I'm confused as to the correct formula for approximate degrees of freedom to use for Welch's t-test. Satterthwaite's (1946) formula is the most commonly cited formula, but Welch gave an alternative in 1947. I'm not sure which is preferable (or used by most statistical software).

Satterthwaite's formula:

$$frac{left(s_x^2/n_x +s_y^2/n_yright)^2}{(s_x^2/n_x )^2/(n_x-1)+(s_y^2/n_y )^2/(n_y-1)}$$

Welch's formula:

$$-2+ frac{left(s_x^2/n_x +s_y^2/n_yright)^2}{(s_x^2/n_x )^2/(n_x+1)+(s_y^2/n_y )^2/(n_y+1)}$$

References:

Satterthwaite, F.E. (1946). "An Approximate Distribution of Estimates of Variance Components".

*Biometrics Bulletin*, 2, 6, pp. 110–114.Welch, B.L. (1947). "The generalization of "Student's" problem when several different population variances are involved".

*Biometrika*, 34, 1/2, pp. 28–35.

**Contents**hide

#### Best Answer

Welcome to CV!

I cannot answer on which one is preferred (they are actually really close so I don't think it matters much), but generally, major statistical software packages use Satterthwaite's method. `SPSS`

and `SAS`

both use it. In `Stata`

, some commands like `ttest`

would allow user to specify Welch's method, but Satterthwaite's is still the default.

And in literature, I have mostly seen Satterthwaite's formula being cited. Time to time it's referred to as Satterthwaite-Welch's degrees of freedom, but the formula cited is Satterthwaite's. I guess having published it one year earlier did matter.

### Similar Posts:

- Solved – Should I use Welch’s (1947) approximate degrees of freedom or Satterthwaite’s (1946)
- Solved – Should I use Welch’s (1947) approximate degrees of freedom or Satterthwaite’s (1946)
- Solved – R’s t.test() unequal variance degrees of freedom
- Solved – When should one use the Welch’s T-test?
- Solved – Finding the degrees of freedom for a confidence interval for the difference of two means