I am pretty sure $20 American can no longer buy a new car, even up in Canada.

That being said there is this.

Thanks for that Dan! Also apparently it’s as Chris Hunter said, the wooden ratio!

Your readership may just have outwitted you. I guessed that this was about crowd sourcing the golden rectangle, so I deliberately picked one that wasn’t particularly close.

Purposefully atypical – I’ll add that to my list of good reasons why I lost the best. Admittedly I’ve got some bad reasons (excuses) too. But the ratio of good to bad is approaching the golden ratio, which I guess only goes to show that I am irrational!

Interesting results but I have a different hypothesis.

I predict the aspect ratio is influenced by the kind of people taking part.

I wonder how many of your sample
a) knew about the golden ratio
b) suspected this was what you were interested in

I’m guessing you would have a lot of maths teachers in your sample and fell into the b) category.

If they were like me, they could play all sorts of havoc with the experiment. I made a few rectangles…one square, one golden rectangle and one with an aspect ratio of about 20:1. That way I could smugly tell myself “Ha! I knew it!” whatever your findings!

Obviously this was for a bit of fun and not up for peer review but before Marc gives you a car, I would support the idea that certain groups of people will tend towards a particular aspect ratio. Asking the ‘right’ people could well have given an average ratio around the golden ratio.

For my undergrad thesis I compared the preference for rectangle ratios of a group of architects with a group of non-architects. I basically found that the longer students had been in architecture school, the more exaggerated the ratio they seem to prefer. As if they were trained to prefer things that stand out, that make you stop and look, that challenge convention. Which I guess architects are trained to do.

Non-architects on the other hand generally preferred the square. Balanced, recognisable, dependable.

Maybe if you took a whole load of school pupils, who hopefully don’t know anything about the golden ratio and who still appreciate conformity, Marc might get the results he predicted.

Just a thought. :)

I just did a lecture on sampling ;-)
And: would you say this is ‘a class of students’?
Still, interesting.
More educated people want to feel more special and non-conformist. Devlin has a lecture online debunking this myth.

PS. Love how the apps Dave and you make prove the point that educationalists and designers should work together. A bit like the Freudenthal Institute…

It would be really interesting to try this same task with a group of adults and/or students who do not have the presumptions of what you were trying to do here. You know, those that were trying to “outwit” you and the experiment. Perhaps people that are outside of your blog/Twitter audience. I’ll volunteer my classes.

Another confession: I deliberately drew an atypical rectangle on the far right side of the page, starting with the lower right corner.

Fascinating. I tell my statistics students that doing a survey project is usually a bad idea, as participants will sometimes try to sabotage your data. In this case, I removed the goofballs at the far end of the histogram (assuming their aspect ratio is exactly 21, and that there are 90 of them). The mean does in fact fall……to 5.14.

But I think there is something out statistically amiss here, but I am having trouble putting my finger on it. Keep in mind that the domain for this “aspect ratio” calculation is from 1 to infinity (or, realistically in this case, 22) . So, it seems unreasonable that we would get that to average out to 1.618, no matter what the task. BUT, then I am also surprised that few people provided an aspect ratio between 1 and 2, so clearly we are wired to define a rectangle as non-regular.

Yeah, I’ll be getting a lot of work done today……thanks a lot….

As a music teacher I’m interested in the relationships between studying music and spatial reasoning in math.. Were you able to determine if there was a pattern to how the rectangles were being drawn (ie bottom left corner, up and across..or top left corner across and down)?? I would predict that those two choices would be the most common..with musicians perhaps deferring to the first!?! Just wondering.. And no I’m not willing to put a car on the line for the answer!

I think the canvas size is the key, but taking the edge to edge values is misleading. Have a look at this and let me know what you think:

https://dl.dropbox.com/u/599673/DM_Rectangle_Icky_Edges.png

From the rectangle app, I thought it was about thinking outside the box. I let some of my more advanced elementary school students experiment with the app. They came up with several variations – the most interesting was the train tracks. To make train tracks (with no dots) you had to “throw” the dots off the border.

How about a quartile range of area, perimeter and diagonal. I am assuming the units are pixels, the area units would be pixels squared, etc.

, Area, Perimeter, Diagonal,
Min, 31, 58, 21.9317122,
Q1, 38590, 980, 377.0026106,
Median, 73455, 1262, 487.6474136,
Q2, 124110, 1590, 621.590291,
Max, 429935553, 82948, 29329.53409,

There’s an alternate way to analyze the data that brings the average closer to the golden mean.

If you look at the data, you’ll see that Dan rotated the rectangles as necessary so that the width/height ratio was always greater than 1 when calculating the aspect ratio. So a tall skinny rectangle was given the same aspect ratio as a short wide rectangle. This is certainly one way to talk about aspect ratios, but it’s not the only way.

Consider a distribution of two rectangles, one with width/height ratio of 20, and one with width/height ratio of 1/20. Dan’s method would count both of these as an aspect ratio of 20, so the mean aspect ratio would also be 20. That’s one way to think about the average aspect ratio. But you can also consider the “average” aspect ratio of these two rectangles to be 1, a square, since they deviate equally from square in different ways (vertically versus horizontally). You can reach this result by calculating the log of the width/height ratio, which gives 3 for one rectangle and -3 for the other, for an average of 0. Then exp(mean(width/height)) is 1, so the average is a square.

If you do this for the 1471 rectangles in sample Dan collected, you get

exp(mean(log(width/height))) = 1.75.

Not exactly the golden mean, but closer. (If you use the median, you get 1.88.)

Also, if you look at the histogram of log(width/height) for the data, you’ll see that it’s fairly symmetrical about the mean. So just as many people were making tall skinny rectangles as short fat ones.

46 squares (including those who have a aspect ratio of 1.0 rounded to 1 decimal place).

It would be interesting to see what the median aspect ratio would be like for a sample of junior school students.

I also don’t think that the aspect ratio of the screen would influence the shape of the rectangle.

Dan, here are my ratios. The average was 2.49, the median was 2.

[clarifying, those are Eddie’s classroom ratios, FWIW. I asked for them on Twitter. –dm]

Dan/Dave,

There’s something I don’t understand about the data.

If I throw out all the points that have negative values for x or y (per our twitter conversation), then I see that the range of x values is 0 to 1528, and the range of y values is 0 to 1731. I can’t reconcile this with the reported value for the aspect ratio of the canvas, which is x = 1.55 y. I would expect the data to reflect this, because I would expect at least one edge of one of the rectangles to be close to all four borders of the canvas. But that seems not to be the case, so ???

I was looking at this because I had an idea to investigate about how the shape of the canvas might affect the results.

BTW, this gives yet another way to calculate the “average” aspect ratio, which is to find the average positions of the four corners, then calculate an aspect ratio from that. If I do that, I get a mean of 2.4.

Dan, it’s your fault for doing this experiment.

But here’s one last question. Why doesn’t my method lead to a mean aspect ratio of 1? Presumably it doesn’t because more people drew horizontal rectangles than vertical rectangles. (Or the horizontal rectangles were more elongated on average than the vertical ones.) What is that telling us? Is it strictly due to human psychology, is it due to the aspect ratio of the canvas, or something else?

Dan: I don’t have any formal reasons to use the geometric mean, but it just feels more correct to use a scale-based mean for dealing with ratio data. If I was looking at coordinate data (upper-left corner, for instance) I would use the arithmetic mean.

“Plus: why? Honestly, what’s the motivation for doing something you wouldn’t ordinarily do just to mess with someone else’s program? Isn’t that like definitional trolling?”

It might have been slightly trolling, but whether you meant to troll or not, the problem is that once you’re conscious of what the experiment is probably after, it’s awfully hard to “just pick something” without deliberately choosing or avoiding the golden ratio. (Or, what you think is the golden ratio.)

But, right, totally this is enough to win the bet. :)

To test if rectangle aspect ratio correlates to browser window aspect ratio, we wouldn’t need to forcibly resize people’s windows.

We could just take advantage of the fact that different people are naturally using different-sized screens, and log their window dimensions along with their response.

I think there are some external factors at play here. You may get different results if you ask people to draw rectangles on paper on on a tablet. On a computer, it is much easier to move the mouse horizontally (which usually happens — not by everyone, of course — to involve moving the wrist) than it is to move it vertically, which (for some) requires extending the fingers inward and outward. In fact, with the ubiquity of widescreen monitors, users might actually slowly be getting trained to have faster horizontal motion than horizontal motion just from everyday computer use.

I don’t have any hard evidence for this, obviously, but I did want to point out the possible constraint imposed by the physical device. One question I might have is, how many people adjusted their rectangle after drawing it? What if you only look at the data from people who adjusted their rectangles?

If the shape of the canvas is a factor, wouldn’t it make more sense that the rectangles would converge to something like a *silver rectangle* (http://en.wikipedia.org/wiki/Silver_ratio#Paper_sizes_and_silver_rectangles)? I know North American standard paper sizes are different, but it might be something along the same lines.

Well in an effort to try to fit the answer to the question… I tried trimming some of your “crazies” out at the end, ratios above 20, but that didn’t change things much, the median is still above 2, and Q1 is about 1.64…. Now if you do a lot of trimming, and get it down to looking at ONLY rectangles with a ratio of 2 or under (about 45% of the rectangles) Then the median is 1.62 which is close to the golden ratio and the average of 1.56 is close to the aspect ratio of screen. But that is some hefty trimming.

Dan
Can you remind me (us?) where this sort of crowdsourcing accuracy theory comes from? I know that this is something I have read on in the past but my brain cannot find an accurate reference. I know that there is a theory that if you ask a large enough sample of people to estimate something, say the population of Fargo, ND then the average guess will be fairly accurate.

Dan

What I vaguely remember is a story about people at a carnival guessing at the weight of an elephant. Individual guesses were terrible but the aggregate guess was surprisingly accurate. Am I making up this memory? Can anyone out there verify or discredit this? As far as this experiment, I worry about the number of people who have openly admitted (claimed?) to have intentionally disrupted its intent.

All that being said, I am suspicious of the theory that a crowdsourced bunch of guesses should be reasonably accurate.

Thanks Michael!

I think the idea of crowdsourcing is a misconceived idea of the Central Limit Theorem developed by Laplace, which is used in statistics and indicates that if you take a large enough sample even if the sample is from data that is skewed, the sampling distribution of the means will be normally distributed and centered at the mean. But this requires the distribution to be a random sample of “factual” information. So this would be true if we were doing something like taking a random sample of the height of adults. But when you start to put their opinions in place, I don’t think you can get correct answers from “bad” information. I may not be making this completely clear, but here is a simple example that you can try with a class. Tell kids to randomly choose one number from a group of numbers that you are going to flash up on a projector, overhead, etc. Put the numbers 1 2 3 4 up on the board.
If they truly randomly picked there should be a fairly even distribution of the numbers so the average of all the numbers should be 2.5 BUT since people are not random number generators, they have a tendency to predominantly pick #3. From the book “Statistics Modeling the World” they indicate about 75% of people will pick 3, about 20% are split between 2 and 4 and only about 5% will pick 1. I have done this in class and have gotten similar results. I’m not sure this completely debunks crowdsourcing, but it does indicate that our minds can have a bias on what type of answers we arrive at.