The Unreasonable Effectiveness of Linear Regression

Everything is a straight line, give or take

The Pip turned 13 last week, so he’s officially a teenager now. He’s been acting like a teenager for about eight months (he’s the youngest in his grade, so most of his friends have been teens for months; one of his friends with a birthday this coming weekend is turning 14), so it’s really just making things official. To mark the occasion, we measured his height, and found he has also broken 5’7” (170cm); this is also not hugely surprising as people keep saying “Oh, my, he’s growing so fast!” and that sort of thing.

Teen Pip

The Facebook post served up from his birthday gave his length at birth (20”), which I used to make a joke on social media using those two points to predict that when he’s my age he’ll be 13’9” tall. We have more data than that, though— we record the kids’ heights on the doorframe between the kitchen and our library, so I snapped a photo of the last several entries, and to my surprise, they fit a line pretty nicely— even if I didn’t include the length at birth, it extrapolated backwards to almost the right value.

I’m a great big nerd so I found this intriguing, and went looking for more data. Which we had on the door frame, and also in a data file from several years back when we did a major home renovation that included replacing that door (so I copied down all the prior measurements). Here are 27 height measurements, going back to when he was approximately two, the earliest we could get him to stand next to the door and be measured.

As you can see, that fits remarkably well to a straight line— the goodness-of-fit parameter (R squared) is 0.968, where 1.00 is a perfect fit. If I dropped the length-at-birth point, it goes up to 0.995, and if I drop the first point from the old door frame, it’s 0.998. That’s good even by physics standards.

I find this fascinating because of the ubiquity of the “He’s really shooting up!” reaction. I even shared the sense that he’s been growing rapidly of late, though maybe not as strongly as many of the other adults who have remarked on it. It’s clear from the graph, though, that his growth rate really hasn’t changed at all— he’s just plugging along steadily adding something like 0.008” per day, for the better part of a decade.

We also have data for SteelyKid, who’s three years older, and I can add that to the same plot:

From that you can see that SteelyKid genuinely has tailed off a bit in terms of overall growth rate— in fact, they’ve changed little enough that they haven’t agreed to stand and be measured for over a year. It’s still an R-squared of 0.937, though (0.982 if I drop the length-at-birth point), which is crazy good for something that has no reason to be a straight line.

Of course, in order for this to be SCIENCE!, we need to be able to predict the future— to basically do a less silly version of my social-media joke about his height down the road. A more realistic question to ask based on these data would be “When will The Pip reach six feet in height?” (He has said for a while now that his goal is to be at least 6’1”, because “If you say you’re six feet even, everybody assumes you’re five-eleven and rounding up. If you say six foot one, they’ll still assume you’re rounding up, but then at least they believe you’re over six feet…” He’s a pretty astute Little Dude.)

For the best linear fit— excluding the length at birth and the first door-frame measurement, the prediction comes out to be a couple of weeks short of his 15th birthday. If we use the version including the low points, it’s about six months earlier.

So, you know, check back in 2026 to see just how well this unreasonably effective linear regression holds up…

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