# Central Limit Theorem only needs sample size, N?

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, $$\mu$$.

• Can someone explain what's unclear about the question? Commented Jun 12, 2019 at 14:38
• @Glen_b I don't understand how "number of sample size" and "number of drawing samples" are different.
– Sycorax
Commented Jun 12, 2019 at 23:05
• You're drawing multiple samples, each of size N (the "sample size"); the other quantity is how many such samples you draw ("number of samples"). I guess it could be clarified a bit with an edit. Commented Jun 12, 2019 at 23:55
• @Sycorax: I've cleaned up the phrasing a little, but besides the OP not having English as a first language (and some major, but not uncommon misconceptions) it seemed clear to me Commented Jun 13, 2019 at 0:40
• @Roy I've just noticed there's a related question here: stats.stackexchange.com/questions/133931/… Commented Jun 13, 2019 at 0:42

1. A single random variable has a distribution; a sample mean from a random sample is a single random variable. Of course if you don't know what the population distribution is you can only see what that mean-variable's distribution looks like 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 of its sampling distribution. 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 population 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.

Of course to simulate them I had to know the population distribution I was assuming; given that, I don't have to simulate; there are other ways to get at distributions of means, even exactly. That said, simulation is often pretty convenient and may be perfectly adequate for many purposes.

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 (which may be important when thinking about properties of tests, for example) you may need considerably larger numbers of sample means to see how the tail behaves.

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 (plus any assumptions you bring).

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. For example strong population skewness can have a substantial impact on the approach to normality (if the population is skewed, the distribution of sample means is also skewed but where the CLT applies, less so with increasing sample size); this was the cause of the non-normality in those plots of means above.

• Thanks for your great reply! I have quick question about it: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. Based on your plot, does it mean you drew 20, 300, 1000000 samples(and get the same number of sample means) and each sample of size is 30, and no matter how many samples you drew (or how many times you drew samples), it has no impact on the dist. of sample means being normality? Or I possibly understand your article in a opposite way...?
– Roy
Commented Jun 12, 2019 at 7:01
• Because I just simulated CLT by Python with uniform dist. with 300 samples(each of size is 10), and it looks quite norm, and so I am little bit confused.
– Roy
Commented Jun 12, 2019 at 7:02
• The shape of the distribution you draw from definitely matters; the uniform is a 'nice' case where n even smaller than 10 is pretty close to normal for most purposes (30 is too high a bar unless you're getting well into the tail). If you had done 1000 samples or 1 (each n=10), the distribution of means is the same, as long as you stick to the same population distribution. If you want to emulate my pictures, try a gamma distribution with shape 0.05 (the scale or rate parameter doesn't matter as long as you don't change it); equivalently you could try a chi-square with 0.1 d.f. Commented Jun 12, 2019 at 7:16
• Note that your sample means from a uniform are nice and normal-looking but are (demonstrably) not actually normal; they are lighter-tailed than the normal (indeed they have a finite range). This non-normality may not matter much, depending on what you're doing with them. Commented Jun 12, 2019 at 7:22
• Wow, yeah, gamma dist. clearly shows what you explained above: the number of sample means has no impact. I wrongly understand CLT, Thanks. And I also found out I thought "point estimation" is based on CLT, and couldn't understand why point estimation use just one sample collection to infer population params. Thanks for your help:)
– Roy
Commented Jun 12, 2019 at 8:06