# Degrees of Freedom for Linear Combination of Variables

I have three mean estimates, all from variables assumed to follow a Normal distribution. The following is a mock example.

x1 is estimated at 4, with a standard error of 5 and 6 degrees of freedom x2 is estimated at 6, with a standard error of 4 and 6 degrees of freedom x3 is estimated at 7, with a standard error of 3 and 6 degrees of freedom

What would be the degrees of freedom attached to (x1+x2+x3) ?

• This depends on the dependence between the three xs. Apr 3, 2022 at 19:07
• How are the estimates made and what exactly do you mean by "degrees of freedom" of such an estimate? (Estimates usually don't have anything known as a "degree of freedom," but often their standard errors do--and even then the meaning of d.f. varies with how the estimate is made.)
– whuber
Apr 3, 2022 at 19:19
• To @RichardHardy: They are independent, I should have stated that. Apr 3, 2022 at 20:35
• To @whuber: Those are the DFs attached to the standard errors. They have been estimated using maximum likelihood. If a parameter X follows a normal distribution, it's maximum likelihood estimate follows a T distribution with N-1 degrees of freedom due to the uncertainty on the variance estimate. Those are the degrees of freedom I'm referring to. Apr 3, 2022 at 20:38
• @PerrySun, please edit the post to include the additional information from the comments. Apr 4, 2022 at 5:01