Is $\overline{x_1,x_2}$ the appropriate way to write the average of $x_1$ and $x_2$?

Question

Simple question of style: if I want to use an overbar to denote the arithmetic mean of two specific numbers, $x_1$ and $x_2$, do I have to enclose them in parentheses, or braces, or anything?

Or is it simply $\overline{x_1,x_2}$ ?

Thanks!

Answer 1

Given that you want a terse label on a figure, the better approach would be to use $(x_1 + x_2)/2$ to convey the arithmetic mean of two specified values $x_1$ and $x_2$, or slightly more concise:

$$ \frac{x_1 + x_2}{2} $$

Of course as an author you are free to redefine an overbar to abbreviate whatever meaning is convenient to your exposition. The trade-off is that you will need to provide that definition to readers because it is unconventional.

What is conventional is the use of an overbar on a random variable to mean its expected value, e.g. $\overline X$ means the expected value $E(X)$.

The concise notation $\overline X$ is attractive when only two outcomes $X = x_1$ and $X = x_2$ are possible and have equal probability. E.g. if we go to the trouble of defining random variable $X$ as a sample based on two observations $x_1,x_2$, then $\overline X$ would mean the arithmetic mean of $x_1$ and $x_2$, as you wished.