11.2: The OneWay ANOVA Test
Learning Objectives
 Understand the shortcomings of comparing multiple means as pairs of hypotheses.
 Understand the steps of the ANOVA method and its advantages.
 Compare the means of three or more populations using the ANOVA method.
 Calculate the pooled standard deviation and confidence intervals as estimates of standard deviations of the populations.
Introduction
Previously, we have discussed analysis that allows us to test if the means and variances of two populations are equal. But let’s say that a teacher is testing multiple reading programs to determine the impact on student achievement. There are five different reading programs and her
We could conduct a series of
Shortcomings of Comparing Multiple Means Using Previously Explained Methods
As mentioned, to test whether pairs of sample means differ by more than we would expect due to chance, we could conduct a series of separate
When more than one
The Steps of the ANOVA Method
In ANOVA, we are actually analyzing the total variation of the scores including (1) the variation of the scores within the groups and (2) the variation between the group means. Since we are interested in two different types of variation, we first calculate each type of variation independently and then calculate the ratio between the two. We use the
When using the ANOVA method, we are testing the null hypothesis that the means and the variances of our samples are equal. When we conduct a hypothesis test, we are testing the probability of obtaining an extreme
To test a hypothesis using the ANOVA method, there are several steps that we need to take. These include:
1. Calculating the mean squares between groups
When we calculate the
where:
When simplified, the formula becomes:
where
Once we calculate this value, we divide by the number of degrees of freedom
2. Calculating the mean squares within groups
To calculate the
Simplified, this formula states:
where
Essentially, this formula sums the squares of each observation and then subtracts the total of the observations squared divided by the number of observations. Finally, we divide this value by the total number of degrees of freedom in the scenario
3. Calculate the test statistic. The test statistic is as follows:
4. Find the critical value on the
5. Interpret the results of the hypothesis test. In ANOVA, the last step is to decide whether to reject the null hypothesis and then provide clarification about what that decision means.
The primary advantage to using the ANOVA method is that it takes all types of variation into account so that we have an accurate analysis. In addition, we can use technological tools including computer programs (SAS, SPSS, Microsoft Excel) and the TI83/4 calculator to easily conduct the calculations and test our hypothesis. We use these technological tools quite often when using the ANOVA method.
Let’s take a look at an example to help clarify.
Example:
Let’s go back to the example in the introduction with the teacher that is testing multiple reading programs to determine the impact on student achievement. There are five different reading programs and her
Method
Solution:
To solve for
Using this information, we find that the sum of squares between groups is equal to
Since there are four Degrees of Freedom for this calculation (the number of groups minus one), the mean squares between groups is
Next we calculate the mean squares within groups
To calculate the mean squares within groups, we use the formula
Using our summary statistics from above, we can calculate that the within groups mean square
And so we have
Therefore, our
We would then analyze this test statistic against our critical value (using the
Technology Note  Excel
Here is the procedure for performing a Oneway ANOVA in Excel using this set of data.
 Copy and paste the table into an empty Excel worksheet
 Select Data Analysis from the Tools menu and choose “ANOVA: Singlefactor” from the list that appears
 Place the cursor is in the “Input Range” field and select the entire table.
 Place the cursor in the “Output Range” and click somewhere in a blank cell below the table.
 Click “Labels” only if you have also included the labels in the table. This will cause the names of the predictor variables to be displayed in the table
 Click OK and the results shown below will be displayed.
Note: The TI83/4 also offers a Oneway ANOVA test.
Anova: Single Factor
Groups  Count  Sum  Average  Variance 

Column 1 




Column 2 




Column 3 




Column 4 




Column 5 




Source of Variation 







Between Groups 






Within Groups 




Total 


Lesson Summary
 When testing multiple independent samples to determine if they come from the same populations, we could conduct a series of separate
t tests in order to compare all possible pairs of means. However, a more precise and accurate analysis is the Analysis of Variance (ANOVA).  In ANOVA, we analyze the total variation of the scores including (1) the variation of the scores within the groups and (2) the variation between the group means and the total mean of all the groups (also known as the grand mean).
 In this analysis, we calculate the
F ratio, which is the total mean of squares between groups divided by the total mean of squares within groups.  The total mean of squares within groups is also known as the estimate of the pooled variance of the population. We find this value by analysis of the standard deviations in each of the samples.
Review Questions
 What does the ANOVA acronym stand for?
 If we are tested whether pairs of sample means differ by more than we would expect due to chance using multiple
t tests, the probability of making a Type I error would ___.  In the ANOVA method, we use the ___ distribution.
 Student’s
t   normal

F 
 Student’s
 In the ANOVA method, we complete a series of steps to evaluate our hypothesis. Put the following steps in chronological order.
 Calculate the mean squares between groups and the means squares within groups
 Determine the critical values in the
F distribution  Evaluate the hypothesis
 Calculate the test statistic
 State the null hypothesis
A school psychologist is interested whether or not teachers affect the anxiety scores among students taking the AP Statistics exam. The data below are the scores on a standardized anxiety test for students with three different teachers.
Ms. Jones  Mr. Smith  Mrs. White 





























 State the null hypothesis.
 Using the data above, please fill out the missing values in the table below.
Ms. Jones  Mr. Smith  Mrs. White  Totals  

Number 



Total 



Mean 



Sum of Squared Obs. 


Sum of Obs. Squared/Number of Obs. 

 What is the mean squares between groups
(MSB) value?  What is the mean squares within groups
(MSW) value?  What is the
F ratio of these two values?  Using a
α=.05 , please use theF distribution to set a critical value  What decision would you make regarding the null hypothesis? Why?
Review Answers
 Analysis of Variance
 Increase or increase exponentially

C 
E,A,D,B,C 
H0:μ1=μ2=μ3
Ms. Jones  Mr. Smith  Mrs. White  Totals  

Number 




Total 




Mean 




Sum of Squared Obs. 




Sum of Obs. Squared/Number of Obs. 





26.35 
4.03 
6.54 
3.40  The calculated test statistic exceeds the critical value so we would reject the null hypothesis. Therefore, we could conclude that not all the population means are equal.
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