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 Math Help > Statistics > 1-Way ANOVA

Here's what you'll find in this section:

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• # Introduction to ANOVA

We have previously discussed methods of testing whether

1. Two populations have equal means (independent two-sample tests).
2. The means from two measurements on one population are the same (paired tests).
Analysis of Variance (ANOVA) allows us to extend this to more than two populations or measurements (treatments/). That is, we can test the following:
1. Are all the means from more than two populations equal?
2. Are all the means from more than two treatments on one population equal? (This is equivalent to asking whether the treatments have any overall effect.)

To set our notation, let I be the number of populations or treatments being compared and let be the I means. Then the hypotheses for testing are

To test these hypotheses, we require a random sample from each population or treatment.

NOTE:\ For computational purposes, the ANOVA equations for the multiple population case and the multiple treatment on one population case are the same. However, the interpretation of hypotheses and results is slightly different. Thus,

1. Multiple populations: is the true mean of population i.
2. Multiple treatments: is the true average response when treatment i is applied.

• One-way ANOVA

# One-way ANOVA

Recall that we have I populations or I treatments on one population and we wish to test whether the means for all I groups are equal. The notation we will use for our random variables and observed data values is:

• : The random variable denoting the jth measurement from the ith population or treatment.
• : The observed value of when the experiment is performed.

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• The Model
• ## The Model

The mathematical model describing analysis of variance can be expressed in two different ways. First,

where is an overall mean, is the effect due to treatment i, and . Note that the treatment effects satisfy the condition . If we define the ith treatment or population mean as , our model can be expressed as

• The ANOVA Table
• ## The ANOVA Table

In this class, we will not be concerned with formulas for computing ANOVA quantities. Instead, we will let Stataquest do that for us. However, we will need to learn how to interpret and test using Stataquest output.

The traditional way to present ANOVA results is in an ANOVA table such as the one given below.

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• Computer Lab for Week 10
• # Computer Lab for Week 10

Applicable StataQuest Commands:

Labs Calculating One-Way ANOVA

Statistics ANOVA One-way

• Concept Lab for Week 10
• # Concept Lab for Week 10

• Ch 17: Between and Within Variation

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### Related pages in this website

The webmaster and author of this Math Help site is Graeme McRae.