Why do we divide?

Question

This may seem like a rather simple concept, but I'm trying to get an intuitive understanding of why we divide. Common formulas in statistics that use division are the calculation of the t statistic and variance. A less well known calculation in evolutionary biology that uses division is strength of female preference for males:

$$\frac{\text{Detectability} \times \text{Honesty} × \text{Benefit from male}}{\text{Costs of exerting choice}}$$

So could someone provide a general, high-level explanation of why we use division?

Answer 1

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.

Answer 2

"Division" is really very broad because there's all sorts of things we divide by other things, for different reasons, but they just about all have, in some way, something to do with 'scaling' the numerator to adjust for something.

As such, my answer won't be 'high level'.

But let's start with one of the most basic examples, the average. We try to find a 'typical' value, and the average is the sum of observations scaled by the count - just adding them would mean we ended up with a number that got bigger the more terms we added - not much use. But as the number of observations in the average grows, it tends to converge on the population mean (under some basic conditions we won't explore) - the scaled quantity is useful, the unscaled one much less so.

The variance itself is basically a kind of average, but an average squared distance from the mean (it's also related to the average squared distance between pairs of observations).

In the case of the $t$, the variability in the numerator depends on the variability in the original data. But by dividing the numerator by its standard deviation, we can compare t-statistics with a standard table, rather than having to recompute a new one for every data set! That is, the unscaled numerator is not as useful (at least not without a lot more effort) as the standardized version. Indeed, we can pretty much judge significance of a t-ratio 'by eye', whereas doing it for a raw t-numerator would not really be possible.