# Proof that the mean is a complete sufficient statistic and the sample variance is an ancillary statistic

I have $$X_1, X_2, ..., X_n$$ that are random samples from the single variate $$N(\mu,\sigma^2)$$. I want to prove that the mean $$\bar{X}$$ and the sample variance $$s_x ^2 = \frac{1}{(n- 1)} \sum_{i=1}^n (X_i - \bar{X})$$ are independent.

As I understand it, I can use Basu's Theorem. Since $$\bar{X}$$ is a complete sufficient statistic, and $$s_x^2$$ is an ancillary statistic, Basu's theorem states that they are indeed independent.

I understand why $$\bar{X}$$ is a complete sufficient statistic logically, but I can't figure out how to show it mathematically. Same goes with the sample variance. Could someone please explain (mathematically) why $$\bar{X}$$ is a complete sufficient statistic, and $$s_x^2$$ is an ancillary statistic? In most sources I've checked this fact just seems obvious and they don't show the proof...

• Complete sufficiency of sample mean follows from the fact that the pdf of $X_1,\ldots,X_n$ is a member of exponential family. And sample variance is ancillary because its distribution does not depend on the parameter $\mu$. See math.stackexchange.com/questions/47350/…. – StubbornAtom Apr 16 at 14:52