What is the difference between $\bar x$ and $\bar X$? I'm reading "Statistics: Principles and Methods" by Johnson and Bhattacharyya (7th edition). I know $\bar x$ is sample mean, $\mu$ is population mean, $s^2$ is sample variance, $\sigma^2$ is population variance, $s$ is sample standard deviation, and $\sigma$ is population standard deviation. However, in the later chapters the authors seem to use capital letters $\bar X$ and $S$ (to denote the sample mean  and  sample standard deviation). Why is this so? Is there a (fundamental) difference between them?
 A: My brief searching through the Statistics: Principles and Methods on books.google.com suggests that it is exactly as @Glen_b's comment describes (and most everyone is guessing): lowercase $\bar{x}$ is a number while $\bar{X}$ is a random variable.
Quick review of what's a random variable:


*

*10, 2.1, $\pi$, etc... are real numbers.

*A real valued random variable is a function from a sample space $\Omega$ to the space of real numbers $\mathbb{R}$.


Example of the sample mean as a random variable and as a number:
Let be $X_1, X_2, \ldots, X_5$ be random variables denoting the result of rolling a fair six sided die five times.
The sample mean of these five random variables is:
$$\bar{X} = \frac{X_1 + X_2 + X_3 + X_4 + X_5}{5} $$
$\bar{X}$ is also a random variable. $\bar{X} = 3.5$ is a possible outcome, $\bar{X} = 2$ is another possible outcome, etc...
Now imagine that we rolled a die 5 times and obtained the series of values $4, 6, 1, 5, 4$. The sample mean for these 5 values is given by:
$$\bar{x} = \frac{4 + 6 + 1 + 5 + 4}{5} = 4$$
The sample mean $\bar{x}$ of these 5 particular numbers is not a random variable. $\bar{x}$ is a single number.
An event occurred where $X_1 = 4, X_2 = 6, X_3 = 1, X_4 = 5, X_5 = 4$ and $\bar{X} = 4$.
General notation notes:
In the context of probability:


*

*Lowercase letters are often numbers.

*Uppercase letters are often random variables.


Of course there's a lot of different notation out there (upper case letters often denote vectors or matrices etc...) so neither of these bullet points are laws set in stone.
References:
Johnson, Richard A. and Gouri K. Bhattacharya, Statistics: Principles and Methods, 6th edition 
