I'm using a Kolmogorov-Smirnov test to examine how normal (or Gaussian) some data sets I have are. I am having trouble with interpreting the $p$-values (surprise surprise).
My present understanding is that if the returned $p$-value is $p > 0.05$ this suggests that the data is normally distributed, while datasets which produce a Kolmogorov-Smirnov test result $p < 0.05$ suggest that the data is not normally distributed.
Firstly, is my present understanding/interpretation correct?
Secondly I performed some very basic simulations. I simply generate data sets with $1000$ points which are normally distributed and perform a Kolmogorov-Smirnov test on each one. If I look at the distribution of $p$-values extracted from this, I see a square distribution with values ranging between 0 and 1.
I don't understand this -- I guess I don't really understand $p$-values as I expected for data which definitely is normally distributed (I generated it to be such) that there would be a median $p$-value.