So this runs the risk of being tangential, but I generally view straight lines in graphs with acute suspicion. This is not the usual expectation: people expect things to keep changing the same way they have been, so they predict a straight line; we have lots of straight lines which come out of diligent aggregation of data like this.
My thinking shifted when I did an electromagnetic theory course for antennas, which contra the rest of engineering school was mostly about Maxwell’s Equations and how to derive them. We relied a lot on the linearity property for those equations, and I was ceaselessly impressed by the stupendous power this gave us.
An unreasonable amount of power. So much power did linearity yield that I look at this first as an explanation for things that we don’t do well. Linearity gives us electricity and computers, precision and control. The impression I got was that anything in a graph that looks like a line isn’t really a line, but is actually just an approximate sum of different curves.
So this is now my prior. Granted, this basically punts the question of straight lines on graphs to ‘why do different curves seem to sum to approximately straight lines so often’ so it doesn’t get me much. My working guess is something like ‘because we expect straight lines, any more growth than that probably gets left on the table.’
If the curves are constructed randomly and independently then in some cases a linear relationship would be implied by the central limit theorem.
Not sure if this is helpful or not—CLT assumptions may or may not be valid in the instances you’re thinking of. I think my brain just went “Sum of many different variables leading to a surprising regular pattern? That reminds me of CLT”.
So this runs the risk of being tangential, but I generally view straight lines in graphs with acute suspicion. This is not the usual expectation: people expect things to keep changing the same way they have been, so they predict a straight line; we have lots of straight lines which come out of diligent aggregation of data like this.
My thinking shifted when I did an electromagnetic theory course for antennas, which contra the rest of engineering school was mostly about Maxwell’s Equations and how to derive them. We relied a lot on the linearity property for those equations, and I was ceaselessly impressed by the stupendous power this gave us.
An unreasonable amount of power. So much power did linearity yield that I look at this first as an explanation for things that we don’t do well. Linearity gives us electricity and computers, precision and control. The impression I got was that anything in a graph that looks like a line isn’t really a line, but is actually just an approximate sum of different curves.
So this is now my prior. Granted, this basically punts the question of straight lines on graphs to ‘why do different curves seem to sum to approximately straight lines so often’ so it doesn’t get me much. My working guess is something like ‘because we expect straight lines, any more growth than that probably gets left on the table.’
If the curves are constructed randomly and independently then in some cases a linear relationship would be implied by the central limit theorem.
Not sure if this is helpful or not—CLT assumptions may or may not be valid in the instances you’re thinking of. I think my brain just went “Sum of many different variables leading to a surprising regular pattern? That reminds me of CLT”.