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All resources I find online state that when you sample from the population, the means form a normal distribution. I also found out that the "mean of the sample means equals the population mean".

This is fair enough. What I do not understand is how this is applied in real life. In real life, don't we usually take just one sample? So doesn't that mean that the rule that the mean sample means equals the population mean not hold when we take one sample? And when we take one sample, won't that not form a normal distribution?

Or is the central limit theorem just a theoretical idea and is not applied in real life?

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  • $\begingroup$ CLT is arguably the most widely applied theorem in all of science. $\endgroup$
    – Bayequentist
    Commented Mar 23, 2020 at 10:00
  • $\begingroup$ Unfortunately, the word sample means something different to a statistician than it means in plain English. A statistical sample of a population means n-tuple realizations. In other words, a sample for a statistician could refer to many blood samples to a biologist. $\endgroup$
    – Carl
    Commented Mar 23, 2020 at 11:14

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You are mixing two different things. One is related to the central limit theorem, the other is related to unbiased estimators.

The sample mean is an unbiased estimator of the population mean, meaning that - on average - the sample mean will estimate the true population mean, that is it won't underestimate or overestimate it.

The central limit theorem states that if you sum some random variables (with some assumptions) their sum will be normally distributed - and since the mean is the sum of random variables divided by some constant - this also applies to the mean. So the sample mean is normally distributed (if you were to sample many times).

We can use this information to make inference about the sample mean - how it's distributed, based on the information gathered from our sample, which is quite useful in many scenarios.

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