I was talking on Friday with two people who've been to Italy recently and saw the leaning tower of Pisa.  One of them was surprised at how short it was:  "about 30 feet tall", she said.  Then the other person, who'd also seen it, agreed that it was surprisingly short, and said it was "only about as tall as the Washington Monument".

"Wait," I said. "You just said it's 30 feet tall; and you just said it's as tall as the Washington monument, which has got to be at least 100 yards tall.  And you agreed with each other."

And they both shook their heads, and said, "No, the Washington Monument isn't anywhere near 100 yards tall."  We all live near Washington DC, and have seen the Washington Monument many times.

I said that the Washington Monument must be taller than that.  One of them said that it was just as tiring to walk to the top of the Leaning Tower as to walk to the top of the Washington Monument; which was odd, since the WM stairs have been closed since before he was born.

They finally agreed that both structures were about 30 yards tall.

The Leaning Tower is 183 feet tall, and the Washington Monument is 555 feet tall.


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In Douglas Hofstadter's article "On number numbness" (published in Scientific American and also in his book Metamagical Themas) he mentions that he did an estimation exercise with one of his (university) classes, asking them to estimate the height of the Empire State Building. The smallest answer: 50 feet. The largest: one mile. (I don't think he asked them for confidence intervals or anything of the kind.)

(A height of 50 feet means about 6 inches per floor.)

One mile is a believable data point for somebody who knows it's really tall and has no clue how tall buildings generally are and has no skills in estimating or reasoning to be able to come up with a good estimate, but I think "50 feet" as an estimate is not a legitimate data point. It is much more likely that somebody was being a smart ass, or wanted to screw up the experiment and make Hofstadter look silly, or had never heard of the Empire State Building (assuming Hofstadter didn't imply that it's a really tall building) and was just guessing for a random building (which it would still be a bad estimate for, but much less so).

The experiment took place in classroom in New York, if I recall.

Everything's up-to-date in Kansas City!

They've gone about as fur as they can go.

They went and built a skyscraper seven stories high,

About as high as a building oughta go!

-- from Oklahoma

with apologies to current residents of Kansas City

There may be some truth to this quote.

Buildings do become increasingly inefficient as they get taller. For one, more and more space and energy has to be devoted to elevators and other ways to move people vertically. (If you've ever played SimTower, you might have some understanding of this phenomenon.) Skyscrapers are a result of too many people and too little horizontal space; in places where land isn't as expensive as in Manhattan, it's usually better (and cheaper) to build outward rather than build upward.

Interesting anecdote, but that's expected for normal, untrained human beings. I'll bet they were "99% sure" as well.

How should one train, to become a trained human being?

The general idea is to get lots of practice and feedback. The practice improves the skill, and the feedback improves the calibration. Lichtenstein & Fischhoff is an early paper on the subject of calibration training, but searching for terms like calibration training and debiasing will turn up many more.

I think you take a few levels in Human Paragon. It doesn't matter how you train as long as you accumulate enough XP doing it.

People seem to be terrible at estimating vertical distances over human heights or a little bit larger.Which makes sense, since there's really no reason for us to have gotten good at it; I don't see how it would be adaptive.

1) is this a situation where signaling of agreement is more important than accuracy to the participants? It seems to be at first glance, as I can see no harm to them getting very wrong answers, but some harm to them failing to agree. I think a preferable outcome would have been agreement on an estimate, but also agreement that there's very low confidence in the estimate.

2) would you post a followup about your friends' reactions when you tell them how far off you all were? Reaction to being wrong seems important when discussing how to be less so in the future.

3) what will you do differently next time a similar situation arises? The obvious first try would be offering a wager.

Maybe. Maybe the best explanation is that a young man who is truly intelligent, as opposed to merely clever, avoids disagreeing with a young woman.

And in line with that, I don't plan to tell either of them how far off they were.

(I was pretty close; I thought it was 150 yards tall.)

I was pretty close; I thought it was 150 yards tall.

Did you remember to worry about being primed, even by the absurd 30 feet?

Between worrying about being primed, and trying to update your opinion in response to others', what are you supposed to do?

I'm curious about the specifics of this situation in trying to figure out the most likely explanation.

Did the two people go to Italy together or separately? Are they romantically involved? If not, is the woman conventionally attractive? And if so, did the exchange roughly fit the pattern of "woman expressing an opinion" and "man agreeing with woman not matter what"?

Separately; not romantically involved (the man is dating someone else); and now I don't remember who the woman was, so I can't answer the rest.

You don't remember who one of the two parties in a recent conversation was even though you do remember the details of the conversation well enough to quote it? That's pretty unusual.

I'm content-oriented.

Honestly, this happens to me far too often.

It has happened to me too. I was surprised not that it happened but that it seemed to happen just 2 days after the event occurred and at what was probably the first telling of the story. When it happens to me, it's generally much later than 2 days and after I've told the story at least a couple of times, so that by that point, the decontextualized story is all that remains of the original memories.

Me too. It gets especially embarrassing when you end up telling someone a story about a conversation they themselves were involved in.

People are just bad at guessing distances and sizes of things that aren't medium-sized physical objects at roughly eye level. Guess how big traffic lights are?


I've walked under traffic lights and looked up at them - from that angle, I'd guess each lens might be ... 0.2 m to 0.5 m diameter, 95% confidence? Which is larger than they feel (note scare-italics) when you're sitting in the car across the intersection.

Apparently they vary in size, but one source claims they can get to be seven feet tall.

I've seen them taken down for repairs, and they're much larger than I'd thought, but not 7 feet tall. Maybe 4' IIRC?

The lenses are either 8" or 12" in diameter in the US: http://mutcd.fhwa.dot.gov/htm/2003r1r2/part4/part4d.htm

So if you're looking at 3 8" lenses in a row vertically, the whole thing is at least 3' tall, probably more.

I just looked through hundreds of photos of traffic lights on google, and not one was of the standard yellow US kind that hangs from a wire.

I just looked through hundreds of photos of traffic lights on google, and not one was of the standard yellow US kind that hangs from a wire.

Inference: your neighborhood (or your belief about it) is not representative and yellow and/or hanging from a wire is not standard. I was surprised by the diversity of results on google. On the first page, there were no yellow-on-a-pole and just two cantilevered yellow ones. Bing put a cantilevered one first and had a couple on wires[dead] in the top 20. But they weren't yellow! They do seem to be in the US. Maybe traffic lights aren't yellow?

ETA: Bing's "similar images" produced a few yellow lights on wires, but it was still dominated by the familiar green and gray.

Ah, that's more authoritative than Wikipedia - thanks for the link!

7' traffic signals, if they in fact exist, are probably reserved for intersections where the lights are hung up unusually high or something. I'm skeptical that the kind you see all the time are really seven feet.

Some traffic signals have several (I've seen up to 6) lights for different directions arranged in a vertical column, I'd guess this is what your source was referring to. For some reason these are really hard to find on google images, but here's one.

I think our algorithm for "tall" is different than our algorithm for "long". Guessing:

We measure "long" by foreshortening and comparison to recognizable objects.

We measure "tall" by angle versus eye level, and the difference in arc gets small quickly.

Here in Canada, we measure "long" in units of time taken to traverse. Mean speed is never given.


How long are pencils in Canada?

My guess is that when you see something like the Washington Monument on a regular basis from a distance, it seems smaller than a tower you've only seen once but got to see close up and even climbed to the top of. Plus, the WM is probably perceived differently as it is a nondescript obelisk rather than a building with floors. Why they thought the leaning tower was so short is odd, but maybe the popularity of foreshortened photographs of people "supporting" it could contribute, especially if they had those photos taken of themselves? Another potential factor is the different width:height ratio between the two structures.

Unless you consider it essential to the story, how about a rewrite to use just one unit type?

I will be misquoting people then, and might remove a source of error. For instance, the woman might have meant 30 yards but said 30 feet.

I'm godawful at estimation. Particularly with regard to time. Perhaps because of my own shortcomings I have very low regard for intuition.