You don't need that in the statement. It works this way: Take independent random variables with same distribution of mean and variance (or a sample of size from the population, if you prefer), where is large (in practical cases, is often enough for the precision needed). Then the CLT tells that the empirical mean behavesalmostlike a normal random variable with mean and variance .

Perhaps the was introduced to consider random variables leading to empirical means (by taking them by ). This way you end up with independent random variables that are approximately normal.