Central Limit Theorem only needs sample size, N?

Question

I think explaining the central limit theorem needs two elements: the sample size and the number of samples drawn.

But nobody seems to talk about the number of samples drawn when they are making some infererence $\mu$ using the central limit theorem and only mention the sample size, $N$ and its distribution, which means they only use one sample group to infer population $\mu$.

I thought, however, there should be lots of samples each of at least 30 elements, and accordingly, lots of sample "means", and their distribution, not just the distribution of one sample group.

Please kindly help me to correctly understand the Central Limit Theorem and inferring the population mean, $\mu$.

Answer 1

1. A single random variable has a distribution; a sample mean from a random sample is a single random variable. Of course you can only observe its distribution by looking at multiple random samples (such as multiple sample means); then as the number of such samples increases the sample (empirical) cdf will approach the population distribution function. The standard error of the sample cdf about the population cdf decreases as the square root of sample size (quadruple the sample size and you halve the standard error).

   In short, the number of samples you take (each of size $n$) has no impact on how close the distribution of sample means is to being normal ... only on how accurately you can see it when you look at a collection of sample means all from samples of the same size.

   To see how close you are to normality at some sample size, you may need a substantial number of sample means. In simulation experiments it is common to look at thousands of such samples so as to get a good sense of the distributional shape.

   

   The picture shows histograms of 20, 300 and 100000 sample means for samples of size n=30 from a skewed distribution. We have some sense of the broad shape in the first one, a somewhat clearer sense of it in the second one, but we get a pretty clear idea of the shape of this distribution of sample means in the third one, where we have a large number of realizations of the sample mean.

   In this case sample means don't have close to a normal distribution; n=30 would not be sufficient to treat these means as approximately normally distributed (at least not for typical purposes).

   If you want a good sense of how the tails of the distribution behave you may need considerably larger numbers of sample means.

   However, when you're dealing with real data, you generally only get a single sample. You have to base your inference (whether you rely on the CLT or not) on that one sample.

2. You may have been misled about what the central limit theorem says.

   The actual central limit theorem says nothing whatever about n=30 nor about any other finite sample size.

   It is instead a theorem about the behaviour of standardized means (or sums) in the limit as n goes to infinity.

3. While it's true that (under certain conditions) sample means will be approximately normally distributed (in a particular sense of approximate) if the sample size is large enough, what constitutes 'large enough' for some purpose depends on several factors. As we see in the plot above, skewness can (for example) have a substantial impact on the approach to normality (if the population is skewed, the distribution of sample means is also skewed but less so with increasing sample size).