# 1.11: Analysis of Variance and the F-Distribution

## The F distribution and Testing Two Variances

Note that the distribution and other tests of variance are not topics on the examination. This does not make them less valid, but they should be retained until after the examination at topics to fill out the remainder of the year.

This may be a little bit tough for students to understand the motivation behind these tests of variance. Another problem is that the F distribution is complex enough that first year students are not exposed to details behind the distribution. The best thing for students to remember is that while means are often easy to find or approximate, the variance is not. The key use of the F test is going to be when two populations are being examined, or when a comparison can be made to a known variance.

## The One-Way ANOVA Test

ANOVA tables are computed frequently in professional statistical analysis. They can be tedious, but in the real world they are nearly always done with the assistance of some computational software. I suspect that it is because of this that ANOVA tables are not a required topic for the AP examination. I think that I have completed a single ANOVA table by hand, ever. This is also what I have my students do and then move quickly on to learning how to use the software program of choice.

ANOVA is a difficult procedure to run on some software packages. At the very least, there is little consistency between any of the software programs. In fact, I was asked to present to the math department when I was an undergraduate the different ANOVA output from different software packages. It took me a very, very long time to prepare the presentation due to the difficulty of learning all of the different syntax and output. I had looked at Mathematica, MAPLE, SPSS, SAS, R, S+ and Excel. I did not have access to Minitab, MATLab (I don’t know for sure if the base package will do ANOVA) or Fathom. The most interesting finding was the extreme limitations of Excel, as well as some errors in calculation. Excel errors are well documented, and many have workarounds, but I can’t ever recommended it as an accompaniment to the stats class. Maple, SPSS and I have heard Fathom, are pretty user friendly and are popular choices. Whatever your package is, make sure to block out time around these chapters to get comfortable with using the software for these computations.

## The Two-Way ANOVA Test

Since there is little difference between a one way and a two way ANOVA test, there shouldn’t be much practice necessary. There are a couple benefits to this. First, it is a way to reinforce some of the topics learned in the one way lesson. Second, by contrasting some of the elements that are different about the two way test, the important parts of the one way test become apparent. Remember that the one of the toughest things about seeing a new topic for the first time is that everything seems of equal importance, and everything seems unique. This is especially true as the theory gets tougher and more and more examples are presented. Students will have trouble at times telling what parts of the problem are specific to the example and which parts are always going to be part of the ANOVA table. Looking at two way tests can help with this.

Another consideration is how much time needs to be spent on setting up data for each program. I know that some programs require entry in cells, or imported form comma separated value files. Other programs require the data in the form of a matrix. Students need to be aware of what the requirements of their software is, but it will also help if they know a little about what possibilities are out there, in case they come across different software at a later date.