# Alternative notation for the arithmetic mean?

Is there any alternative to the convention of using an overline for denoting the arithmetic mean?

In my opinion overlines are confusing. For instance, is $\overline{\mathbf{a}}$ a vector or a scalar? The convention is to write vectors in bold face, but on the other hand the mean of a vector is a scalar.

Moreover, it's inconsistent with the function(argument) notation, which seems to apply to every other statistical function but the mean. So, how does one write the mean in this style, $\text{mean}()$, $\text{avg}()$...?

• I don't see the confusion. $\mathbf{a}$ is a vector in the notation "$\overline{\mathbf{a}}$", so it is bold. The notation "$\overline{\mathbf{a}}$" is not a bold symbol, it is "bar of something bold". – Stéphane Laurent Dec 20 '15 at 12:41
• Personally I don't like bold characters because you can't use it when you write with a pen. I prefer $\bar y_\bullet$ for the mean of a vector $(y_i)$. When there are two indexes, such as $(y_{ij})$, this allows to write $\bar y_{\bullet j}$ and $\bar y_{\bullet\bullet}$ (and in fact the bar is not necessary with this notation). – Stéphane Laurent Dec 20 '15 at 12:48
• @StéphaneLaurent Because people have been writing far longer than they have been typesetting, there are plenty of ways to indicate bold symbols. A standard way, I believe, is to write a tilde beneath the character. – whuber Dec 20 '15 at 14:16
• $a_{\hat{\mu}_1}$ where $a$ is a field, and $\hat{\mu}_1$ arithmetic average? Just an idea. – Herpes Free Engineer May 29 '18 at 20:34

## 1 Answer

For some historical material, see here

Angle brackets are sometimes used, but not much in mainstream statistics in my experience. Thus we might have $\langle x \rangle$. I associate this notation with physics.

I've seen on occasion the text $\text{ave}(x)$, for example in writings by J.W. Tukey when mixing words and more conventional algebraic notation. (To me $\text{avg}()$ is an ugly abbreviation.)

Peter Whittle has used $A()$ for an averaging operator in various editions of his text Probability (various changes of name and publisher since the first edition in 1970, Harmondsworth: Penguin). There is some similarity with the much longer established and much more widely used $E()$ for expectation. A key difference is that the latter would be not used in practice to refer to empirical calculations, whereas $A()$ could be.

A deeper discussion of this point might refer to whether we have here a function, a functional or operator. I surmise that all senses could be valid, but not necessarily in the same cases.

• I've also occasionally seen $\hat{\mathbb E}$, where the interpretation is that it's an estimator of the expectation, the most common of which is the arithmetic mean. – Dougal Feb 21 '16 at 17:56