Calculating Effective Degrees of Freedom

When performing uncertainty analysis, it is important to calculate the degrees of freedom associated with the estimation of uncertainty. However, determining the total degrees of freedom is not simply the summation of the independently calculated degrees of freedom. Instead, we must use the Welch Satterthwaite approximation equation to calculate the effective degrees of freedom. In this article, we will introduce you to the Welch Satterthwaite approximation equation and show you how to apply it to your analysis.

Degrees of Freedom

Before getting ahead of ourselves, it is important to address degrees of freedom. In statistics, degrees of freedom is the number of values in the final calculation which are free to vary. In other words, it is the number of ways or dimensions an independent value can move without violating constraints.

To calculate degrees of freedom, we subtract the number of relations from the number of observations. For determining the degrees of freedom for a sample mean or average, we would subtract one (1) from the number of observations, n.


Effective Degrees of Freedom

Now that we have explained degrees of freedom, let’s look at effective degrees of freedom and the Welch Satterthwaite approximation equation. When performing uncertainty analysis, we are evaluating and combining multiple variances characterized by various probability distributions. Due to the increased complexity, many times the degrees of freedom is inappropriate or undefined. Therefore, we calculate the effective or equivalent degrees of freedom, for inference purposes, to approximate the actual degrees of freedom. This is accomplished using the Welch Satterthwaite equation.


Applying the Equation

Using the equation above and the table below, we can see how to easily apply the equation to our uncertainty calculations. Refer the colored boxes. Each box is identified by color and symbol. Plug the values into the equation and calculate the effective degrees of freedom.



Now that we have explained effective degrees of freedom and the Welch Satterthwaite equation, feel free to try it out for yourself and include it in your uncertainty budgets. I hope that I have made this task a little easier for those who have struggled with this. If you have any questions, please feel free to contact me.

Want to learn more about the Welch Satterthwaite approximation equation, check the original papers published by F.E. Satterthwaite and the B.L. Welch.

The Generalization of `Student’s’ Problem when Several Different Population Variances are Involved
B. L. Welch
Vol. 34, No. 1/2 (Jan., 1947), pp. 28-35
Published by: Biometrika Trust
An Approximate Distribution of Estimates of Variance Components
F. E. Satterthwaite
Biometrics Bulletin
Vol. 2, No. 6 (Dec., 1946), pp. 110-114
Published by: International Biometric Society
About the Author

Richard Hogan

Richard Hogan is the CEO of ISO Budgets, L.L.C., a U.S.-based consulting and data analysis firm. Services include measurement consulting, data analysis, uncertainty budgets, and control charts. Richard is a systems engineer who has laboratory management and quality control experience in the Metrology industry. He specializes in uncertainty analysis, industrial statistics, and process optimization. Richard holds a Masters degree in Engineering from Old Dominion University in Norfolk, VA. Connect with Richard on LinkedIn.


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