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The CLT states that "if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population, then the distribution of the sample means will be approximately normally distributed."

I have two questions:

  1. Assuming I want n=30 observations if I make a laboratory experiment, does it make a difference whether I do the same experiment with n=5 6 times each, or whether I conduct one experiment with n=30? I guess this question concerns the relation between number of repitions and n.
  2. Does the CLT mean that variables observed or measured in the experiment should, when averaged, more or less be equal to the unknown population mean?
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This is the complicated thing about CLT. There are two "n"s here: there is the $n$ of your experiment: this must be large(ish). Then there is the $N$ of the replications of the experiment. In actual science you only have $N=1$, but you speak of hypothetical replications of $N=\infty$ to infer what the distribution of your estimates would be like. Using the variability of the data in your $n=30$ sample, the CLT allows you to connect the dots: the variability of the data is related to the variability of the estimate by a factor of $\sqrt{n}$ and tends toward a normal distribution.

Regarding your question of $n=5$, $N=6$ if you use the right method, you should get the same results. In some settings, you have experiments being replicated with some sources of error or deviation (for instance, time of day or temperature may vary and contribute to error), and so you lose some power accounting for the differences among the replicates (the number of which is denoted $N$) using hierarchical models.

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