Can I use a Z-score with skewed and non-normal data? I've been working with some process cycle time data and scaling using the standard z-score in order to compare between parts of the full cycle time. 
Should I use some other transformation since the data are heavily right-skewed/non-normal? ('outliers' can never take negative time and often take much longer than 'average')
Using the z-score still seems to "work" ...
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# R code    
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mydata <- rweibull(1000,1,1.5)
hist(mydata)
hist(scale(mydata))

 A: If X is highly skewed the Z statistic will not be normally distributed (or t if the standard deviation must be estimated. So the percentiles of Z will not be standard normal.  So in that sense it does not work.
A: The R code will work, but the z-score will be about as meaningful as the sentence "Grapes are phoning the fountain pen lightly."  It's a valid sentence, but doesn't convey anything meaningful.
Judging by your R code, it seems like you think your data is Weibull distributed.  In that case, I'd just use the Weibull statistic and not scale anything unless you absolutely have to.  Even though z-scores are taught in every intro statistics class, that doesn't mean you should use them all the time, and especially not if you don't have symmetric data.
A: If the population is not normally distributed. In that case, the distribution of bar(X) {sample mean} approaches a normal distribution as as per central limit theorem; for large sample size. Though theoretically we say we are using Student's-t but for higher values of n (sample size or degree of freedom), t distribution & Z distribution are nearly equal. 
A: YOUR DATA DOESN'T HAVE TO BE NORMAL FOR A Z-TEST. (TOWNEND,2002) HOWEVER, THE VARIANCES SHOULD BE APPROXIMATELY EQUAL. TO  CHECK THAT CARRY OUT AN F-TEST ON YOUR TWO DATASETS, AND IF YOUR VARIANCES ARE APPROXIMATELY EQUAL, THE Z TEST RESULT IS USEFUL. IF NOT, TRANSFORM THE DATA. 
