Degrees of freedom

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## The origin of degrees of freedom

In most research it is impossible to include every single object of the population in the research. A sample is taken and the response is being analysed. If the response is large enough a statistical test should give a pretty good vision of the population based on the response.

However, the estimate is based on the response size. If 2 out of 10 people indicate that they are willing to use a product, ie 20% of the respondents, it can be assumed that 20% of the entire population is willing to use that product. If 1 person changes their mind to use the product, the estimates for the total population will change by 10%. That is why a larger response is better. If 20 out of 100 respondents are willing to use the product, the estimate for the total population is still 20%. But now, if one person changes their mind, the estimate for the population will only change 1%. Therefore, the greater the response, the better the estimate for the total population.

## The use of degrees of freedom in statistical tests

Statistical tests show a more stable result when more respondents or objects are included. In general, an indicator for the stability of the result of the statistical test is equal to the total number of respondents or objects in the analysis minus 1. This indicator is called the degrees of freedom, or simply DF (sometimes DoF) and is calculated (again in generally) as n - 1. The DF are noted with the Greek character ν.

In many textbooks, the concept of degrees of freedom is described as the extent to which a variable can vary freely in a test. This may give the idea that a greater amount of degrees of freedom leads to less stable results. However, the opposite is true: the greater the number of degrees of freedom, the more stable the result of a statistical analysis will be. That is why it is good to have a large sample response. In addition, it is easier with more respondents to find statistically significant results, which is based on the stability of estimating aspects in the population.

The number of degrees of freedom varies per test. Some tests have only one number for the degrees of freedom, others have two, and still others have no numbers at all. That makes it a bit confusing. How the number of degrees of freedom should be calculated, however, is indicated with every test. So it can be learned.

## Reporting the degrees of freedom

Finally, a remark. The number of degrees of freedom must be reported when the results of a statistical test are presented.

You can read more about the degrees of freedom on our webpages about the z-, t-, chi square and F-distribution.

## Mission

My goal is to teach you how to conduct good research.

Because:

Good research provides you with better information.
With better information, you can make better decisions.
With better decisions, you can create a healthier, wealthier and freer world,
for people, fauna and flora, for current and future generations.

That is why I think it is important that  you know how to do your research well.