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 built up a linear interpolation for Charlotte’s population between 1970 and 1980. In principle, I could extend this beyond those years, and project what the population was before Census Day 1970 or after the census of 1980. Generally, we’d call that — the values the interpolating polynomial takes on, outside the range of data we started with — an extrapolation rather than an interpolation, but they’re pretty closely tied together. If we understand one we’re doing pretty well understanding the other.
[ 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.
Just because there are in principle uncountably many possible equations for any line doesn’t mean we ever actually see any of them. Actually, we just about always pick one of a handful of representations. They’re just the convenient ones. I’m going to say there’s four patterns that actually get used, because I can only think of three that turn up, as long as we’re sticking to Cartesian coordinate systems and aren’t doing something weird like parametric descriptions, and I want to leave some hedge room for when I realize I overlooked the obvious. The first one — that I want to talk about, anyway, and just about the first one anyone encounters — is called the slope-intercept form, and it’s probably what someone means if they do talk about “the” equation for a line.
[ According to the WordPress statistics, trapezoids are just the hook bringing people into here. I didn’t realize there was such a big community of people who need trapezoid information. If I did I’d have played up my search engine terms more. ]
If anyone had doubts about using polynomials as a generally good thing I hope either the doubts or the doubters are quieted now. My next couple goals are simple ones: I want to set up polynomials to interpolate what the population of Charlotte, North Carolina, was around 1975. That is, I’ll be creating at least one equation of the form where somehow the right choices of numbers for , et cetera will mean if I put the right number in for x I’ll get out of it an estimate of the population. I’ve got symbols. I need to figure what I want them to mean.
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.
[ I’m grateful to all for the help in reading my pages here. I’ve not quite reached 3,000 hits, but it’s within sight. If you do know of people who might be interested in either what I’m doing now — and it should be clearer after today’s post — or articles I’ve written in the past, please let them know, or let me know if I could be doing better at reaching interested audiences. ]
I left off the list of places I’d lived the city of Charlotte, North Carolina. There’s justice in my doing so. We lived there only for a couple years, when I was extremely young. I have only a few memories of the place, most of them based on the popcorn machine they had in my preschool program. I don’t know what else I got out of that, but I certainly appreciated seeing popcorn pop. Also I had two brothers born then. But, mostly, I can’t say that Charlotte made much of an impression on me. I couldn’t identify any major features of it from memory, and challenged to point to it on a map I might point at Delaware instead, or wander off to find a soda. Plus, I last lived there somewhere around 1975. I can accept that the population of South Amboy, New Jersey, may not have changed very much since the mid-1970s, but not that Charlotte’s hasn’t.