Central limit theorem, taking repeated measurements

Question

I'm using a book from R.J.Barlow, a stat book for science students. I was confused by some of the wordings in the book.The first two paragraph are some true statements.The bottom paragraph confused me most.

Suppose you took n measurements right? e.g. 1.0,2.0,3.0,4.0? If n =4 , 4 measurements and you took average of them to get a mean, How could this mean fluctuate you only have a fixed mean = 2.5 ? Is it saying that they repeat taking these 4 measurements? if YES by how many times? "That's where the confusion comes in.

Is the xi mentioned here the variable or a set of variables i.e.sample from a population? Is there anything to do with sample mean? sample variable? Population mean?

I got completely confused by some of the wordings. Thanks for any help!

Answer 1

Okay, so first the $<\bar{x}>$ is the expectation, i.e., $<\bar{x}>=E(\bar{x})$.

Why can the mean fluctuate? This happens if you repeat measurements. Suppose you have $N<\infty$ persons selected randomly from the population and ask them whether they like icecream (this is what you "measure"). For person $i$, $x_i=1$ if they like icecream, $x_i=0$ if not. Clearly, the measurements depend on who you ask. If, by chance, you take a sample of $N$ persons who don't really like icecream, then your average $$\bar{x}=1/N \sum_{i=1}^N x_i$$ will be small, when by chance you get persons who like icecream, then $\bar{x}$ will be larger. This is why the mean of your sample (measurements) fluctuates: you randomly draw measurements.

In your example, the mean would fluctuate, because the next 4 measurements will likely not have mean $2.5$. What the text is saying is that the mean is sometimes above the true mean in the population $\mu$, and sometimes above, depending on your $N$-sample. But on average your sample mean is equal to the population mean, i.e., $E(\bar{x})=\mu$. This is why we take measurements in the first place: it allows us to infer $\mu$ on average. In my example, we can learn how many people in the population like icecream, without asking all of them.