# Why are all CLT problems using a single random sample when the CLT requires "repeatedly drawing random samples"?

In our course notes, it says: "The CLT states that if random samples of size $n$ are repeatedly drawn from any population with mean $\mu$ and variance $\sigma^2$, then when $n$ is large the distribution of the sample means will be approximately normal."

I bolded "repeatedly" because that's the part I'm confused about. In the examples and the homework problems, it seems that even when we have $n$ supposedly large enough ($>30$), we're still only talking about a single sample of size $n$.

For example, one of the problems states: "Suppose a population has mean $\mu=5$ and standard deviation $\sigma = 2$, then supposed a random sample of size 38 is selected. What is the probability that the sample mean is between $4$ and $6$?"

Then we go on to use the CLT to solve the problem, supposedly because $n > 30$ allows us to use the CLT. But from that previous paragraph in italic I quoted, it seems that the CLT only works when many many samples of size n are drawn from a population, even when $n > 30$.

Here we're only randomly selecting a single sample, yet it seems to be good enough to use the CLT. Why is that?

EDIT: When I say "all CLT problems" I mean all the ones I ran into, including the famous swan problem, in which we randomly select $n>30$ swans once.

• You ask about the meaning of probability (explained as a long-run phenomenon) in a context where just a single result (such as the sample mean) is available. Here is a perfect analogy. When we say a coin is "fair" we mean that in a sufficiently long run of repeated independent flips, the proportion of heads will be close to $1/2$. If I ask you about the chance that one flip of this fair coin will be heads, if you are rational you will answer "1/2". But that's only a statement about one flip, not a long run of flips! Why was your answer of "1/2" good enough?
– whuber
Jul 11 '16 at 18:19
• Possible duplicate of Why does the central limit theorem work with a single sample? Jul 11 '16 at 18:20
• @whuber, you have a knack for understanding the source of people's confusions :) and you're right, that's exactly the same thing; for some reason I couldn't see it. I think stuff is thrown at us so quickly (summer course) that I'm getting confused with all the terminology. Jul 11 '16 at 18:56
• @Greenparker, you're right, it's a very similar question. Thanks for the link. Jul 11 '16 at 18:58
• You're confused about the meaning of the word "sample." The trials $X_1,\ldots,X_n$ contitute one sample of size $n.$ $\qquad$ May 27 '19 at 17:41

The CLT is a statement about fluctuations of averages. Specifically, a single sample of size $n$ will give you an expression:
$$\mu_n:=\frac{X_1+\cdots+X_n}{n},$$
which gives you an estimate of the true mean (average). Here $\mu_n$ is a random variable, with some complicated distribution. The CLT states that as $n$ gets larger, $\mu_n$ will be closer and closer to the true mean $\mu$, with fluctuations that look like a normal distribution, centered on the mean with variance $\sigma^2/n$. Said another way, $\mu_n$ is approximately distributed like $N(\mu,\sigma^2/n)$. In other words, if you were to repeat your sampling a bunch of times, and plot a histogram of the $\mu_n$'s that you drew each time, you'd get something that looks like a normal distribution with the above parameters.
Thus it makes perfect sense to ask questions like: what is the probability that $\mu_n>2$? In this case we'd use the CLT to approximate the distribution of $\mu_n$ as a normal distribution. This means that you could numerically verify the CLT by repeatedly drawing samples of size $n$ and computing $\mu_n$, thereby plotting a histogram and calculating probabilities from it.
• @AlexR. I think that makes sense, thank you. Does it matter whether we know that the population is normally distributed if $n$ is large enough? In the "swan problem", we are told "the weight of an swan is normally distributed", then told we get a random sample of $30+$ swans, and we go on to use the CLT. but if our sample of size $n$ is already $>30$, does it even matter what the original population's distribution looks like? For ex if we weren't told the swan weights are N distributed, would it matter? Jul 11 '16 at 18:53