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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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marked as duplicate by whuber Oct 25 '12 at 15:53

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  • $\begingroup$ Not really. Clarify for your self what is $Y$. See for example: en.wikipedia.org/wiki/Variance $\endgroup$ – djhurio Oct 25 '12 at 7:44
  • $\begingroup$ 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 $\bar{y}$? $\endgroup$ – whuber Oct 25 '12 at 10:03
  • $\begingroup$ @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. $\endgroup$ – Saber CN Oct 25 '12 at 14:43
  • $\begingroup$ @whuber I added the definitions to the post, please take a look at them. $\endgroup$ – Saber CN Oct 25 '12 at 14:46
  • $\begingroup$ 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 :-). $\endgroup$ – whuber Oct 25 '12 at 15:20