# Notation for sample mean: $\overline{Y}_{\cdot{j}}=\frac{\sum\limits_{i=1}^N Y_{ij}}{N}$

I sometimes see the following notation, e.g., for the mean of the jth variable over N observations: $\overline{Y}_{\cdot{j}}=\frac{\sum\limits_{i=1}^N Y_{ij}}{N}$
My question is: What is this little $\cdot$ in the subscript of $\overline{Y}$ called? Interpunct? Middle dot? Centered dot? And when is it used? Is it just a matter of taste or are there any conventions?

• Don't know if it has a particular name, but this dot usually signifies that a certain dimension of the data has been "summed out". If you sum $Y_{ij}$ over $i$, then you change $\overline{Y}_{ij}$ to $\overline{Y}_{.j}$ to show this because this measure does not change within $i$ anymore.
– Andy
Commented Jun 25, 2013 at 10:52
• I have only heard it referred to as "dot notation" Commented Jun 25, 2013 at 11:22
• Okay, I listened to Andre Silva and posted my half-way answer. If someone knows the name of this dot (if it has a particular name - I never actually imagined it had one) please feel free to add it. Thanks!
– Andy
Commented Jun 25, 2013 at 11:51
• If every minutely different use of notation needs a different name, we really are in trouble... More seriously, I have seen $+$ used with the same role, although perhaps more often for totals than for the associated means. $+$ has two very simple advantages: it's less likely to be overlooked than a very small dot and it does carry the implication of addition (summation). Commented Jun 25, 2013 at 11:55
• I would call it a dot subscript. Commented Jun 25, 2013 at 12:16

Whenever you have data which is grouped in some way, you can calculate the group means in the way you wrote it. For instance, let $i$ denote individuals and $j$ is years, then the expression $$\overline{Y}_{.j} = \frac{\sum_{i=1}^{I} Y_{ij}}{I}$$ gives you the mean of all individuals by year. This number will not change within $i$ but it changes for each year $j$. To show that in this expression $i$ has been summed out, you usually write $\overline Y_{.j}$
Vice versa, if you sum out the time component, you get $$\overline{Y}_{i.} = \frac{\sum_{j=1}^{J} Y_{ij}}{J}$$ which gives you the individual specific mean over time. This value changes between individuals but not anymore over time.
Finally, if you sum out both $i$ and $j$ $$\overline{Y}_{..} = \frac{\sum_{i=1}^{I}\sum_{j=1}^{J} Y_{ij}}{IJ}$$ you get the overall mean, which is sometimes referred to as the mean of means, or the grand mean.
• Good, but $N$ does not have constant meaning. The divisors should be $I$, $J$, $IJ$. Commented Jun 25, 2013 at 12:46