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I am on the hypothesis testing for two populations unit. I need some intuitive explanation as to why this formula is used. My statistics professor put this up on the board but he didn't explain why its true.

For a T distribution, the formula for the degrees of freedom is:

$$ \large \mathrm{df} = \frac{ \left(\frac{s_{1}^{2}}{n_1} + \frac{s_{2}^{2}}{n_2} \right)^{2} }{ \frac{\left(\frac{s_{1}^{2}}{n_1}\right)^2}{n_{1} - 1} + \frac{\left(\frac{s_{2}^{2}}{n_2}\right)^2}{n_{2}-1}}$$

Here $s_1, s_2$ are the sample standard deviations and $n_1,n_2$ are the sample sizes.

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    $\begingroup$ This is something of a big issue, actually. There isn't really a good test for the difference of two means with unequal variances (en.wikipedia.org/wiki/Behrens%E2%80%93Fisher_problem). In fact, the test statistic you are using is only approximately distributed as a t distribution. $\endgroup$ – B R Apr 25 '12 at 2:34
  • $\begingroup$ This doesn't really give any intuition as to why that formula is used to calculate the degrees of freedom, but perhaps it indicates why your professor would not want to spend time on it. $\endgroup$ – B R Apr 25 '12 at 2:38
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    $\begingroup$ This is based on matching moments of a linear combination of independent chi squared random variables to a gamma distribution and then using plug-in estimators. It is covered, for example, in B. L. Welch (1947), The generalization of Student's problem when several different population variances are involved, Biometrica, vol. 34, no. 1/2, pp. 28-35. See pages 31 and 32 in particular. $\endgroup$ – cardinal Apr 25 '12 at 3:18
  • $\begingroup$ There is no reason to use \Large in the math display equations. I've left it as \large only because it makes it a little easier to see the denominator terms. $\endgroup$ – cardinal Apr 25 '12 at 3:53
  • $\begingroup$ Maybe stattrek.com/estimation/difference-in-means.aspx?tutorial=stat can help. Under the subsection titled "If you use a t score, you will need to compute degrees of freedom (DF)." $\endgroup$ – Kirthi Raman May 4 '12 at 12:11
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This question is essentially answered. The comment of user "cardinal" that mentions the B.L. Welch (1947) paper provides all there is to it, as regards where the formula comes from. Welch derives the exact mathematical solution for the general problem of calculating the degrees of freedom when the samples are more than two, then he develops an approximate solution through a Taylor expansion, and then mentions the resulting approximate df-formula for the special case of two samples. There is no deep intuition behind the formula, just patient (but healthy) mathematics. Welch's style and notation is rather old-fashioned - for educational purposes, another paper of his "The Significance of the Difference Between Two Means when the Population Variances are Unequal", Biometrika, Vol. 29, No. 3/4 (Feb., 1938), pp. 350-362, focuses on the two-samples case, and is a bit more accessible.

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