Quadratic Stuff In North Carolina


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.

The form of a quadratic interpolation is y = a_0 + a_1\cdot x + a_2 \cdot x^2, where x is the number of years since the first of April, 1950, and where now a0, a1, a2 are coefficients, some fixed numbers whose value we may not know offhand, but which we can solve by some method. (My routine joke when getting to this point in class is to say we find the coefficients by our favorite method of Gaussian elimination, which never amuses my students. I’m not much amused by it either. It’s just something to say to pass the time.) There’s now three coefficients we have to find, though, and that means we’ll need three data points to find these three coefficients. Since 1975 was of particular interest it seems obvious to use the data from 1970 and 1980; but, for the third data point, should we use 1960 or 1990? Well, let’s try 1960 to start. This means that we want the coefficients that solve together the set of equations

702383 = a_0 + a_1 \cdot 10 + a_2 \cdot 10^2	\\  840347 = a_0 + a_1 \cdot 20 + a_2 \cdot 20^2	\\  971391 = a_0 + a_1 \cdot 30 + a_2 \cdot 30^2

Using my favorite method of Gaussian elimination — making the computer do it, in this case by Maple — finds that a0 is 557499, a1 is 14834.4, and a2 is -34.6. And this implies that the population of Charlotte on April 1, 1975 was:

557499 + 14834.4 \cdot 25 - 34.6 \cdot 25^2 = 906734

The linear interpolation projected the population there to be 905,869, so we’re looking at almost perfect agreement. That’s reassuring. It’s nice to get just about the same result from different approaches to the problem.

Although … why not work this out with 1990 instead of 1960? If we get about the same population for that case we might be more confident that we’ve got Charlotte’s population in 1975 nailed down, to something right around 906,000. Supposing that we want to match perfectly the data that in 1970 Charlotte had 840,347 people, in 1980 971,391, and in 1990 1,162,093, we have the system:

840347 = a_0 + a_1 \cdot 20 + a_2 \cdot 20^2	\\  971391 = a_0 + a_1 \cdot 30 + a_2 \cdot 30^2	\\  1162093 = a_0 + a_1 \cdot 40 + a_2 \cdot 40^2

which gives the solution

y = 757233 - 1810.1 \cdot x + 298.29 \cdot x^2

For April 1, 1975, this projects a population of 898,411.75; apparently, someone was sneaking out the back door. That isn’t too very different — it’s a difference of about one percent the size of Charlotte, whatever it was exactly — although since it’s below that nice round number of 900,000 it looks alarmingly out of step. Which one’s right?

And the short answer is, they both are. All three are, if we count the linear interpolation. We make different assumptions between the two quadratic interpolations, one about what the population must be in 1960, 1970, and 1980; the other about what it must be in 1970, 1980, and 1990. They should agree pretty well between 1970 and 1980, and be not too wildly divergent between 1960 and 1970 and between 1980 and 1990. Outside those ranges, anything might happen.

It does, too; the 1960-70-80 projection looks very nearly like a straight line, but if projected far enough out it says the population of Charlotte will reach a maximum number and then start to decline again; someday in the far distant future Charlotte empties out entirely. The 1970-80-90 projection grows without bounds as we project farther into the future, implying a day when the entire population of the universe is within the city bounds. And, more, if we look backwards in time we eventually find the population bottoming out and growing again, back in time, so that in some distant past the city was even more populated than it is today. This is why we don’t take simple projections that far away from data too seriously, but, it does show how different the projections are.

Since I have the data on hand, though, why not throw in a third projection? This one I’l base on the data from 1960, 1980, and 2000, for the trio of equations

702383 = a_0 + a_1 \cdot 10 + a_2 \cdot 10^2	\\  971391 = a_0 + a_1 \cdot 30 + a_2 \cdot 30^2	\\  1499293 = a_0 + a_1 \cdot 50 + a_2 \cdot 50^2

and this has the solution

y = 664963.5 + 505.8\cdot x + 323.615 \cdot x^2

That projects a 1975 population of 879,867.875. The person sneaking out gets a little farther out in this model. If we really felt like it we could come up with some more quadratic projections, each based on three data points out of the five given here — bonus points on the next quiz to whoever can say how many more there are — and without bothering to work them all out I’d say they’ll all be around 900,000. That we’ll get — how many? — different numbers from these projections is a good thing; just running our calculations gives us these hypnotic decimal points, and that can fool us into thinking we know the population more precisely than we actually do. It’s probably fairer to say the population was about 900,000, with a margin for error of thousands, maybe tens of thousands, one way or another.

I’ve included a picture showing the population of Charlotte on the census days, as the X marks. The red curve that looks like a line is the quadratic projection based on the 1960, 1970, and 1980 data; you’ll see that’s where the projection and the actual data match exactly. The blue curve is the quadratic projection based on the 1970, 1980, and 1990 data. The maroon curve is the one based on the 1960, 1980, and 2000 data. And if they demonstrate anything it’s that we should be pretty good at projecting the population of Charlotte near 1980, using these three curves.

The population of Charlotte, North Carolina, as a quadratic interpolation/extrapolation, based on three different sets of Census data.  The census data from 1960, 1970, 1980, 1990, and 2000 are included for comparison.
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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

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