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| Reviewed: |
Nov 13, 2006 by Statistics Editorial Board |
| Overview: |
This tutorial is a group of linked web pages that introduces One-Way Analysis of Variance (ANOVA). The first page describes what ANOVA is and how to do it using an example on studying and test scores. It explains the difference in Between Groups (Treatment) variance and Within Groups (Error) variance, along with the F-statistic. The second page gives the formula for the Total Sum Square (SStotal), the Treatment Sum Square (SSamong), and the Error Sum Square (SSwithin) and provides a method for working through the formula. The third page gives an example of an experiment and shows the steps used to obtain the ANOVA table. The final page provides a calculator which allows users to enter their own data (3 groups with 8 observations each), click on buttons representing the various steps, and watch the calculation unfold. This is part of the larger site, Virtual Statistician. |
| Learning Goals: |
After using this tutorial, students should:
1) Understand when to use ANOVA;
2) Understand the ANOVA decomposition;
3) Understand the relationships between Total, Within (Error), and Between (Treatment) Sum Squares;
4) Be able to perform an ANOVA;
5) Be able to calculate the F-statistic. |
| Target Student Population: |
This would be appropriate for students in an introductory statistics class at the college level or advanced high school level or in any statistics class in which ANOVA is discussed. This would also be appropriate for those needing a review of ANOVA. |
| Prerequisite Knowledge or Skills: |
Necessary statistical background includes basic probability, understanding of mean and variance, ability to perform hypothesis tests, and knowledge of the F-distribution. Necessary mathematical background includes basic algebra and familiarity with sigma-notation. |
| Type of Material: |
This tutorial is set up as a group of linked web pages that includes a Java-based calculator. |
| Recommended Uses: |
This material could be used as an introduction to or review of ANOVA. It could be assigned to students for reading or given to a student who missed class or needed extra instruction. The calculator could be used to perform computations as part of a lecture/demonstration or homework. It provides a lead in to multiple comparisons. |
| Technical Requirements: |
This material requires a computer with an Internet connection and a Java-enabled web browser. |
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| Strengths: |
This tutorial is a good, well written, and factually correct introduction to ANOVA. The discussion of ANOVA stresses the variance decomposition, and develops intuitive understanding of Between versus Within group variability. The example is fully worked out and easy to follow. The calculator is well-labeled with the appropriate steps. |
| Concerns: |
The assumptions underlying the procedure are not stated.
The pages and calculator do not discuss what happens when the data set sizes change.
The formulas given for the Sum Squares are not intuitive; therefore it will be difficult for students to understand the meaning of these statistics.
In paragraph 5 of the description, the author states analysis of variance and the F score it yields is a ratio of explained variance versus error. It does not explain that it is the ratio of the Mean Square Treatment (Between) and the Mean Square Error (Within). Students might not pick this up, unless they try to calculate F from what is given.
Some terms are not defined. For instance, the formulas for Sum Squares and Mean Squares are given on the Steps page, but they are not discussed on the description page. Students will be able to calculate each one without knowing what it means.
On the example page, the statistics M1, M2, and M3 are given in the calculations, but it is not stated that these are the means of the three groups.
***While a note at the top of the calculator page tells the user to be careful to press the calculator buttons in order, it is possible to press them out of order. If this happens, the results obtained are wrong.
There are typographical errors throughout the tutorial. For instance, the second formula given on the steps page and the calculator page should be SS(among), not SS(total).
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Potential Effectiveness as a Teaching Tool |
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| Strengths: |
The tutorial has a natural flow: discussion, then steps, then an example, then application (calculator).
Formulas are displayed beside the computations. The calculator is set up as a table and has numbered buttons to lead the user through the process of performing an ANOVA calculation, which is helpful for remembering and computing the formula.
The description is written in easy to understand terms and provides connections to other statistical topics. |
| Concerns: |
There is no flexibility with respect to the number of groups or the number of observations in each group because the calculator requires the user to have three sets of eight data points each. Unbalanced designs are not available.
The fact that the user is led to press the calculator buttons in order does not require him or her to understand the steps or even to read the numbers obtained.
The description page is several paragraphs of text. Often, students who encounter a lot of text on a web page tend to skip reading it and go directly to the applet, calculator, or chart.
Because the formula given for calculating Sum Squares is not intuitive, students may have a hard time remembering it without the formula in front of them. |
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Ease of Use for Both Students and Faculty |
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| Strengths: |
The interface is easy to follow. Steps in sequence are labeled, and links are clearly marked. The text is easy to read, and the calculator has well labeled instructions making it easy to operate. A novice would have no problem using this resource.
The information is provided all in one place. Spacing of the steps on the page reinforces the order in which to perform them. |
| Concerns: |
While the description, steps, and example pages each have links to each other and to the calculator, the calculator page doesnt have a link to any of the other three.
The formulas on the steps page and the calculator page are so small that the subscripts are hard to read.
If the user enters fewer than eight data points in a set on the calculator page, the buttons for the sum of the data, the square of the sum, and the mean yield "NaN".
Students cannot use the Tab key to go from one box to another in calculator because, on this page, Tab takes you to a completely different column.
Users can type into boxes that are supposed to be populated by the calculator.
***While a note at the top of the calculator page tells the user to be careful to press the buttons in order, it is possible to press them out of order. If this happens, the results obtained are wrong. |
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| Other Issues and Comments: |
There are a few grammatical typos on the description page. |
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