We use division for several reasons, some purely mathematical and some more statistical. I am not clear that there is a deep explanation lurking anywhere for a simple fact of statistical life.

Here is a very elementary start:

1. Division as when producing a mean, i.e. mean = total / frequency, is a prototype of a division which scales for greater comparability. The mean heights of 10 males and 5 females give a simple example. The total height of 5 females and the total height of 10 males are not as directly informative as the means. That said, there are occasions when totals are of interest. The total number of bags carried by a party of 10 males and 5 females on a journey is the total that needs to be carried somehow.

2. Division as when producing a $t$ statistic is a prototype of a division which removes dependency of results on units of measurement. A $t$-statistic is typically a coefficient estimate divided by its own standard error: numerator and denominator have the same units of measurement, which cancel. Otherwise we would often be dealing with odd units. In an old example in Fisher's book, Statistical methods for research workers, he regresses wheat yield in bushels/acre against rainfall in inches/season; the regression coefficient thus has units of (bushels $\times$ season / acre $\times$ inches). If we use $t$ statistics instead, we have a scale that can be compared across regressions much more easily. It is also true that units of measurement are needed to interpret many results.

Pearson correlation is an example where both kinds of division are used, so that the result is not dependent on either sample size or whatever units of measurement were used, often a local or capricious choice.

Statistical people have often missed out on any teaching of dimensional analysis. A little dose is highly illuminating:

D. J. Finney. 1977. Dimensions of statistics. Applied Statistics 26: 285-289 is an excellent paper in this territory. http://www.jstor.org/stable/2346969

There are any number of books on the general principles of measurement, which touch on many issues here (and much, much more). Most I know seem to lapse into admiring their own mathematical sophistication, but others may have positive recommendations.

We use division for several reasons, some purely mathematical and some more statistical. I am not clear that there is a deep explanation lurking anywhere for a simple fact of statistical life.

Here is a very elementary start:

1. Division as when producing a mean, i.e. mean = total / frequency, is a prototype of a division which scales for greater comparability. The mean heights of 10 males and 5 females give a simple example. The total height of 5 females and the total height of 10 males are not as directly informative as the means. That said, there are occasions when totals are of interest. The total number of bags carried by a party of 10 males and 5 females on a journey is the total that needs to be carried somehow.

2. Division as when producing a $t$ statistic is a prototype of a division which removes dependency of results on units of measurement. A $t$-statistic is typically a coefficient estimate divided by its own standard error: numerator and denominator have the same units of measurement, which cancel. Otherwise we would often be dealing with odd units. In an old example in Fisher's book, Statistical methods for research workers, he regresses wheat yield in bushels/acre against rainfall in inches/season; the regression coefficient thus has units of (bushels $\times$ season) / (acre $\times$ inches). If we use $t$ statistics instead, we have a scale that can be compared across regressions much more easily. It is also true that units of measurement are needed to interpret many results.

Pearson correlation is an example where both kinds of division are used, so that the result is not dependent on either sample size or whatever units of measurement were used, often a local or capricious choice.

Statistical people have often missed out on any teaching of dimensional analysis. A little dose is highly illuminating:

D. J. Finney. 1977. Dimensions of statistics. Applied Statistics 26: 285-289 is an excellent paper in this territory. http://www.jstor.org/stable/2346969 (David Finney, 1917$-$2018)

There are any number of books on the general principles of measurement, which touch on many issues here (and much, much more). Most I know seem to lapse into admiring their own mathematical sophistication, but others may have positive recommendations.