Three Conventions

Roman and Greek Letters

To clearly distinguish estimates and parameters we (sometimes) use

Roman letters for estimates

like

$\bar{x}$	for the sample mean
$s_X$	for the sample standard deviation

and

Greek letters for parameters

like

$\mu_X$	for the population mean
$\sigma_X$	for the population standard deviation

Note:

When there is only one variable $X$ under discussion, we often drop the subscripts, writing $s$, $\mu$, and $\sigma$ rather than $s_X$, $\mu_X$, $\sigma_X$.

Hats

There is also an entirely different convention for the same thing. To clearly distinguish estimates and parameters we (at other times) use

letters decorated with ``hats'' for estimates

like

$\hat{p}$	for the sample proportion
$\hat{\theta}$	for a generic estimate (sample characteristic)

and

undecorated letters for parameters

like

$p$	for the population proportion
$\theta$	for a generic parameter (population characteristic)

Capital and Small Letters

A fairly subtle convention, not nearly as important as the two preceeding (so if you have to skip something in this review, skip this), distinguishes

capital letters, like $X$, $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}$, and $S_X$, for random variables

and

small letters, like $x$, $\bar{x}$, and $s_X$, for observed values of those random variables.

The subtile point is that after a random variable is observed, it is no longer random. When I calculate that the sample mean of my data is 2.716, then I write

$$\bar{x} = 2.716$$

because 2.716 is not random. It's just a plain ordinary number. In contrast, the equation

$$X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}= 2.716$$

may be either true or false depending on what the value of $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}$ turns out to be when observed. In fact, for a continuous probability model

$$\mathop{\rm pr}\nolimits (X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}= 2.716)$$

is zero because continuous probability models give positive probability only to intervals not single numbers (p. 236 in Wild and Seber).

It's often not clear whether you should use $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}$ or $\bar{x}$. Sometimes it depends on context which has not been made clear enough to decide.

There are, however, two places where capital letters are required:

in probabilities and expectations like $\mathop{\rm pr}\nolimits (X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}> 10)$ and $E(X{\mkern -13.5 mu}\overline{\phantom{\text{X}}})$, and

in subscripts indicating random variables like $s_X$ and $\sigma_X$.