Degrees of freedom (DF) is a mathematical equation used in mechanics, physics, chemistry and statistics. The statistical application of degrees of freedom is quite broad and students can expect to need to calculate degrees of freedom early on in statistics coursework. Accurately calculating the degrees of freedom you have in an equation is vital since the number of degrees lets you know how many values in the final calculation are allowed to vary. Since statistics attempts to be as precise as possible, the degrees of freedom calculation is done often and contributes to the validity of your outcome. Practical usages of degrees of freedom might include statistically analyzing baseball positions.

## Determine Statistical Test

Determine what type of statistical test you need to run. Both t-tests and chi-squared tests use degrees of freedom and have distinct degrees of freedom tables. T-tests are used when the population or sample has distinct or discrete variables. In the financial world, one discrete variable is each stock price because it is not changing at all times. Instead, a discrete variable in the stock market only changes when a transaction occurs. In contrast, a continuous variable is something that has a value at all times. For example, light emission or sound are both considered continuous variables. Chi-squared tests are used when the population or sample has continuous variables. Both tests assume normal population or sample distribution of the data.

## Visual Degrees of Freedom Data Table

If you have trouble conceptualizing what degrees of freedom means in your data set, picture a two-by-two table where the sum of the numbers in each row and column must equal 100. If you knew the values of three of the cells you would also know the value of the fourth. In this example you would have N-1 degrees of freedom or three degrees of freedom (4-1=3).

## Identify Independent Variable Number

Identify how many independent variables you have in your population or sample. If you have a sample population of N random values then the equation has N degrees of freedom. If your data set required you to subtract the mean from each data point--as in a chi-squared test--then you will have N-1 degrees of freedom.

## Critical Value Table

Look up the critical values for your equation using a critical value table. Knowing the degrees of freedom for a population or sample does not give you much insight in of itself. Continuing the financial world example, an alpha can be defined as the intrinsic movement of a specific stock removed the overall effect of the market. Rather, the correct degrees of freedom and your chosen alpha together give you a critical value. This value allows you to determine the statistical significance of your results.

#### References

- Statistics How To: Degrees of Freedom, What Are They?
- Ron Dotsch: Degrees of Freedom Tutorial
- Mathematics Stack Exchange: What is Degree of Freedom in Statistics?
- Statistics by Jim: Degrees of Freedom in Statistics
- iSixSigma: Can You Explain Degrees of Freedom and Provide an Example?
- Surgical Critical Care: Analysis of Continuous Variables Comparing Means