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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

beep-boop
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Kartopukus
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1 Answers1

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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

beep-boop
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  • What is the difference between X and $ \bar X $ ? – Kartopukus May 27 '14 at 15:22
  • $X$ 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. – beep-boop May 27 '14 at 15:24
  • One last thing, is there any difference between $ \bar x $ and $ \bar X $ ? – Kartopukus May 27 '14 at 15:27
  • 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 $\bar X=\frac{X+Y+Z}{3}$. – beep-boop May 27 '14 at 15:31
  • does the same relationship applies s and S for std dev? – Kartopukus May 27 '14 at 15:33
  • Yep- exactly. $S$ is just the standard deviation of a load of random variables. You might see it written as $S=\sqrt{\frac{1}{n-1}[\sum\limits_{i=1}^{n}X_i^2-n \bar X^2]}$ – beep-boop May 27 '14 at 15:35
  • and what about $ \mu _ \bar X = \mu $ and $ \sigma_ \bar X = \sigma / \sqrt n $ , since you omitted them in the answer, are they incorrect notations? – Kartopukus May 27 '14 at 15:35
  • @Kartopukus Yeah- they're also correct. Some alternative notations include writing $\mu_{\bar X}=E[\bar X]$ and $\sigma_{\bar X}^2=Var[\bar X]$, but that's just a matter of personal preference on the writer's part. – beep-boop May 27 '14 at 15:38
  • Thank you for your answers, thanks to you I'm now able to understand the main idea clearly. – Kartopukus May 27 '14 at 15:42
  • Sure. I was pretty much in the same position as you when starting statistics, as the notation, more than anything, can throw people off. Once you get your head around the notation, it's a lot easier! If you've any more questions, just comment below. – beep-boop May 27 '14 at 15:43