I want to compare the distribution of the data from my model with normal distribution (since some previous works state that in comparison with normal dist. my data should have thicker tails). I decided to use QQ plot. Now, I am wondering whether I should compare it with normal distribution that has the same mean and same standard deviation as my data. Should I?
The form of the qq-plot should not depend on the mean and standard deviation of your data, so there is no need to standardize. This is true when using qqplotting to compare the data distribution against a normal distribution. When using qqplotting against other theoretical distributions, the question arises again and becomes more interesting, since for many other distribution families the form "shape" of the distribution do vary with parameters.
Specifically, a qqplot is plotting sample quantiles (usually on the y axis) against theoretical quantiles (usually on the x axis). If you standardize the sample, the resulting sample quantiles will be a linear function of the sample quantiles before standardizing. This will not change the shape/appearance of the plot, it will only change the labeling on the y axis. A simple example:
opar <- par(mfrow=c(1, 2), no.readonly=TRUE) set.seed(7*11*13) x <- rnorm(100, 2, 3) qqnorm(x) qqnorm(scale(x)) par(opar)