Formatting: $\overline{X_1}$ or $\overline{X}_1$?

Question

To represent the means of a two random variables $X_1, X_2$, should the bar also cover the subscript, i.e.

should it be $$\overline{X_1},\overline{X_2}$$ or $$\overline{X}_1,\overline{X}_2$$

and why?

The first option seems to be the logical choice but the second option looks neater.

---

Addendum: If we decided that the first option is the appropriate one, what would the second option then represent?

Answer 1

In this situation, the first option certainly seems more “logical,” i.e. you can interpret its meaning right away.

However, you can also justify the second by considering $X$ to be tuple of two random variables (with components $X_1$ and $X_2$) and the mean of the tuple to be the tuple $\overline{X}$ with components $$ \overline{X}_i = \overline{X_i} $$ for every relevant index $i$ (i.e. $i \in \{1, 2\}$). This is what I would expect the second notation to denote – so the second would really be the same as the first.

After all, considering their visual similarity, it would almost seem malicious to use them for different things.

Answer 2

The notation $\overline{X_i}$ is more correct if you indicate $X_i=Y$ as a random variable.

Often you can find the notation $\overline{X}_n$ which means the sample average of the rv $X$ based on a $n$ size simple random sample.

This is taken from Mood Graybill Boes, McGraw Hill

In other words,

$$\overline{X_n}$$

indicates the sample mean of the rv $X_n$ but

$$\overline{X}_n$$

indicates the sample mean of the rv $X$ calculated on a $n$ sized random sample from $X$

Answer 3

$$\overline{X_n}=\frac1n\sum_{i=1}^nX_n$$ is the average of the samples $X_n$.

$$\overline X_n=\frac1m\sum_{i=1}^mX_{m,n}$$ is the average of the $n^{\text{}th}$ distribution.

In practice, both notations are probably used interchangeably (though illogically).