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?

• Surely they explain their notation! Have you looked closely through the text to find out what they say? (Although there are some conventions about notation, they are not universal, so it is best to consult your original source--the text--rather than hoping that reasoned speculation will resolve your question.)
– whuber
Jul 14 '15 at 22:46
• Trust me I am studying this book very seriously...I cannot find any explanation in there for this...my only guess is that X-bar is the sample mean of a random variable and S is the sample sd of a random variable.
– kits
Jul 14 '15 at 22:55
• kits -- my first guess would be that the $\overline{X}$ represents the random variable defined as $(X_1+X_2+...+X_n)/n$ and $S$ would be the corresponding random variable $\sqrt{\sum_i (X_i-\overline{X})^2/(n-1)}$, but as whuber says, I expect that this is explained somewhere in the book. These quantities are distinct from the observed sample values $\bar{x}$ and $s$ which can be thought of as samples from the distribution of the (upper-case) random variables. Jul 15 '15 at 1:36
• Also $\mu^2$ should probably be $\sigma^2$. Apr 8 '17 at 13:19
• @Michael Chernick. yes. Sir. Apr 8 '17 at 14:43

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

• +1 Thank you, Matthew for sharing your knowledge, always so didactically. You know... I find the distinction between a random variable as a function, and the actualization into concrete values after an experiment a bit blurry when a well-defined experiment (tossing the coin) is expressed, as in this case, with subindices to signify multiple individual tosses. This is convenient and probably necessary (e.g. time series), but it is as though one was re-defining the function. Apr 8 '17 at 19:13
• @AntoniParellada I'm not sure I fully follow? How does indexing the random variables seem to redefine them? Apr 8 '17 at 19:29
• Or rather, it may seem as though each index refers to an individual outcome. Don't sweat it... If it doesn't resonate, it's just me as a hobbyist. Apr 8 '17 at 19:32
• @AntoniParellada Ahhh got it. Yeah, I'm using each $X_1$, $X_2$, $X_3$ here as different random variables. They could be $X$, $Y$, $Z$. I agree it can get a bit confusing sometimes what indices refer to. Eg. you can also use an $n$ dimensional vector $\mathbf{x}$ to define a function from $n$ possible outcomes to the space of real numbers. You'd use $x_i$ to refer to the $i$th row of vector $\mathbf{x}$. And $\mathbf{x}$ is basically a random variable: it's a function from $\{1, 2, \ldots, n\}$ to $\mathbb{R}$. Apr 8 '17 at 19:42