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.