As I understand, enough data ($\geq 30?$) allows us to (boldly) assume that the data is normally distributed ($\mu, \sigma^2$) by CLT. Thus, the MLE of $\mu$ is $\bar{X}$, the sample mean, which can be used as an estimator for the mean of the sample since it is unbiased.
I am seeking another perspective on how the sample mean is derived such that it is related to a probability distribution over the "errors".
Here is a paragraph of what I'm trying to understand (specifically, in bold):
The mean is the result of a probability model over the "errors". One expects the values scattering around a common center, and this centre is determined as the value for which the observed data has the hhighest likelihood. This likelihood has to be calculated from a probability distribution for the deviations of the observed values from this hypothetical center. Maxwell, Herschell and others derived an approprite probability distribution from the simple assumptions that the expectation about the errors is symmetric (positive and negative errors of the same absolute size are expected with the same probability), and that all errors are congeneric (they share the same probability distribution). The result of this derivation is the normal distribution. Calculating the max. Likelihood using the normal distribution leads to the mean as the "best guess" for this assumed center. This is quite a different interpretation than just being the "center of the data".
- What is this probability model over the "errors"?
- "Deviations of the observed values from this hypothetical center" means the model is $\mathbf{y} = \boldsymbol{\mu} + \boldsymbol{\epsilon}$, where $y_i, \mu_i, \epsilon_i$ are the $i$-th observation, hypothetical center and error, respectively?
- How is the interpretation of the resultant unbiased estimator of the mean different to "center of the data"? Is it because the weighted average over the sample points under a uniform distribution (equal weights) which means significantly larger values of the sample points would unnecessarily boost the sample mean which may not be the case?
Ok I googled the text you wanted to understand. The text talks about the difference and similarities between the mean and the median. This is important to understand this text, because the writer describes the difference between the two concepts. Honestly in my opinion the writer does this not in the best way possible
I'll go over you questions point by point.
the probability model over the errors is simply the probability distribution of the errors. The errors in this are the errors of the model described in your second question.
Yes that is the "model"
So this is the important part. what he means is that the mean is in many ways NOT the center of the data. The mean is for the normal distribution equal to the median and his whole thing about deriving the normal distribution is weird. The normal distribution is not unqiue in this and I think this is just confusing. The mean is not the "center of the data" because while in distributions that are not symmetrical the mean is equal to the median; the mean is related to the total size of the observations. the median is not. so for example if I have a dataset that looks like this :x=[1,2,3,4,5,6,1000] the "center of the data" should not be larger than six. However the mean is much larger than six because it takes in to account the size of the data. The median on the other hand is 4 because the median does not take into account the size of the observations. The point he is making is that the mean only lies visibly in the center if the data is nicely distributed. while the median always lies there. Furthermore the mean has it's own nice definition that has nothing to with being in the middle.
Lastly I want to make the point that CLT only says that if $z_n=\sqrt{n}(\bar{x}-\mu)$ then $z_n$ converges to the standard normal distribution. This point about medians and means has nothing to do with this since both of these concept are defined in a consistent way for all applicable distributions and the normal distribution isn't really special in this case.