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I'm self teaching myself statistics and got stuck with notations; I'm working with multiple sources and notions seems to differ and confuse me a little, so I decided to write them down first.

$ \mu = $ mean of the population
$ \sigma = $ std deviation of the population

$ \bar X = $ mean of the sample
$ S = $ std deviation of the sample

$ \mu _ \bar X = \mu $ mean of the sample mean
$ \sigma_ \bar X = \sigma / \sqrt n $ std deviation of sample mean, aka mean standard error

It will be a silly question, but are all of these are correct? Are there any alternative notations for these?

Thanks

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$ \mu = $ mean of the population
$ \sigma = $ std deviation of the population

$\rho_{XY}=$ population correlation coefficient (between $X$ and $Y$)

$\sigma_{XY}=$ population covariance between $X$ and $Y$

$ \bar x = $ mean of the sample
$ s = $ std deviation of the sample

$r_{XY}=$ sample correlation coefficient (between $X$ and $Y$).

$s_{XY}$= sample covariance between $X$ and $Y$

$\bar X$ is the (arithmetic) mean of a group of random variables (i.e. $\bar X=\frac{X_1+X_2+...+X_n}{n}$). $\bar X$, itself, is a random variable.

$S$ is the standard deviation of a group of random variables. $S:=\sqrt{\frac{1}{n-1} \sum\limits_{i=1}^{n}(X_i-\bar X)^2}$ $S$,itself, is a random variable.

So, to conclude, we use Greek letters to denote the population (standard deviation, mean, whatever) and Roman letters to denote the sample (standard deviation, mean, whatever). Capital letters, in statistics, usually denote random variables.

For a much-more-comprehensive list than this one, see: http://en.wikipedia.org/wiki/Notation_in_probability_and_statistics

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  • $\begingroup$ What is the difference between X and $ \bar X $ ? $\endgroup$
    – Kartopukus
    Commented May 27, 2014 at 15:22
  • $\begingroup$ $X$ is a random variable, following some distribution; $\bar X$ is also a random variable, following some distribution, but $\bar X$ is the mean of a load of random variables. i.e. $\bar X=\frac{X_1X_2+...+X_n}{n}$ where $X_1, ..., X_n$ are random variables. $\endgroup$
    – beep-boop
    Commented May 27, 2014 at 15:24
  • $\begingroup$ One last thing, is there any difference between $ \bar x $ and $ \bar X $ ? $\endgroup$
    – Kartopukus
    Commented May 27, 2014 at 15:27
  • $\begingroup$ Sure. $\bar x$ is the sample mean (or just the mean). It's just a number. e.g. for $1,2,3$, $\bar x=2$. $\bar X$ is a random variable; it's the mean, yes, but of random variables, not numbers. So, if we've got random variables $X,Y,Z$, then $\bar X=\frac{X+Y+Z}{3}$. $\endgroup$
    – beep-boop
    Commented May 27, 2014 at 15:31
  • $\begingroup$ does the same relationship applies s and S for std dev? $\endgroup$
    – Kartopukus
    Commented May 27, 2014 at 15:33

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