Normality testing with very large sample size? Hypothesis testing such as Anderson-Darling or Shapiro-Wilk's test check normality of a distribution. However, if the sample size is very large, the test is extremely "accurate" but practically useless because the confidence interval is too small. They will always reject the null, even if the distribution is reasonably normal enough.
How should I test normality when sample size is very large, other than visualizing histograms?
The motivation is that I want to automate checking normality of large data set in a software platform, where everything needs to be automated, not manually visualized and inspected by humans.
One thing that came across me is that instead of using Shapiro-Wilk test, I calculate kurtosis and skewness of the distribution, and if they are $\pm 1.0$, I can assume that my large dataset is "reasonably" normally distributed.
Is my approach correct, or is there any other alternatives?
 A: Continuation from comment:  If you are using simulated normal data from R,
then you can be quite confident that what purport to be normal samples
really are. So there shouldn't be 'quirks' for the Shapio-Wilk test to detect.
Checking 100,000 standard normal samples of size 1000 with the Shapiro-Wilk
test, I got rejections just about 5% of the time, which is what one
would expect from a test at the 5% level. 
set.seed(2019)
pv = replicate( 10^5, shapiro.test(rnorm(1000))$p.val )
mean(pv <= .05)
[1] 0.05009

Addendum.
By contrast, the distribution $\mathsf{Beta}(20,20)$ "looks" very
much like a normal distribution, but isn't exactly normal.
If I do the same simulation for this approximate model, Shapiro-Wilk
rejects about 7% of the time. Looked at from the perspective of
power, that's not great. But it seems Shapiro-Wilk is sometimes
able to detect that the data aren't exactly normal. 
This is a long
way from "always," but I think $\mathsf{Beta}(20,20)$ is closer
to normal than a lot of real-life "normal" data are. (And the link
says always may be "a bit strongly stated." I suspect the greatest
trouble may come with samples a lot bigger than 1000, and for some
normal approximations that are quite useful--even if imperfect.)
"Not every statistically significant difference is a difference
of practical importance." Sometimes, people who should know better seem to forget
that when doing goodness-of-fit tests.
set.seed(2019)
pv = replicate( 10^5, shapiro.test(rbeta(1000, 20,20))$p.val )
mean(pv <= .05)
[1] 0.07152


A: As @gg pointed out in a comment, this entire discussion in pointless without defining how normal-like does data have to be for us to consider it "normal enough". In practice, I often like the following criteria:


*

*Skewness close to 0, maybe a (-1,1) range, or that you feel more comfortable with depending on "how normal-like is normal enough". 

*Kurtosis close to 3 (or excess kurtosis close to 0) High kurtosis is often a greater issue than low kurtosis as it leads to more outliers.

*Median not far away from the mean

*QQ-plots are your friends!

A: 
...However, if the sample size is very large, the test is extremely "accurate" but practically useless because the confidence interval is too small. They will always reject the null, even if the distribution is reasonably normal enough...

What if you take a sub-sample of size 100 or 300 from the large sample consisting of several thousands or more. If I'm not mistaken then taking the sub-samples will reflect the same distribution but will work better with the common normality tests.
