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TECEP Principles of Statistics: Study Guide & Test Prep13 chapters | 91 lessons | 11 flashcard sets

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Lesson Transcript

Instructor:
*Betsy Chesnutt*

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

In statistics, the F-ratio is used to determine if there are differences between groups in an experiment. In this lesson, learn how to calculate the F-ratio and interpret the result.

James works for an advertising agency, and his team has developed four potential new advertisements for a product. As part of the evaluation process, James shows the ads to a focus group of potential customers and has each person rate (on a 0-10 scale) how likely they are to purchase the product after watching each ad. The data he collected is shown in the table:

How can he tell if one ad was more effective than another? Simply looking at the mean score of each is not enough because there might be a lot of variability in the data, and the means might appear to be different, when statistically, they aren't. A statistical test that he could use to determine if there were any differences between the four groups is called analysis of variance (or ANOVA). **ANOVA** is a test used to detect differences between continuous variables when there are more than two groups, so it would be ideal to use in this situation.

As part of performing an ANOVA on this data, James will need to calculate an F-ratio. The **F-ratio** is defined as the ratio of the between group variance (MSB) to the within group variance (MSW).

F = between group variance / within group variance = MSB / MSW

The calculated F-ratio can be compared to a table of critical F-ratios to determine if there are actually any differences between groups or not.

Let's look in a little more detail at all the steps involved in calculating and interpreting the F-ratio.

To calculate the variance within the groups, first calculate the sum of the squared deviations from the mean for each group. To find the sum of the squared deviations for a single group, follow these steps:

- Calculate the mean of each group by adding up all the values in each group and dividing by the total number of measurements in each group (
*n*).

- Subtract each individual measurement in the group from the group mean. This gives you the deviation of each measurement from the group mean.

- To find the sum of the squared deviations, square all of the deviations you just calculated and then add them all together. This number is the sum of the squared deviations.

Once you've done this for one of the groups, repeat the process for all the other groups. You should end up with a sum of squared deviations for each one of your groups.

Then, you can calculate the total within group variation like this:

Let's look at how to apply that to our advertising example. For each of the four groups, the sum of squared deviations is 7.33 (for Ad #1), 7.33 (for Ad #2), 10.83 (for Ad #3), and 3.00 (for Ad #4). The number of measurements in each group is 6, and the total number of measurements (N) is 24, and the number of groups (g) is 4.

To calculate the F-ratio, you also need the between group variance. This is a little easier to calculate than the within group variance.

- Calculate an overall mean by adding up all the group means and dividing the sum by the number of groups. For our example, the overall mean is 5.63.

- Subtract each group mean from the individual mean and square these differences.

- Multiply the difference you get for each group by the number of measurements in that group and add all these together.

- Finally, divide by (g - 1).

For our advertising example, that would look like this:

Now that you know MSB and MSW, simply divide MSB by MSW to find the F-ratio:

F-ratio = MSB / MSW = 18.02 / 7.123 = 2.53

What does the F-ratio tell you? Remember that the purpose of performing an ANOVA is to determine if there are any differences between the groups. The F-ratio tells you whether you should accept or reject the null hypothesis.

The **null hypothesis** in this situation is that there are no differences between groups. If you reject the null hypothesis, that means there are differences between groups. Basically, if you want to find that something works, you hope to reject the null hypothesis.

How does the F-ratio determine whether you should accept or reject the null hypothesis? You have to compare it to a critical F-ratio, which you can look up in a statistical table like this:

Each table corresponds to a particular significance level. This table is for a significance level of alpha = 0.05. To find the correct critical value in the table, you need to determine the number of degrees of freedom in the numerator and the denominator. For the between group variation (the numerator), there are 3 degrees of freedom (number of groups - 1). This is df1 in the table and is listed at the top of each column. For the within group variation (the denominator) there are 20 degrees of freedom (N - g; the total number of results minus number of groups). This is df2 in the table and is listed in front of each row.

So, to find F-critical, locate 3 degrees of freedom on the top and 20 degrees of freedom on the left side. Find the value at which these two intersect:

You can see that F-critical is 3.10 in this situation. Remember that the F-value you just calculated was 2.53. Since this is less than F-critical, you should accept the null hypothesis and reject the alternative. At this significance level, there are no differences between the groups.

- If F-ratio > F-critical, reject the null hypothesis.
- If F-ratio < F-critical, accept the null hypothesis.

After performing this statistical analysis, James cannot say that there is a difference in the effectiveness of the ads his company has produced.

An analysis of variance, or **ANOVA**, is a test used to detect differences between continuous variables when there are more than two groups. It looks at the ratio of the between group variance (MSB) to the within group variance (MSW). As part of performing an ANOVA, an F-ratio will be calculated. The **F-ratio** is the ratio of the between group variance to the within group variance. It can be compared to a critical F-ratio, which is determined by rejecting or accepting the **null hypothesis**, which determines whether or not there are no differences between groups.

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TECEP Principles of Statistics: Study Guide & Test Prep13 chapters | 91 lessons | 11 flashcard sets

- Analysis Of Variance (ANOVA): Examples, Definition & Application 6:47
- Two-Way ANOVA: Definition & Application
- Sum of Squares: Definition & Application 4:50
- Using ANOVA to Analyze Variances Between Multiple Groups 9:14
- Using ANOVA to Analyze Within-Group Variance
- How to Calculate the F-Ratio 6:54
- Go to TECEP Principles of Statistics: ANOVA

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