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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.
But it’s not fair to completely ignore Charlotte just because I don’t know what its population density was around 1975. All I need is to find what its population was that year, and what its area was — I understand these municipalities outside New Jersey are able to grow, somehow consuming this strange “unincorporated territory” that’s been allowed to accumulate — and divide the one into the other. That division is likely the hardest part.
The trouble is I don’t know what Charlotte’s population was in 1975. I could find what it was in 1970, and in 1980, thanks to the decennial Census, but 1975 is as far from a census as it’s possible to get.
What I want to do is use the data I have available about the city’s population to make an estimate of what the population would likely be for this other date. This is the basic problem of interpolation, figuring out reasonable values for some quantity based on data which doesn’t directly address the question. These kinds of problems turn up when one has one quantity expressed as a function of another; in this case, it’s the population of Charlotte as a function of time. And I have data about what the value of the population is at particular times. Interpolation would be estimating what the population is at some time in-between the times for which I have data. (Similarly, extrapolation would be estimating what the population is for times after the latest available data — say, estimating what the population is likely to be for 2015, or 2020 — or for before the earliest available data set.) If you imagine plotting data points on a graph, the interpolation amounts to connecting the dots into some reasonable-looking curve. Extrapolation amounts to drawing the curve off to either side away from the data.
I talked about making estimates about the population in this interpolation. Any kind of interpolation or extrapolation requires making estimates; it’s just not possible to be perfectly correct. There’s more ways to connect any possible set of dots than anyone can imagine, and if all we have are a handful of data points we can’t rule any out.
According to CensusScope.org, which for all I know is right, in the 1970 Census Charlotte had a population of 840,347 people, and in 1980 it had a population of 971,391. So what might it have had in 1975?
Well, how do I know that a hundred million people didn’t move in to Charlotte on the 1st of January, 1975, and all but just shy of a million of them move back out right after Christmas? I know that from common sense reasons; such a population migration would have produced many side effects, including hastily built and now abandoned houses (and other buildings), stories of the year Everybody Moved To Charlotte, scholarly texts about this strange flocking to and away from North Carolina, and a great number of people hopelessly trapped when they set an elbow down on the counter at Waffle House and couldn’t lift it again. Similarly, I can be confident that the whole population of Charlotte didn’t move out the 1st of the year and move back in just after Christmas, because the side effects of that would be remarkable too. Plus my parents would probably have mentioned it.
However, that doesn’t show up in my data, which — at the moment — consist of just the information that “in 1970, Charlotte had 840,347 people” and “in 1980, Charlotte had 971,391 people”. Based just on that information, I can’t logically exclude any possibility for 1975.
But I do. I have expectations about what plausible values for the 1975 population might be. I’d be very resistant to it being below the 1970 population or above the 1980 population, for example. I’d also be resistant to an interpolation which gave me a very different population for 1974 or 1976 than for 1975.
Much of what’s involved in making interpolations and extrapolations, then, is looking to what kinds of interpolated values are believable, and whether they can be tested. So I’ll look next into the simplest possible interpolation.