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I have two scales measuring the same construct, and I would like to know which scale is distributed more closely to a normal distribution. There are various possibilities for comparing a distribution with a normal distribution (e.g., kolmogorov-smirnov test, etc.), but is there any possibility to compare the deviance from normality between two scales directly?

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    $\begingroup$ At a fixed sample size, this would be a matter of comparing statistics that measure deviation from normality (whether Shapiro-Wilk, Anderson-Darling, or whatever you feel captures what you want to capture). It would be difficult to compare across different sample sizes without a much more careful identification of what you're trying to achieve, which might help you identify a suitable metric. Why is it important to compare how normal they are? $\endgroup$ – Glen_b -Reinstate Monica Jul 25 '14 at 2:04
  • $\begingroup$ The two scales were used in the same sample, so sample sizes are equal. I want to do this comparison because it was claimed that scale A yields a distribution which is closer to a normal distribution than scale B. Therefore I am looking for a possibility to compare the deviation from normality between the two scales directly, ideally with a significance test... $\endgroup$ – jkelgs Jul 25 '14 at 14:54
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The R package nortest contains 5 normality tests that you can conduct to test the normality of your samples. The question "which scale is distributed more closely to a normal distribution?" is wrongly formulated in my opinion, unless you can come up with a satisfactory distance in your context.

Please, find a simulated example below:

library(nortest)

set.seed(123)
data = rnorm(100)

ad.test(data)
cvm.test(data)
lillie.test(data)
pearson.test(data)
sf.test(data)
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