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.


  1. Hi,

    I am Marco from Hong Kong. I have a question for the above equation.
    I see in your example, your calculated degree of freedom is 10.8, the value is in between the degree of freedom of the contributed components (9,4,2,2, power200of1).

    But I calculate the effective degree of freedom for my own test, in which 38 is obtained, but the degree of freedom of the contributed components are 9,29,30,9,9.
    my question is, is the valve of 38 is reasonable ? Thanks.

    • Hi Marco,

      The calculated value of effective degrees of freedom is weighted by the magnitude of uncertainty and the number of degrees of freedom. So, the degrees of freedom with the greatest magnitude of uncertainty will be given preference in the final calculation.

      If you would like to send me your calculation, I will be glad to look it over for you for a small fee. You can email me at

  2. Hi Richard,
    In your example you have used k = 2.000.

    According to GUM G.4.1 Note 1 I would have truncated the value of Effective Degrees of Freedom to 10. Then according to Table G.2 k = 2.23 @ 95 %.

    According to GUM E.2.1 an understatement of uncertainties might cause too much trust to be placed in value reported.

    • I agree with Mr. Rune. I thought the purpose of calculating the effective degrees of freedom was to find the correct k value.

    • I agree with Mr. Rune. Based on the above example, the k value shouldn’t be 2.000 since effective degrees of freedom is 10.8. Excerpt from student’s t table, the correct k value for 10 and 11 degrees of freedom is 2.23 and 2.20, respectively at 95% confidence level. Either you use k value of 2.23 or 2.206 (interpolated value), expanded uncertainty will be equal to 1.5%, not 1.4%.

      • It is true that table G.2 in GUM states that the k factor when the effective degrees of freedom is 10 should be 2.23. But earlier in section G.6.6 (which is the section that table G.2 is in) it is stated that:

        “…[if] the uncertainty of uc(y) is reasonably small because its effective degrees of freedom is of a significant magnitude, say greater than 10…adopt k = 2 and assume that U = 2uc(y) defines the interval having a level of confidence of approximately 95 percent”

        So isobudgets is correct in using K=2 when the effective degrees of freedom is 10.8.

    • Hi Yahwant,

      It lets you know how many degrees that the estimated value can vary. Typically, the larger the number of degrees of freedom, the more confidence you should have in the estimate.

      I hope this helps.

      Best regards,


Leave a Reply

Your email address will not be published. Required fields are marked *