However weird the linear interpolation of Charlotte, North Carolina’s population may be outside the range from 1970 to 1980, it seems to do nicely enough between those years. And that’s as we might expect, since we used the actual population data from the census days of 1970 and 1980 to form this interpolation. But we don’t have to make a linear interpolation. We could in principle use any function, but let’s try a simple one. This would be a quadratic polynomial, one where the variable x gets raised all the way to the second power, and one that brings back faint memories of the quadratic formula, which is one of the rare pieces of mathematics for which I have a work-related anecdote. Ask sometime if you’re interested.
[ I cannot and do not try to explain it, but yesterday was a busier-than-average day around these parts, with a surprising number of references coming from an Entertainment weekly article about the House series finale for some reason. In this context a “surprising” number is “any number other than zero” since I don’t know why anyone would go from there to here. I watched House, sometimes, sure, and liked it, but kind of drifted away when there was other stuff to do, you know? ]
That’s enough time spent establishing the heck out of the idea of a polynomial. Let’s actually put one in place. My goal back when was estimating what the population of Charlotte, North Carolina, was around 1975. I had some old Census data from 1970 and 1980 giving its population on the first of April, the earlier year, as 840,347; and the first of April, 1980, as 971,391.
[ Oh, wow. Yesterday’s entry had way fewer hits than average. I also put an equation out right up front where everyone could see it. I wonder if this might be a test of Stephen Hawking’s dictum about equations and sales. Or maybe I was just boring yesterday. I’d ask, but apparently, nobody found me interesting enough yesterday to know for comparison. ]
It shouldn’t be too hard to translate the the idea “I want to know the population of Charlotte at some particular time” into a polynomial. The polynomial ought to look something like y equals some pile of numbers times x’s raised to powers, and x somehow has to do with the particular time, and y has something to do with the population. And it’s not hard to do that translating, but I want to talk about some deeper issues. It’s probably better explaining them on the simple problem, where we know what we want things to mean, than it would be explaining them for a complicated problem.
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?
[ 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.