I have a couple of other thoughts about these piecewise constant functions which I’ve been using to make interpolations. The basic idea is simple enough; we pretend the population of Charlotte was a constant number, the 840,347 it happened to be on the 1970 Census Day, and then leapt upwards at some point to the 971,391 it would have on the 1980 Census Day. Maybe it leapt up immediately after the 1970 Census; maybe immediately before the 1980; maybe at the exact middle moment between the two; maybe some other day. Are those all the options we have?
[ We didn’t break 3,100 yet, and too bad that. But over the day I did get my first readers from Turkey and the second from the United Arab Emirates that I’ve noticed. Also while my many posts about trapezoids are drawing search engine results, “frazz sequins” comes up a lot. ]
I think I’ve managed, more or less, acceptance that a piecewise constant interpolation makes the simplest way to estimate the population of Charlotte, North Carolina, when all I had to work with was the population data from the 1970 and the 1980 censuses. In 1970 the city had 840,347 people; in 1980 it had 971,391, and therefore the easiest guess to the population in 1975 would be the 1970 value, of 840,347. We suppose that on the 1st of April, 1970 — that Census Day — the population was the lower value, and then sometime before the 1st of April, 1980, it leapt up at once by the 131,044-person difference. Only … how do I know the population jumped up sometime after 1975?
[ I’d like to thank all who’ve read me or passed on links to me for getting my total hit count above 3,000. In fact, as I write this, the total seems to be 3,033, which is a pleasantly 3-ish number. I suppose that it’s ungrateful to look for 4,000 right away, but after all, I do hope to be interesting or useful, and both of those seem to correlate pretty strongly with being read. In any case, I’ll see how long it takes to reach 3,100, and be silent about that if it’s a number of days too embarrassing to mention. ]
The task I’ve set myself is finding an approximation to the population of Charlotte, North Carolina, for the year 1975. The tools I have on hand are the data that I’m fairly sure I believe for Charlotte’s population in 1970 and in 1980. I have to accept one thing or I’ll be hopelessly disappointed ever after: I’m not going to get the right answer. I’m not going to do my job badly, at least not on purpose; it’s just that — barring a remarkable stroke of luck — I won’t get Charlotte’s actual 1975 population. That’s the nature of interpolations (and extrapolations). But there are degrees of wrongness. Guessing that Charlotte had no people in it in 1975, or twenty millions of people, would be obviously ridiculously wrong. Guessing that it had somewhere between 840,347 (its 1970 Census population) and 971,391 (its 1980 Census population) seems much more plausible. So let me make my first interpolation to Charlotte’s 1975 population.