Friday, February 20, 2009

Casual Fridays: How random are we?, by Dave Munger

This week's Casual Fridays study was inspired by this comment on the Random Number thread:

When a freshman at Penn State too many years ago to count, the intro psychology prof did an amazing demonstration. I wonder if anyone knows the answer to this which I have long forgotten.

He said he had written the numbers 1 through 5 in random order on a piece of paper. He then asked the very large class to read his mind and write down his number order.

When the class compiled the answers, more than 50% of the class had his order, and so proved that telepathy was possible!!!

The class was ecstatic, until he then told us that humans more often than not arrange those numbers in that particular sequence that he had.

Does anyone know what that sequence order is?

I have puzzled over this for years since.

I thought we might do Bobbysoxer a favor by uncovering the number sequence, if indeed it exists. I was skeptical: half the class? A sequence of five items, in random order? Since there are 120 possible unique sequences of the numbers 1 through 5, a random distribution of responses would mean fewer than one percent of answers would be in any given order. Chances that HALF the responses would be the same seem remote.

That's not to say that responses will be truly randomly distributed. But I doubted that any single combination of digits would even approach 50 percent.

Our survey asked respondents to randomly arrange the digits 1 through 5 -- and also 1 through 4, and 1 through 3. Perhaps with a smaller range of choices we'd get something approaching Bobbysoxer's memory of the event.

First let's take a look at the distribution of responses for the digits 1 through 5. How "random" were our answers?

The most popular response was "12345", selected by 5.3 percent of our 1,409 respondents. Yes, technically, sometimes a "random" arrangement of those digits would come out in that order, but I suspect this reflects the percentage of our readers who are cheeky and want to make a point about "randomness," not the amazing mystery number Bobbysoxer remembers from college. Second most popular was "54321" with 3 percent of the responses. The next three sequences were tied: "34251", "53421", and "52314", each with 1.8 percent of the total response. That's hardly the 50 percent response that would have been so impressive to a class of intro psych students.

I suspect either the professor did something else to lead the students to respond in the way they did, or Bobbysoxer's memory isn't entirely accurate. What's clear from our study is that there's no "particular sequence" that most people arrange a sequence of five digits in.

That said, there were still some interesting patterns in the data. Take a look at this:


This shows the percent of people choosing each possible first digit in the sequence. They were significantly more likely to choose 3 or 5 than the other digits.

What about four-digit sequences? Were there any trends there? Once again, nothing as dramatic as what Bobbysoxer recalled. The most common response was "4231", with 6.7 percent of responses. This isn't anywhere near 50 percent, and only 2.5 percent above the expected 4.2 percent in a random sample. Was there a pattern with first digits?


This time, 1 was the least common first digit, with just 18.9 percent of an expected 25 percent responses. Interestingly, if you subtract out the 6 percent of respondents who answered "1234", answers starting with 1 sink to a very low 12.9 percent of the total -- about half of what would be expected due to chance.

What about 3-digit sequences? Here we see a bit more of a pattern to the answers.


With just six possible combinations, "231" garnered 22.6 percent of responses, and answers starting with 2 accounted for 43 percent of answers. But still, even with only three digits to arrange, no one answer accounts for anything near the 50 percent Bobbysoxer recalls.

But as with four-digit numbers, there seems to be a definite reluctance to start with the number 1.

So is it possible that Bobbysoxer's teacher somehow led his students to pick a particular sequence? It might be, although I'm having a hard time coming up with exactly how it might have been done. Perhaps the class had to fill out a form with a course number or some other numeric item before responding to the "mind-reading" challenge.

I didn't try to lead the respondents in a particular direction with this study, but another part of our study may have revealed just how suggestible people can be. We also asked respondents to "pick a random word and type it below."

Out of 1,409 responses, you wouldn't expect many repeats, given the fact that there are over 70,000 words in frequent use in the English language (math/stats whizzes: can you compute the expected number of repeats?). Yet there were in fact 409 repeats, including several words that were produced ten or more times. The most common word was "random," repeated 40 times (and of course, that word appeared immediately above the answer box). Next most popular was "pickle", with 20 repetitions ("pick" was also in the instructions for the question). Other common choices were "banana" and "monkey" (we use Survey Monkey to host our surveys). Food items were chosen dozens of times (we had previously asked how often readers went to the grocery store).

And take a look at this graph showing how frequently words were chosen starting with each letter:


The red bars show our survey responses, while blue shows the average incidence of starting letters in the English language. As you can see, our respondents chose words starting with B, C, and P significantly more frequently than those words actually appear in English. The discrepancy may again be due to our prompt -- it starts with a "P" word, "pick," and ends with a "B" word, "below."

I didn't find any correlation between the responses to the other "random" questions and how random the number sequences were, but in case you're interested, here are some of those results:


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