This morning as I walked to the studio I was doing what geeks do best: pondering a slightly esoteric mathematical quandary.
Ingraining the American spirit of optimism at a young age, and under dubious circumstances, our schools always taught rounding numbers in a peculiar way. You always round your decimal values to the nearest integer. That part makes sense. But what if the decimal is .5 — exactly half? In my education, at least until late in high school (or was it college?), we were always taught to round up! The glass is half full. Optimism.
Eventually — far later than it should have been, I think — the concept was introduced that always rounding .5 up is not really that accurate, statistically speaking. It might be nice in the case of a single number to be an optimist and think a solid half is good as a whole, but in aggregate this thinking introduces a problem.
If you have a whole lot of numbers, and you’re always rounding your halves up, eventually your totals are going to be grossly inaccurate.
Of course, the same would happen if you were ever the pessimist and always rounded down.
The solution, I later learned, was to round halves up or down, depending upon the integer value that precedes them. Which way you go doesn’t really matter, as long as you’re consistent, but as it happens, I learned it as such: if the integer is odd, round up; if it is even, round down.
In my work, I write a lot of PHP code. Most of it is of the extremely practical variety; I’m building websites for clients, after all. But every once in a while I like to indulge my coding abilities in a bit of frivolous experimentation, and so today I produced a little PHP script that generates 10,000 random numbers between 1 and 100, with one decimal place, and then it shows the actual sum and average of those numbers, along with what you get as the sum and average if you go through all 10,000 numbers and round them to whole integers by the various methods described above. Try it for yourself!
Any time the rounded average is different from the “precise” (and I use that term somewhat loosely) average, it is displayed in red. Interestingly, and not at all surprisingly, when you always round halves in one direction or the other, at least one of those directions will (almost) always yield an incorrect average. Yet if you use the “even or odd” methods, both of those methods will almost always yield a correct average.
It’s all about the aggregate.