I want to avoid misusing normality tests where a large enough sample size will highlight any slight non-normality. I want to be able to say that a distribution is "normal enough".
When the population is non-normal the p-value for the Shapiro-Wilk test tends to 0 as the sample size increases. The p-value isn't helpful in deciding if a distribution is "normal enough".
I think a solution would be to measure the effect size of the non-normality and reject anything which is more non-normal than a threshold.
The Shapiro Wilk test produces a test statistic $W$. Is this a way to measure the effect size of the non-normality?
I tested this in R by doing a shapiro wilk test on samples drawn from a uniform distribution. The number of samples ranged from 10 to 5000, the results are plotted below. The value of W does converge to a constant, it doesn't tend towards $1$. I'm unsure if $W$ is biased for small samples, it seems to be low for small sample sizes. If $W$ is a biased estimate of effect size that could be a problem if I want to accept anything under $W=0.1$ as "normal enough".
My two questions are:
Is $W$ a measure of effect size of non-normality?
Is $W$ biased for small sample sizes?