice

其存活率達 52.5 %

for patients who arrive at hospital in poor health,

那麼如果你的親人入院時 健康良好呢？

with a survival rate of 52.5%.

非常奇怪的是， 醫院 B 仍是較好的選擇

And what if your relative's health is good when she arrives at the hospital?

其存活率超過 98 ％

Strangely enough, Hospital B is still the better choice,

所以若醫院 B 在這兩組都有較高存活率， 為何卻是醫院 A 有較高的整體存活率？

with a survival rate of over 98%.

這是我們碰巧遇到的 一個「辛普森悖論」的情況

So how can Hospital A have a better overall survival rate

同一套數據依據其分組方法， 能呈現出相反的走向

if Hospital B has better survival rates for patients in each of the two groups?

這經常發生在當已收集的數據中 隱藏了一個「條件變項」

What we've stumbled upon is a case of Simpson's paradox,

有時也稱為「潛在變項」

where the same set of data can appear to show opposite trends

它是另一個隱藏因素， 會顯著地影響結果

depending on how it's grouped.

在此，隱藏因素是 兩組病人的相對比例

This often occurs when aggregated data hides a conditional variable,

即入院時健康情況好或壞

sometimes known as a lurking variable,

辛普森悖論並不限於假設的情境

which is a hidden additional factor that significantly influences results.

它在真實世界時有出現

Here, the hidden factor is the relative proportion of patients

有時甚至在重要的情況

who arrive in good or poor health.

一個英國的研究發現

Simpson's paradox isn't just a hypothetical scenario.

吸煙者比非吸煙者有較高存活率

It pops up from time to time in the real world,

這研究長達二十多年

sometimes in important contexts.

然而，當把參與者按年齡分組

One study in the UK appeared to show

便顯現非吸煙者的 平均年齡明顯地比較高

that smokers had a higher survival rate than nonsmokers

因此，較可能在研究期間死亡

over a twenty-year time period.

這正是因為非吸煙者 普遍較長壽的緣故

That is, until dividing the participants by age group

在此，年齡分組是潛在變項

showed that the nonsmokers were significantly older on average,

這對正確解讀數據非常重要

and thus, more likely to die during the trial period,

另一例子是

precisely because they were living longer in general.

佛羅里達州死刑案件的研究分析

Here, the age groups are the lurking variable,

似乎顯示因謀殺罪的黑人或白人 被判死刑的情況，並無種族差異

and are vital to correctly interpret the data.

但當按受害人的種族來分組， 就截然不同了

In another example,

無論受害人的種族如何， 黑人被告都較可能被判死刑

an analysis of Florida's death penalty cases

白人被告在整體上 被判死刑的機率稍微較高

seemed to reveal no racial disparity in sentencing

是因為涉及白人受害者的案件

between black and white defendants convicted of murder.

比涉及黑人受害者的 較有可能被判死刑

But dividing the cases by the race of the victim told a different story.

而謀殺案又多數發生在相同種族之間

In either situation,

那我們要如何避免掉入這種悖論呢？

black defendants were more likely to be sentenced to death.

不幸的是， 並沒有一個適合各種情況的答案

The slightly higher overall sentencing rate for white defendants

數據能夠以各種方法進行分組

was due to the fact that cases with white victims

而整體數據有時 能給我們一個更準確的描述

were more likely to elicit a death sentence

相較於誤導或任意分組的數據

than cases where the victim was black,

我們唯一能做的是仔細研究 統計數據所描述的真實情況

and most murders occurred between people of the same race.

並考慮當中是否存在「潛在變項」

So how do we avoid falling for the paradox?

否則，我們便很容易受到 運用數據達到目的人的操弄了

Unfortunately, there's no one-size-fits-all answer.

翻譯：Crystal Yip

Data can be grouped and divided in any number of ways,

and overall numbers may sometimes give a more accurate picture

than data divided into misleading or arbitrary categories.

All we can do is carefully study the actual situations the statistics describe

and consider whether lurking variables may be present.

Otherwise, we leave ourselves vulnerable to those who would use data

to manipulate others and promote their own agendas.