I need help with the following:
So, in my statistics class, I have covered that if $X_1, ..., X_n$ are independent NORMAL variables, then $X_1$ and $X_1 - \bar X$ would be independent.
What if the variables $X_1$ to $X_n$ are not from the normal distribution, but still are independent, are the two above statistics still independent?
Comment: Showing that $\bar X$ and the deviation $X_1 - \bar X$ are uncorrelated is a simple exercise. But uncorrelated implies independent only for normal data. [This result can be used to show that the sample mean and SD are independent for normal data.]
Below is an illustration that may help you visualize the distinction. The first plot of the deviation against the mean is for normal data and the second is for exponential data. About 9000 samples of size 10 are plotted in each graph.