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• Let's say you want to find out

• if the beverage that people drink

• affects their reaction time.

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

• The first group gets water to drink.

• The second group gets some sugary fruit juice,

• and the third group gets coffee.

• Now you test everyone's reaction time.

• And you want to know if there's any difference

• in reaction time between the groups.

• The null hypothesis says that the mean reaction time

• for all three groups is the same.

• If there were only two groups,

• you could use a t-test to find out

• if there's a difference between them.

• But when you have three groups or more,

• you need to use a different approach--

• the analysis of variance.

• When you do the experiment,

• the scores won't all be the same.

• The total variation of all the scores

• is made up of two parts:

• The variation within each group,

• because the people in each group

• have different reaction times,

• and the variation between the groups,

• because the drinks you gave each group are different.

• Here's an example.

• Look at this set of scores.

• They've been sorted into order

• to make it easier to see the patterns.

• You can see that there's a lot of variation

• within each group;

• some people are faster and some are much slower.

• But all the groups look pretty much alike;

• there's not much variation between the groups.

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

• is due to the people, and the drink

• didn't make much of a difference.

• You can't reject the null hypothesis;

• which is that the type of drink doesn't have

• any effect on reaction time.

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

• In this case, all the scores within each group

• are very close to one another.

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

• but the groups are very different from one another.

• There's a lot of difference between the groups.

• In this case, you would reject the null hypothesis.

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

• So here's the idea behind analysis of variance:

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

• and the within-groups variance.

• Take the ratio of between-groups

• to within-groups variance,

• and the larger this number is,

• the more likely it is

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

• and that you should reject the null hypothesis.

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

• Now look at these numbers.

• You probably can't tell

• if there's a significant effect

• because it's not clear whether there's

• more variance within groups or between groups,

• or how much.

• The calculations show that the ratio is 4.27,

• which has a probability of .04,

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

• With these numbers, the drink you give the people

• does have an effect on their reaction time.

• What's that "2,12" doing there?

• Those are the degrees of freedom

• for variance between groups

• and variance within groups.

• And here's how you calculate the degrees of freedom

• when you report results for analysis of variance.

• This trick of separating the variance

• not only when you have three or more groups,

• it also works when you have multiple variables.

• For example, if you test three groups

• for reaction time in the morning,

• and you test another three groups in the evening,

• an analysis of variance can tell you

• if there's a significant effect

• for the type of drink,

• or if the time of day makes a difference,

• or if there's some interaction.

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

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

• You figure hοw much of the total variance

• comes from between the groups,

• and how much comes from within the groups.

• If most of the variation is between groups,

• there's probably a significant effect;

• if most of the variation is within groups,

• there's probably not a significant effect.

Let's say you want to find out

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# 分散分析 (ANOVA) (Analysis of Variance (ANOVA))

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羅紹桀 に公開 2021 年 01 月 14 日