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I was looking at an example of difference of two means test where $t$-statistic is calculated as follows: $$t=\frac{(\bar x_1-\bar x_2)-D}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$$ and degrees of freedom for the $t$-distribution is obtained using following formula: $$v=\frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{\left(s_1^2/n_1\right)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}}$$ My questions are:

1) What is degrees of freedom? (I've known it as the number of items that can be chosen freely.)
2) How the formula for $v$ has been derived?

I've found a reasonable answer to 2), by reading an article written by Michael Allwood, using following facts -

$$\chi_n^2=Z_1^2+Z_2^2+...+Z_n^2$$ $$t_n=\frac{N(0,1)}{\sqrt{\frac{\chi_n^2}{n}}}$$ $$E[\chi_n^2]=n\,\,\,\,\,\,\,var(\chi_n^2)=2n\,\,\,\,\,\,\,and\,\,\,\,\,\,\,var(aX)=a^2var(X)$$

In this case answer to 1), degrees of freedom, is just a multiplier and can be found using algebra and calculus.

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    This question is a bit of a tall order since degrees of freedom is one of those notoriously difficult to explain topics and this is not an ideal example to explain it on. In case it doesn't get answered, for your first question, start here: http://blog.minitab.com/blog/statistics-and-quality-data-analysis/what-are-degrees-of-freedom-in-statistics and for your second, here https://en.wikipedia.org/wiki/Behrens%E2%80%93Fisher_problem#Welch.27s_approximate_t_solution . – spaceisdarkgreen Apr 22 '17 at 02:31
  • @spaceisdarkgreen proof for 2) is real pain –  Apr 22 '17 at 23:47

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