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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, $\text{mean}()$, $\text{avg}()$...?

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    $\begingroup$ 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". $\endgroup$ – Stéphane Laurent Dec 20 '15 at 12:41
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    $\begingroup$ 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). $\endgroup$ – Stéphane Laurent Dec 20 '15 at 12:48
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    $\begingroup$ @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. $\endgroup$ – whuber Dec 20 '15 at 14:16
  • $\begingroup$ $a_{\hat{\mu}_1}$ where $a$ is a field, and $\hat{\mu}_1$ arithmetic average? Just an idea. $\endgroup$ – Herpes Free Engineer May 29 '18 at 20:34
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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.

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  • $\begingroup$ 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. $\endgroup$ – Dougal Feb 21 '16 at 17:56

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