## Degrees of Freedom - A New Definition

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### Degrees of Freedom - A New Definition

Doc: I have a research question. I just finished a research class and during the semester we had several discussions of the research term Degrees of Freedom. Although the professor tried many times to explain the concept, many of the students in the class, including me, always had questions, primarily because the Degrees of Freedom computation is different for each statistical method. The discussion always ended the same way when the professor would tell us that the Degrees of Freedom is "the number of values in the final calculation of a statistic that are free to vary". That's still confusing to me so I searched the Internet for more information.

On a few websites that that discuss your new 9th Edition of the research book you wrote with Joe Dominick, there are mentions that the it includes a new definition of Degrees of Freedom - the first new re-definition since the concept was developed in the early 1920s. Unfortunately, I don't have a copy of your book, so I would like to ask if you would include your new definition in your column. I would really appreciate it. Thanks in advance. - Anonymous

Anon: You don't have a copy of our book, eh? Well, OK, but before I include my new definition, I would like to provide a little background on why I developed a new definition for Degrees of Freedom.

I'm not sure if you know, but whenever most authors of academic textbooks prepare a new edition of a text, the publishing company contracts with several professors to provide a review of the text and make suggestions for things to include in a new edition. In our case, we received seven very comprehensive reviews from professors from around the world who use our text in their research classes. One professor commented on the difficulty many students have in understanding Degrees of Freedom and suggested that we develop a new definition for our new 9th edition that became available in January 2011.

I thought the professor's suggestion was interesting because I have found the same student confusion with Degrees of Freedom while teaching on-and-off since 1976. The problem was that I never gave much thought to developing a new, and hopefully clearer, definition for a concept that has been around since the early 1920s. The reviewer's comments provided the encouragement I needed.

The new definition in the 9th edition is on pages 305-308 and includes a table I developed to demonstrate the effect Degrees of Freedom has on a data set. The complete discussion is a bit too much to include here in the column, so I'm only going to include an edited version of the complete explanation and new definition.

Most statistics historians would probably agree that the concept of degrees of freedom was first developed by British mathematician Karl Pearson (1857-1936) around 1900, but was refined, or more appropriately, corrected, by R. A. Fisher, in an article published in 1922. Both Pearson and Fisher shared a hatred of being wrong (and they also reportedly shared a hatred for each other), and in their quest for correctness, they realized that something was needed in statistical calculations to compensate for possible errors made in data collection, analysis, or interpretation. The fear of making an error is actually the foundation for the development of degrees of freedom.

The philosophy behind degrees of freedom is very simple—fear of being wrong. Since most research studies analyze data from a sample where the results are projected to the population, there is a need to make a slight adjustment to the sample to compensate for errors made in data collection and/or interpretation. This is because population parameters (the "real" data) are rarely, if ever, known to researchers. When calculating statistics for a sample, there is a need to have results that are somewhat conservative (corrected, adjusted) to compensate for any errors that may be present

The confusion surrounding degrees of freedom might be reduced if the concept had another name or definition. From our previous discussion, we can say that, in essence, the "key" to degrees of freedom is not that data are free to vary, but rather the concept relates to an adjustment made to data to provide a slightly more conservative estimate of the data to compensate for the possibility of errors in data collection, analysis, or interpretation. Therefore, our formally stated definition for degrees of freedom is:

There is the edited discussion. As you can see, the new definition highlights the fact that Degrees of Freedom is a reduction in sample size to compensate (somewhat) for research error. I hope that helps, but let me know if you have any other questions.

Note: I need to include that the definition for Degrees of Freedom has a 2011 copyright by Wadsworth Cengage Learning, which means that you're not supposed to use the information in any form without permission from the company.

(Want to comment on this question? Click on the POSTREPLY button under the question.)

On a few websites that that discuss your new 9th Edition of the research book you wrote with Joe Dominick, there are mentions that the it includes a new definition of Degrees of Freedom - the first new re-definition since the concept was developed in the early 1920s. Unfortunately, I don't have a copy of your book, so I would like to ask if you would include your new definition in your column. I would really appreciate it. Thanks in advance. - Anonymous

Anon: You don't have a copy of our book, eh? Well, OK, but before I include my new definition, I would like to provide a little background on why I developed a new definition for Degrees of Freedom.

I'm not sure if you know, but whenever most authors of academic textbooks prepare a new edition of a text, the publishing company contracts with several professors to provide a review of the text and make suggestions for things to include in a new edition. In our case, we received seven very comprehensive reviews from professors from around the world who use our text in their research classes. One professor commented on the difficulty many students have in understanding Degrees of Freedom and suggested that we develop a new definition for our new 9th edition that became available in January 2011.

I thought the professor's suggestion was interesting because I have found the same student confusion with Degrees of Freedom while teaching on-and-off since 1976. The problem was that I never gave much thought to developing a new, and hopefully clearer, definition for a concept that has been around since the early 1920s. The reviewer's comments provided the encouragement I needed.

The new definition in the 9th edition is on pages 305-308 and includes a table I developed to demonstrate the effect Degrees of Freedom has on a data set. The complete discussion is a bit too much to include here in the column, so I'm only going to include an edited version of the complete explanation and new definition.

Degrees of Freedom

[/b]Most statistics historians would probably agree that the concept of degrees of freedom was first developed by British mathematician Karl Pearson (1857-1936) around 1900, but was refined, or more appropriately, corrected, by R. A. Fisher, in an article published in 1922. Both Pearson and Fisher shared a hatred of being wrong (and they also reportedly shared a hatred for each other), and in their quest for correctness, they realized that something was needed in statistical calculations to compensate for possible errors made in data collection, analysis, or interpretation. The fear of making an error is actually the foundation for the development of degrees of freedom.

The philosophy behind degrees of freedom is very simple—fear of being wrong. Since most research studies analyze data from a sample where the results are projected to the population, there is a need to make a slight adjustment to the sample to compensate for errors made in data collection and/or interpretation. This is because population parameters (the "real" data) are rarely, if ever, known to researchers. When calculating statistics for a sample, there is a need to have results that are somewhat conservative (corrected, adjusted) to compensate for any errors that may be present

The confusion surrounding degrees of freedom might be reduced if the concept had another name or definition. From our previous discussion, we can say that, in essence, the "key" to degrees of freedom is not that data are free to vary, but rather the concept relates to an adjustment made to data to provide a slightly more conservative estimate of the data to compensate for the possibility of errors in data collection, analysis, or interpretation. Therefore, our formally stated definition for degrees of freedom is:

An intentional and predetermined reduction in sample size to provide a conservative data adjustment to compensate for research error.

[/b]There is the edited discussion. As you can see, the new definition highlights the fact that Degrees of Freedom is a reduction in sample size to compensate (somewhat) for research error. I hope that helps, but let me know if you have any other questions.

Note: I need to include that the definition for Degrees of Freedom has a 2011 copyright by Wadsworth Cengage Learning, which means that you're not supposed to use the information in any form without permission from the company.

(Want to comment on this question? Click on the POSTREPLY button under the question.)

Roger Wimmer is owner of Wimmer Research and senior author of Mass Media Research: An Introduction, 10th Edition.