Degrees of freedom is a mathematical equation used primarily in statistics, but also in mechanics, physics, and chemistry. In this lesson, explore how degrees of freedom can be used in statistics to determine if results are significant.
The degrees of freedom in a statistical calculation represent how many values involved in a calculation have the freedom to vary. The degrees of freedom can be calculated to help ensure the statistical validity of chi-square tests, t-tests and even the more advanced f-tests. These tests are commonly used to compare observed data with data that would be expected to be obtained according to a specific hypothesis.
For example, let's suppose a drug trial is conducted on a group of patients and it is hypothesized that the patients receiving the drug would show increased heart rates compared to those that did not receive the drug. The results of the test could then be analyzed to determine whether the difference in heart rates is considered significant, and degrees of freedom are part of the calculations.
Because degrees of freedom calculations identify how many values in the final calculation are allowed to vary, they can contribute to the validity of an outcome. These calculations are dependent upon the sample size, or observations, and the parameters to be estimated, but generally, in statistics, degrees of freedom equal the number of observations minus the number of parameters. This means there are more degrees of freedom with a larger sample size.
Formula for Degrees of Freedom
The statistical formula to determine degrees of freedom is quite simple. It states that degrees of freedom equal the number of values in a data set minus 1, and looks like this:
df = N-1
Where N is the number of values in the data set (sample size). Take a look at the sample computation.
If there is a data set of 4, (N=4).
Call the data set X and create a list with the values for each data.
For this example data, set X includes: 15, 30, 25, 10
This data set has a mean, or average of 20. Calculate the mean by adding the values and dividing by N:
Using the formula, the degrees of freedom would be calculated as df = N-1:
In this example, it looks like, df = 4-1 = 3
This indicates that, in this data set, three numbers have the freedom to vary as long as the mean remains 20.
Knowing the degrees of freedom for a population or for a sample does not give us a whole lot of useful information by itself. This is because after we calculate the degrees of freedom, which is the number of values in a calculation that we can vary, it is necessary to look up the critical values for our equation using a critical value table. These tables can be found in textbooks or by searching online. When using a critical value table, the values found in the table determine the statistical significance of the results.
Examples of how degrees of freedom can enter statistical calculations are the chi-squared tests and t-tests. There are several t-tests and chi-square tests that can be differentiated by using degrees of freedom.
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For instance, if a sample size were 'n' on a chi-square test, then the number of degrees of freedom to be used in calculations would be n - 1. To calculate the degrees of freedom for a sample size of N=9. subtract 1 from 9 (df=9-1=8). With this information, use the appropriate row of a chi-square distribution table by looking for the location of the line for 8 degrees of freedom. Once the line for the desired degrees of freedom is located, its row will provide lots of information to help examine and analyze the data. Similarly, for other statistical measures like t-tests, the analyst would have to calculate, or be given, the appropriate degrees of freedom to use in order to analyze the data.
Degrees of freedom also happen to show up in the formula for the standard deviation. Standard deviation is a statistical value used to determine how far apart the data in a sample (or a population) are. It is also used to determine how close individual data points are to the mean of that population or sample. The population includes all members of a defined group that information is being studied or collected on for data driven decisions. A sample is part of the population. When calculating standard deviation for a sample, use n - 1 degrees of freedom.
Degrees of freedom calculations are used in many disciplines, including statistics, mechanics, physics and chemistry. It is a mathematical equation that tells how many values can vary and can help to determine if results are statistically significant.
The most commonly encountered equation to determine degrees of freedom in statistics is df = N-1. Use this number to look up the critical values for an equation using a critical value table, which in turn determines the statistical significance of the results.
The degrees of freedom are calculated to help ensure the statistical soundness of tests, such as of chi-square tests and t-tests.
What are Degrees of Freedom?
Degrees of freedom indicate how many values have the freedom to vary
Degrees of freedom are often used with critical value tables to interpret a test's results
Degrees of freedom are used in calculating the standard deviation of a data set
The equation for degrees of freedom is df= N-1
When you are finished, you should be able to:;
Explain what degrees of freedom are and what they represent
Calculate the degrees of freedom for a data set
Recall the role of degrees of freedom in determining standard deviation and critical values for a data set
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