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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?

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    $\begingroup$ How many observations do you have? The form of the qq-plot should not depend on standardization of the data, but that is the most practical way of computing it. So: standardize your data, and compare the qqplot to a standard normal distribution. You can also make a simulated envelope. See stats.stackexchange.com/questions/96553/… $\endgroup$ – kjetil b halvorsen May 4 '15 at 17:50
  • $\begingroup$ I have around 1,000 observations. The think is that I should compare non-standardized data (the non-standardized data are mentioned in all previous studies) $\endgroup$ – virusdotcom May 4 '15 at 17:53
  • $\begingroup$ So, make a qqplot against a normal distribution with same mean and variance. There are functions for plotting a simulated envelope in MASS. $\endgroup$ – kjetil b halvorsen May 4 '15 at 17:55
  • $\begingroup$ Thanks, @kjetilbhalvorsen. So I would make a mistake by doing that (making a QQ plot of the sample quantiles of my data versus theoretical quantiles from a normal distribution with the same mean and variance as in case of my data)? $\endgroup$ – virusdotcom May 4 '15 at 18:06
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    $\begingroup$ Possible duplicate of Interpreting QQplot - Is there any rule of thumb to decide for non-normality? $\endgroup$ – kjetil b halvorsen Apr 21 '17 at 18:32
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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)

two qqplots side by side

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