if the beverage that people drink

一般人喝下飲料後

affects their reaction time.

在體內平均的反應時間

So you set up an experiment with three groups of people.

因此你設計一個有三組人的實驗

The first group gets water to drink.

第一組獲得水喝

The second group gets some sugary fruit juice,

第二組獲得含糖果汁

and the third group gets coffee.

而第三組則獲得咖啡

Now you test everyone's reaction time.

現在你來測試大家的反應時間

And you want to know if there's any difference

而你想知道的是，其間有任何差異

in reaction time between the groups.

三組之間的反應時間

The null hypothesis says that the mean reaction time

因此先建立虛無假設平均反應時間

for all three groups is the same.

三組都相同

If there were only two groups,

如果只有兩組的情況下

you could use a t-test to find out

你可以使用 t檢定來找出

if there's a difference between them.

其間所存在的差異

But when you have three groups or more,

但如果你有三組或三組以上

you need to use a different approach--

你必須使用另一種不同的方法--

the analysis of variance.

變異數分析

When you do the experiment,

當你開始實驗

the scores won't all be the same.

所有的分數都不一致

The total variation of all the scores

所有分數的總變異量

is made up of two parts:

都由兩部分所構成

The variation within each group,

各組內的變動

because the people in each group

因為每一組內的人

have different reaction times,

都有不同的反應時間

and the variation between the groups,

加上組與組之間的變異

because the drinks you gave each group are different.

由於你給每一組的飲料都不一樣

Here's an example.

以下即是一例

Look at this set of scores.

來看一下這一組的分數

They've been sorted into order

它們已照順序排列

to make it easier to see the patterns.

以易於看出其間的模式

You can see that there's a lot of variation

你可以看到它們的差異很大

within each group;

在各組之內

some people are faster and some are much slower.

有些人很快，有些人則很慢

But all the groups look pretty much alike;

但各組看起都很相像

there's not much variation between the groups.

雖然組與組之間差異不大

In this case, you'd say that most of the difference

在這個案例中，你只能說最大的差異

is due to the people, and the drink

來自於人，至於飲料

didn't make much of a difference.

並沒有產生很大的差異

You can't reject the null hypothesis;

你無法拒絕虛無假設

which is that the type of drink doesn't have

主要是這類的飲料

any effect on reaction time.

在反應時間上並沒有任何效果

Now let's look at a different set of numbers.

現在，我們來看一下不同組的數字

In this case, all the scores within each group

在這個案例中，各組內的所有分數

are very close to one another.

彼此非常接近

There's not a lot of variance within each group,

各組內的變異不大

but the groups are very different from one another.

但組與組之間差異卻很大

There's a lot of difference between the groups.

使得各組之間差異就變大

In this case, you would reject the null hypothesis.

在這個案例子，你可以拒絕虛無假設

In this case, the type of drink makes a big difference.

在此案例中，飲料的種料造成差異

So here's the idea behind analysis of variance:

因此，在變異數分析之後的想法是

Figure out how much of the total variance comes from the between-groups variance

找出來自組與組之間的變化總量

and the within-groups variance.

以及組內的變化量

Take the ratio of between-groups

再計算組間

to within-groups variance,

和組內變異量的比例

and the larger this number is,

比例值越大

the more likely it is

它越有可能

that the means of the groups really *are* different,

組與組的平均數真的「是」有些差異

and that you should reject the null hypothesis.

而依此，你就可以拒絕先前設定的虛無假設

In the examples, it was obvious where the variance was.

在這個例子中，變異數非常明顯

Now look at these numbers.

現在看看這些數字

You probably can't tell

你可能無法分辨

if there's a significant effect

其中有任何顯著效果

because it's not clear whether there's

因為其中並不明顯

more variance within groups or between groups,

在各組內或組與組間，是否有較多的變異

or how much.

或是變異有多大

The calculations show that the ratio is 4.27,

計算結果顯示比率為4.27

which has a probability of .04,

機率是0.04

so in this case, you can reject the null hypothesis.

因此在這個例子中，可以拒絕虛無假設

With these numbers, the drink you give the people

以這些數字來判斷，你給人們喝的飲料

does have an effect on their reaction time.

在他們的反應時間確實有效

What's that "2,12" doing there?

那這裡「2,12」要做什麼呢?

Those are the degrees of freedom

它們是所謂的自由度

for variance between groups

即組間的變異

and variance within groups.

和組內的變異

And here's how you calculate the degrees of freedom

此處你該如何計算自由度

when you report results for analysis of variance.

當你要報告變異數的結果

This trick of separating the variance

這是分離變異數的小技巧

not only when you have three or more groups,

不管在你有三組或三組以上

it also works when you have multiple variables.

或是有更多重的變異數時

For example, if you test three groups

例如，你要檢定三組

for reaction time in the morning,

在清晨時的反應時

and you test another three groups in the evening,

或是檢定另外三組在傍晚的反應時間

an analysis of variance can tell you

變異數分析就能告訴你

if there's a significant effect

其間是否顯著

for the type of drink,

對飲料的種類

or if the time of day makes a difference,

或者是因每天時段不同差生的差異

or if there's some interaction.

又或者是反應本身

For example, coffee might be more effective in the morning than in the evening.

例如，咖啡本身在清晨時段就比較傍晚強烈

So to recap, here's the main idea of analysis of variance:

來回顧一下，這裡變異數的主要想法是

You figure hοw much of the total variance

你已體會到總變異量的多寡

comes from between the groups,

來自於組間

and how much comes from within the groups.

以及有多少來自於組內

If most of the variation is between groups,

如果大部分的變異來自於組間

there's probably a significant effect;

造成顯著的機率比較高

if most of the variation is within groups,

如果大部分的變異來自於組內

there's probably not a significant effect.

顯著的機率就會降低