Still making (good) progress with my knowledge on statistic. Sorry if I ask lots of questions about this recently but I really like math.stachexchange and really appreciate the quality of the answers I can find here.
My questions are pretty generic and simple (and I searched before asking).
- In the Central Limit Theorem definition you can find on wikipedia, it says that "the distribution of the difference between the sample average Sn and its limit µ, when multiplied by the factor √n (that is √n(Sn − µ)), approximates the normal distribution with mean 0 and variance σ2.".
It doesn't explain why we weight (Sn - µ) by √n? and I would like to know.
- I understand the fact that as the number of samples increases then, the difference between Sn and µ decreases, so the definition overall makes sense to me. Now it doesn't clearly say what σ2 is. I know what σ2 means as "it's the variance", what I want to know is what it refers to in this context. Does it designate the "variance" of the population (which you may or may not know?).
EDIT: yes σ2 here is the variance of the population.
- When you want to visualise the distribution of statistics, is the plot "variance" as a function of "mean" is the usual way of plotting this distribution?
EDIT: For this last question, while making progress in my knowledge of statistics. I was actually confused by an exercise in which I had computed man/variance for 1000 samples and plotted them. I was confused because in this particular example the distribution of points on the graph very much looks like a normal distribution while I believe it's actually purely accidentally in this case. But I got really confused because I was learning about sampling distribution at the same time, the CLT and normal distribution. It's all clear now.
Thank you.
The Law of Large numbers tells us that $\bar{X}_n\xrightarrow{a.s.}\mu$. See http://en.wikipedia.org/wiki/Law_of_large_numbers. When you weigh it by $\sqrt{n}$, then it tends to a standard normal variable variable and not a constant, (the $\sqrt{n}$ somehow increases the uncertainty of the limit intuitively). For a more elaborate explanation of CLTs look into this book: http://www.amazon.com/History-Central-Limit-Theorem-Probability/dp/0387878564.
As for your second question, I do not understand it properly, please state it clearly, what do mean by "distribution of statitsics".