they help.

但這常會產生問題，舉例來說我們以一種療法治療一種會同時影響人與貓的疾病

Like, suppose we're treating a disease that afflicts both people and cats, and among 1

如果我們以此療法治療一隻貓和四個人身上，那一隻貓與一個人會痊癒但三個人會死亡

cat and 4 people we treat, the cat and 1 person recover and 3 people die.

如果有四隻貓和一個人沒被治療的話，那會有三隻貓痊癒但一隻貓與一個人死亡

And of 4 cats and 1 person we don't treat, three of the cats recover while the person

在真實世界中這些數字可能會是300或100，但無論如何我們會以小數字來方便討論它

and 1 cat die.

所以在我們的例子中100%被治療的貓會痊癒但只有75%沒被治療的貓會

In the real world, these numbers might be more like 300 and 100, or whatever, but we'll

而25%被治療的人會痊癒但沒被治療的人都不會痊癒

keep them small so they're easier to keep track of.

這讓此療法似乎增加痊癒的機率

So, in our sample, 100% of treated cats survive while only 75% of untreated cats do, and 25%

但如果我們匯總資料來看，被治療的人和貓有40%存活

of treated humans survive while 0% of untreated humans do.

但沒被治療的人和貓會有60%痊癒

Which makes it seem like the treatment improves chances of recovery.

這讓此療法似乎減少了痊癒了機率

Except that if we aggregate the data, among all people and cats treated, only 40% survive,

所以哪個才是對的呢？

while among all people and cats left on their own, 60% recover.

這是辛普斯被論的一種例證

Which makes it seem like the treatment reduces chances of recovery.

一個統計的悖論使不同分析方法在相同的資料中得出相反的結論

So which is it?

但單單統計學不能幫助我們解決它

This is an illustration of Simpson's paradox , a statistical paradox where it's possible

我們必須跳脫出統計學並了解這個情境中的因果關係

to draw two opposite conclusions from the same data depending on how you divide things

舉例來說如果我們知道人類得病的情況較為嚴重而因此較有可能規定要被治療

up, and statistics alone cannot help us solve it – we have to go outside statistics and

那麼就算療法會增加存活率

understand the causality involved in the situation at hand.

被治療的還是會比沒被治療的存活率低(不管人還是貓)

For example, if we know that humans get the disease more seriously and are therefore more

因為每個被治療的人都比較容易死(病的比較重)

likely to be prescribed treatment, then it can make sense that fewer individuals that

另一個例子是不管人類病的多重都比貓會容易得到治療因為沒人想為貓的健康付出

get treated survive, even if the treatment increases the chances of recovery, since the

但五個人類中有四個死亡但五隻貓中卻只有一隻死亡

individuals that got treated were more likely to die in the first place.

使此療法變得的確不是好選擇

On the other hand, if we know that humans, regardless of how sick they are, are more

所以當在做控制實驗時

likely to get treated than cats because no one wants to pay for kitty healthcare, then

你必須確保不讓任何關係事件影響你對待實驗的方式

the fact that 4 out of 5 humans died while only 1 in 5 cats died suggests that, indeed,

如果做非控制實驗時

the treatment may be a bad choice.

你必須把所有的外部的影響都計算在內

So if you're doing a controlled experiment, you need to make sure to not let anything

以一個較為現實的情況來說

causally related to the experiment influence how you apply your treatments, and if you

威斯康辛州八年級學生已有多次標準化成績高於德州學生

have an uncontrolled experiment, you have to be able to take those outside biases into

所以你可能會認為威斯康辛州教的比德州好

account.

但當已不同種族分開──通過根真柢固的社會經濟差異是一個主要的表準化測式時

As a more tangible example, Wisconsin has repeatedly had higher overall 8th grade standardized

德州學生在所有項目表現得比威斯康辛州出色

test scores than Texas, so you might think Wisconsin is doing a better job teaching than

黑人德州學生分數比黑人威斯科辛州高

Texas.

西班牙裔與白人學生也一樣

However, when broken down by race – which, via entrenched socioeconomic differences is

會有這樣的差異是因為威斯康辛州在比例上黑人和西班牙裔學生遠低於德州

a major factor in standardized-test scores – Texas students performed better than Wisconsin

而比例上白人學生多於德州

students on all fronts: black Texas students scored higher than black Wisconsin students,

所以整體來說並不是威斯科辛州比德州有更好的教育

and likewise with hispanic and white students.

只是在比例上有更多人擁有較好的社會經濟而以

The difference in the overall ranking is because Wisconsin has proportionally far fewer black

所以了解統計結果的根本原因才會有有效的公共意義

and hispanic students and proportionally more white students than Texas – so the takeaway

在一些情境中也有一些漂亮的圖形可以用於解釋辛普森悖論

should not be that Wisconsin has better education than Texas!

有兩個分開趨勢，都朝一種方向，但人口的趨勢是朝另一種方向

Just that it has (proportionally) more socioeconomically advantaged people.

像是更富裕讓人不快樂，而更富裕讓貓不快樂

In some situations there's also a nice graphical way to picture Simpson's paradox: as two separate

但如果一開始貓就比人富裕與快樂

trends that each go one way, but the overall trend between the populations goes the other

那整體的趨勢就錯誤地指出更富裕讓你更開心

way.

在這個例子中當一隻貓讓你更快樂，但也跟更富裕產生關連

Like, maybe more money makes people sadder, and more money makes cats sadder, but if cats

當然，你可以曲解這個圖表使得整體來說，更富裕讓你變成一隻貓

are both much happier and richer than people to start with, the overall trend appears,

這讓我覺得對於說明盲目使用統計資料而沒有上下文來說謊或得到不正確的結論有幫助

incorrectly, to be that more money makes you happier.

這當然不是說統計都會變得字相矛盾或令人困惑

Of course, you can also misinterpret this graph to show that, overall, more money makes

──統計很有可能都會從一開始就讓人信服

you a cat, which I think helps illustrate very well the ability to lie or reach incorrect

像是當人和貓都會因為給他們更多錢而不快樂，而貓比人都要窮和快樂

conclusions by blindly using statistics without context!

使得總體趨勢不在自相矛盾──更富裕會更不快樂

Of course, this is not to say that statistics are always going to be paradoxical or confusing

但要小心像辛普森悖論一樣的悖論可能存在

– it's quite possible that everything will just make sense from the get-go, like if people

且我們常常需要更多上下文去了解統計的真正義涵

and cats both get sadder when you give them more money, and cats are both poorer and happier

像這樣的數學和物理影片，你可能不會訝異我在製作時練習了很多其中有關問題

than people, then the overall trend is no longer paradoxical: more money = more sadness.

而製不影片的贊助商brilliant.org也想要幫助你有厲害的問題解決能力

But it's important to be aware that paradoxes like Simpson's paradox are possible, and we

因為不幸的看影片不需要解決問題

often need more context to understand what a statistic actually means.

練習似乎是最好了解一個主題的方式

Given the mathiness of my videos, it may not surprise you to hear that I get a lot of practice

而brilliant.org提供你許多如機率、邏輯、數學用於定量金融等的付費的培訓班

with math & physics problems while working on them, and this video's sponsor, Brilliant.org,

與令人上癮的題目如如果地球的一半被慧星撞掉了

wants to help you stay sharp on your problem solving, too! (since, unfortunately, watching

那月球的軌道會變怎樣

videos doesn't require as much problem solving).

這就像是MinutePhysics 會有的內容...但你必須到brilliant.org來解決它(或是許多其它的)

Practice is pretty much the best way to really get to know a subject, and Brilliant.org is

而想要造訪時，請使用網址brilliant.org/minutephysics讓Brilliant知道你來自這裡

ready to give you plenty with premium courses in probability, logic, and math for quantitative

finance.

Plus addictive puzzles: for example, “if half of the earth is blown away by the impact

of a comet, what happens to the orbit of the moon?”

It almost sounds like a MinutePhysics video… but you're going to have to go to Brilliant.org

to solve it (or one of their many others) – and when you do, use the URL brilliant.org/minutephysics

to let Brilliant know you came from here.