Normal distribution with integer numbers

Question

I'm running into some quite interesting problems over here. It is about creating a normal distribution with integer numbers in the range of 1 to 5 (1,2,3,4,5). Technically, it is a Poisson distribution with the shape of a normal distribution.

My question: When I create a distribution as mentioned above, tests for normality fail (p < 0.01)(Shapiro Wilk Test, Kolomogorov Smirnov Test), as I created a pool of normally distributed numbers which I rounded:

xRND<-round(rnorm(179,mean=2.9,sd=1)) table(xRND) xRND 0 1 2 3 4 5 6 2 14 41 67 45 9 1

Is there any test that helps me to check for a normal distribution shape?

Best regards, St.

Answer 1

From my prospective the Shapiro Wilk Test is the test with the best goodness. Working with rounded values will have an impact to the result. There is a method called Sheppard-correction to calculate a adapted W-value (test statistic) considering rounding of the tested values:

with ω as the rounding difference. (here the link to the German source)