I have a continuous variable which is skewed (possibly because of a ceiling effect) in a very large sample. My colleague argues that due to the central limit theorem we can treat this as normally distributed. It is data from a population study with over 1000 subjects. However, if some of the variables indeed have ceiling effects, does the central limit theorem apply? The means would be normally distributed, but they would not be the real means for this cohort.
You can treat the sampling distribution of MEANS as being approximately normally distributed (Central Limit Theorem). This is of practical importance in hypothesis testing and fitting confidence intervals. The Central Limit Theorem does NOT apply to the measurements themselves. If your measurement variable is skewed, then it is skewed, no matter how many samples you have.