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How does the standard error work?

What is the estimate of sample mean variance and how do you get it? Are my understandings of the following correct?

$var(Y)=\frac{1}{N} \sum (X_i-\mu_Y)^2$ population variance, how much the individuals vary from the mean.

$var(\hat Y)=\frac{1}{n-1}\sum(x_i-\bar y)^2$ sample variance, how much the individuals of the sample vary from the sample mean.

$var(\bar y)=\frac{var(Y)}{n}$ sample mean variance, how much the sample mean can vary from the true mean.

$var(\hat{\bar y})=$ estimate of sample mean variance

$Y={X_1...X_N}$ is a population with N individuals

$y={x_1...x_n}$ is a sample of the population with n observations

$\mu$ is the mean

$\bar y=\sum x_i/n$ is the mean of the sample.

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  • In order for this question to be answerable, could you please tell us what the definitions are and the relationships among $Y$, the $x_i$, $N$, $n$, $\mu$, and $\bar{y}$? – whuber Oct 25 '12 at 10:03
  • Not really. Clarify for your self what is $Y$. See for example: http://en.wikipedia.org/wiki/Variance – djhurio Oct 25 '12 at 07:44
  • @djhurio $Y={X_1...X_N}$ is a population with N individuals, $y={x_1...x_n}$ is a sample of the population with n observations. $\mu$ is the mean, I should had used a subscript to separate them. $\bar y=\sum x_i/n$ is the mean of the sample. – Saber CN Oct 25 '12 at 14:43
  • @whuber I added the definitions to the post, please take a look at them. – Saber CN Oct 25 '12 at 14:46
  • Thanks. You're almost there. What is $\sigma^2$? What is your formula for "$var(\hat{\bar y})$"? What is the distinction between $\mu_y$ and $\bar{y}$, if any? BTW, it gets awfully tough reading formulas that mix x's, y's, and $\mu$'s willy-nilly when referring to comparable things: the more mathematical symbols there are in a question, the more important it is that their lexical appearance reflects underlying relationships. Otherwise readers (aka would-be answerers) have to invest too much time just understanding what is being asked and wondering whether there are any typos :-). – whuber Oct 25 '12 at 15:20
  • @whuber $\sigma^2$ should be $var(Y)$ from my understanding, $\mu_y$ should be the same as $\bar y$. I don't know the formula for $var(\bar y)$, it's also what I am trying to ask in this post. I know this is quite messed up, took them from my notes and didn't organize them. – Saber CN Oct 25 '12 at 15:35
  • Because you have now (implicitly) supplied a formula for $\sigma^2$ and your question equates the variance of $\bar{y}$ with $1/n$ times this, then doesn't your post already answer the question posed in your latest comment? Or are you asking for how this formula is determined? If so, then please remove the extraneous (and irrelevant) three-quarters of the definitions so that readers can focus on what you really want to know. – whuber Oct 25 '12 at 15:37
  • @whuber I see, you are saying $var(\hat {\bar y})=var(\hat Y)/n$. Few questions, will I need 1/(n-1) to make it unbiased? what is the definition of $var(\hat {\bar y}$ in words? If $var(\bar y)$ is the how much the sample mean vary from the true mean, then $var(\hat {\bar y}$ seems like how much the sample mean vary from the 'sample mean' because we are using $var(\hat Y)$? – Saber CN Oct 25 '12 at 15:47
  • You will be interested in questions concerning [tag:standard-error]. Prominent among them are [Difference between SE and SD](http://stats.stackexchange.com/questions/32318), [Intuitive explanation for dividing by n-1](http://stats.stackexchange.com/questions/3931), and [How does the SE work?](http://stats.stackexchange.com/questions/33547). At least one of the answers in the latter answers your question, so I will close yours as an exact duplicate. – whuber Oct 25 '12 at 15:53

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