How to empirically show that a certain quantity approximately follows a normal distribution? To motivate some theoretical work, I need to show that two certain variables (say $X$ and $Y$) approximately follow a normal distribution in actual datasets. I have one large dataset with about $300$ samples (yes, it is large in the field I work in), which I can use to do this. The theoretical work would address other studies with samples sizes in the range $10-40$, which are more usual.
Now, I was thinking to take samples of size $20$ or so from the larger dataset, and show with normality tests that they usually are normal. Would this sound ok?
If there are some better choices (should I use the whole sample somehow insteand?), I would like to hear them.
 A: Typical significance tests, including all the usual goodness of fit tests, do not address the question "approximately follows a normal distribution"; at a large sample size even quite small deviations from normality would lead to rejection, even though the distribution may be easily close enough to normal for whatever purpose you have in mind.
The question is nearer to one of effect size (is the deviation from normality large enough to matter?), which doesn't really relate to sample size. 
[Indeed, for some purposes, how much normality matters goes down with sample size -- e.g. it may matter a lot more at n=20 than it does at n=50, say. In that case, a hypothesis test will be much more likely to reject when it matters least.]
You haven't made clear what exactly might constitute "near enough" for your problem (or indeed what your problem actually is).
One can of course make an assessment via a Q-Q plot, which does at least give a visual impression nearer to some kind of effect size, but the visual deviation from normality may be of a kind that doesn't especially matter for your particular problem.
So to try to address your problem in a more sensible way, perhaps via simulation, one would need to know why you need approximate normality, and what it is that this non-normality may be affecting that you need to worry about.
