Nice post today. Thanks for reminding us of how sensitive the GOF tests are. With small sample sizes the issue of assessing which distribution is a good approximation to the data is tricky. Even normal probability plots can fail us when we have small sample sizes. For your small datasets only the exponential data is not well approximated by the normal distribution; both the gamma and lognormal data hug the normal line well.

ods graphics /byline=footnote;
ods select QQPlot;
proc univariate data=Rand normal;
by Distribution;
var x;
qqplot / normal(mu=est sigma=EST color=Red) square;
run;

I’m glad you are bringing up this point: “The GOF tests reject the hypothesis that R is normally distributed, merely because the values in R are rounded to the nearest 0.1 unit.” This is important not only in the GOF situation but also in the analyses we perform. For example, when using process behavior charts rounding can cause out-of-control signals for an in-control process. Dr. Wheeler calls this “chunky data” and has shown how the measurement increment affects the range-estimates of standard deviation.