### Let's Be Rational Here

Saturday, 29 September 2012Years ago, when I was in graduate school, I got into an argument, about a real-world crime statistic, with another student who didn't have much math-sense. The mathematics itself is very simple, and yet at least one implication of it seems to run counter to the intuïtions of many people.

Let's say that a population `p` is divided into groups, each `i`-th group with population `p`_{i} `p` = ∑(`p`_{i}) And let's say that the `i`-th group has a propensity `c`_{i} to commit crimes, such that `c`_{i} · `p`_{i} gives the sum of the crimes committed (however measured) by members of that population.

If criminals from within each group draw their victims with each person having an equal chance of victimization regardless of his or her own group, then the proportionate share of victims that they draw from the `j`-th group will be `p`_{j} / `p` The total number of crimes then committed against the `j`-th group by members of the `i`-th group will then be (`p`_{j} / `p`) · (`c`_{i} · `p`_{i}) and the *ratio* of `i`-on-`j` crime to `j`-on-`i` crime will be [(`p`_{j} / `p`) · (`c`_{i} · `p`_{i})] / [(`p`_{i} / `p`) · (`c`_{j} · `p`_{j})] = `c`_{i} / `c`_{j} So, if `c`_{i} = `c`_{j}, then the ratio of `i`-on-`j` crime to `j`-on-`i` crime will simply be **1:1**.

The other graduate student had been *sure* that, if group `i` were the smaller group, then the ratio should be larger than **1:1**, because group `j` furnished more potential victims. The *proper* intuïtion here is that, if one group is larger than another, then it furnishes proportionally *both* more potential victims *and* more potential victimizers; or, to say the same thing differently, if one group is smaller than another, then it furnishes proportionally both fewer potential victims and fewer potential victimizers.

If we see a very different ratio, then the difference implies that one group has a greater propensity to criminality than the other, or that one group is seeking (or avoiding) the other in its acts of criminality, or both.

It should be noted that members of the `j`-th group may be sought or avoided for reasons other than their being members of that group *as such*. For example, members of the `j`-th group may happen to have more portable wealth. Still, if one sees a ratio of, say, about **50:1**, then it's hard to explain this lop-sided ratio in terms simply of the `j`-th group having more wealth, or of the `i`-th group simply having a greater propensity to criminality. With a ratio like that, one should expect that members of the `j`-th group are *indeed* being targetted for being in that group, by members of the `i`-th group.