The Central Limit Theorem states that the distribution of the sample means approaches normal regardless of the shape of the parent population.
Sample means (s) will be normally more distributed around (u) than the individual readings (Xs). As n – the sample size – increases, then the sample averages (Xs means) will approach a normal distribution with mean (u).
So, don’t worry if your samples are all over the place. The more sample sets you have, the sooner the averages of those sets will approach a normal distribution with a mean of (u).
The spread of the sample means is less (narrower) than the spread of the population you’re sampling from.
So, it does not matter how the original population is skewed. The means of the sample sets will approach a normal distribution.
The standard deviation of the smaple means equals the standard deviation of the population you’re sampling from divided by the sqrt of the sample size:
σ(xbar) = σ(x) / SQRT(n)