Another test for Normality , or just Q-Q plot? I am considering this test for  univariate normality which I believe, but not sure, may be just like the Q-Q test.
The procedure is like this:


*

*We compute sample mean, standard error (S.E)

*Then compute percent of sample values within $ \pm 1,2,3 $ S.E's from the sample mean.

*We then compute a chi-squared goodness of fit between expected ( using 68-95-99 % rule ) and actual.


Is this a known test? If not, how could I find out how this statistic is distributed? This is not too deep, so I don't expect it to be something new, but I am curious as to how to figure out the distribution of this statistic.
Maybe his is something really simple, in which case maybe I can delete the question. My apologies if this is so.
 A: Your description of the procedure is unclear. If you calculate the proportions in overlapping intervals those quantities will be dependent (above that caused by conditioning on the total), violating the chi-square assumption (though it should be possible to account for this issue).
If you have disjoint intervals then the statistic should be close to a chi-square as whuber notes (even though the boundaries are chosen based on the data). 
I have seen something like this used before, but its power is surprisingly low (even after you take account of the fact that the chi-squared is already known to have fairly low power as a goodness of fit test) and it suffers from considerable bias against alternatives that are presumably of interest (that is, with power below the significance level) even at fairly large sample sizes.
[The chi-square has better power and less bias when the intervals are nearer to equiprobable but power is still fairly low even then]
I should mention that there's an issue with values outside $[-3,3]$, which should also be addressed.
Further note that testing goodness of fit for an assumption of normality is not particularly useful. See for example the discussion in Is normality testing essentially useless?; I particularly recommend Harvey's answer as concise and clear but covering what I see as the main issues.
A: There are a whole bunch of goodness-of-fit methods out there. However, I am particularly interested in this new method based on "kernel Stein  measure". It is difficult to explain it here, so please see the original paper
http://proceedings.mlr.press/v48/chwialkowski16.pdf 
