Disclaimer: I did this experiment for fun, and I wrote this paper for fun. It is not to be taken too seriously, and by no means scientific. This contains my own immediate interpretations. I might disagree with myself completely in a few years, and I wouldn't mind, as I had fun doing this and I think I've learnt from it. If you don't agree with my theories, that's fine; you're free to use any data from the spreadsheet yourself.

Pick a number...

The idea

It all started during a Monday's maths lesson.
Well, it didn't all start there, but I'm going to start telling from that point anyway.
None of us seemed to be paying much attention to the teacher, I'm not sure what was going on in class, but we were talking about TI-BASIC programs.
Now, I don't have much programs on my TI as I don't want to download programs from the internet and I lost all of my own when the batteries ran out some time ago. I do have a simple little program called RANDOM, though, which I created for a giveaway.
It's quite simple, really:
:Input "FROM ",A
:Input "TO ",B
So I ran it and input -10 for A and 10 for B. After asking the people around me for guesses, I ran the program. Nobody had guessed the outcome correctly.

The next day, while I didn't have any lessons going on and was studying maths on my own, I started wondering about the guessing of yesterday. If asking a big group of people for their estimate, would a pattern arise? Of course they would need to be asked independently, as in the case of Monday's game, people probably didn't want to choose a number somebody else chose already as they wanted to win exclusively.

Besides the little game of sorts I did during mathematics, there was another Monday experience fuelling this idea. On the EmpireMinecraft forums a friend of mine held a survey for school, using Google Forms. The survey itself was entirely unrelated to my idea, but I think having done that survey the evening before made me consider creating a Google Form myself, to see if a pattern would arise.

After considering for a few moments that Tuesday (ah, I was studying maths, wasn't I? Ah, that can wait), I decided I would try to realise an experiment of some sorts using Google Forms. Besides -10 to 10, I would include some other intervals too. I even spent a bit of time (not wasting study time anymore, I was using the lavatory) thinking about different intervals.
That Tuesday, however, I didn't have much time left when I finally came home after hours of studying (I had actually been kind of productive, I'll have you know), so I postponed the experiment to the following day.

The intervals

Wednesday evening, I went about creating the Google Form. I had never used the service before, but it is intuitive enough, so I didn't lose much time trying to figure out the possibilities.
I started with the interval [1, 10]. It seemed like a logical choice for me as the first thing most people think of when needing to choose someone randomly is thinking of a number within that interval themselves and having people take turns guessing. At this point, I actually spent some time considering how to word my question. I could've said "Pick a number in the interval [1, 10], but I figured not everybody would understand that right away. I Googled "Pick a number from 1 to 10", but noticed 'Google' suggested otherwise:

'between' was apparently the more natural choice.
However, from my point of view, "Pick a number between 1 and 10" suggests the interval being <1, 10>, excluding the numbers 1 and 10 themselves, as opposed to [1, 10].
So I decided to try and add a bit of clarification at the start of the survey.
"Please pick a number within the interval given. (inclusive, so between 0 and 2 includes 0, 1 and 2)"
However, later on we'll notice not everyone seems to have picked up on that.

Anyway, on to the next interval: I was of course going to include [-10, 10], as that was the original interval I used Monday.
Then I also included [11, 101], which I thought of Tuesday. Why? Hm, it simply seemed more interesting to me than [1, 100], this excluding the numbers one through 10. I think I included 101 because it seemed nicer to me. Intuition, you know?

Another idea I had was having an interval that included more negative than positive numbers. My hypothesis being that even though there were more negative numbers, people would be more likely to choose the positive ones. However, I didn't quite remember my original intention while creating the form and went with the interval [-43, 51]. Oh well, that works too, I suppose.

My last interval chosen was [0, 1]. I was actually quite curious as to the results of this one, as - I thought - the only possible answers would be 0 and 1, and I was really wondering which one would be more popular.

One option I'm very glad about Google Forms offering it: the ability to shuffle the question order. I wanted to be able to exclude bias introduced by question order.
On Facebook, the site I first shared this on, the preview showed only the first three questions, in a static order, but I don't think many people noticed that.

Gathering responses

After opening the form for responses I shared it on Facebook, with my Facebook friends. I was hoping to get a decent response from people there, as they were on Facebook anyway so willing to spare a bit of time, and they knew me, and thus would likely want to help me out.

Shortly after, I created a thread on the EmpireMinecraft forums linking the form, with a short introduction similar to the one on Facebook, but in English.
I shared it in a few smaller places too, but I think at first most responses came from Facebook, and a smaller but surely substantial portion from EmpireMinecraft.

I had hoped for about 20 responses at least, so I could make some graphs to look at that didn't have most beams at 0.
And the results relieved me, as when I went offline for the night half an hour later I had already got 12 people that filled it out.

When I woke up Thursday morning, I obviously wanted to check the spreadsheet that contained the responses, as I expected to have gathered some more over night.
I was quite surprised to find 69 responses. Not because of the number 69, hush…
But I was hoping for 20, and had already exceeded that number by quite a way!

I kept bumping (bump = Bring Up My Post) my thread on EmpireMinecraft every now and then and after every bump I'd get a new stream of responses. The stream got less dense every time I did it, though, as many of the people who got to see it had already submitted a response before.
I also sent the form to some friends who hadn't seen it on Facebook yet, and at this point the point of 100 responses was in sight.
At night, I did hit that milestone.

Throughout the next day, I got less than 10 responses.
Until, in the evening, I e-mailed my mother the form (which she had filled out before herself) with a little introduction, which she could sent to family.
And that did some stuff.
Apparently one of my aunts posted about my experiment on Facebook, which somebody with a lot of followers shared.
I haven't been able to confirm this, however.
But one thing's for certain: it worked.
I got over two dozen new responses, and they were all quite welcome.

Eventually, having kept bumping my thread every now and then, I now have 164 responses. That is a lot more than I had expected, honestly.
At the time of writing, I have not closed the form yet, I will do so when I publish this document, but I'm not sharing it anywhere anymore so I doubt I'll get much new data.

Validity of results

Shortly after opening the form, when the first results started rolling in, I noticed there was something I hadn't thought through very well: there was nothing that specified the need to use whole numbers.

Interestingly, for the [0, 1] interval, many people chose decimal numbers. Probably because they took the "between" literally, not having read and/or understood the clarifier at the top.
This wasn't a problem, though.
What was a bit of a problem is that some people were using the Dutch format for numbers, as you can see above, using a comma instead of a period for the decimal separator. Google doesn't recognise "0,25" as a number, so I had to correct all of these cases by hand.

And there were more of these little things I needed to change manually, to make Google understand that it was in fact a number.

Hi, lottie1664!

However, there were also some cases where I could not make a number out of the response given.

Sorry, but π is not counted as a number. It's not a rational number, anyway.

And writing it using the Latin alphabet surely won't work.
Some cases seemed to be more deliberately invalid, though, with numbers that fell outside the interval:

Or no numbers at all.
This person managed to get invalid answers for all but one of the intervals:

But by far most of the answers were entirely valid, fortunately.

Processing of data

Friday afternoon, having hit 100 responses already, I tried to make it so that the data would get processed into frequency tables and charts.
I needed some help from the internet, as I don't have too much experience with Excel/Spreadsheets, but in the end, it worked wonderfully.
This is the table for the [1, 10] interval, for example:

And when new responses enter, it gets updated automatically. If you want to see the functions I used, you can check out the Spreadsheet, a link is included below.

I also made all of these into charts. I had to fiddle around a bit before I had them configured like I wanted, but it worked in the end. I've got to say, Google Spreadsheets is a good product.


Now, onto the actual results! … Or did you scroll all the way down here without reading the rest? Ah, I suppose you have the choice to read whatever you want.
At this point I will in fact flip the switch and stop the form being open for responses, so the charts won't change while I describe them.

We saw the frequency table already above, but here's the chart for the [1, 10] interval:

Striking right away is the high frequency of 7s. And this could definitely have been expected, as some people did.
7 is used a lot in the Bible, which might be where much of its popularity comes from. For many people, 7 has become a sort of 'lucky number', and that's apparent in this chart.
6, on the other hand, was one of the least answered numbers, and much lower than any in its vicinity. That is probably because it is associated as 'one less than 7', and thus 'incomplete', of sorts. 666 is associated as the 'Number of the Beast', which also originates from the Bible, and might very well be an explanation for it being unpopular here.
10 was only submitted 3 times for this interval. Within all intervals, it seems like the border numbers are unpopular, also because, as mentioned before, not everyone seems to have understood that these numbers do in fact count as valid within the interval.
1, however, is much more popular than 10, and even more popular than 2. Probably because it is the first natural number, and '1' is associated with winning and being the best.

On to the next interval.

7 is again quite popular, likely for the same reasons as mentioned above. It is however not the most popular option this time around.
0 beats it by quite a long shot. I think this is likely to be caused by it being the middle number. It's exactly between -10 and 10. I've got another theory, but I'll explain that later.
Something that is very apparent in this chart is that the positive numbers are generally favoured above the negative numbers. In total, 37 negative whole numbers were submitted, compared to 97 positive ones. That's quite a difference.

These next two intervals were quite big, and as such it might not be as easy to get information out of them.

In this chart it might not be entirely clear what beam belongs to which number, but luckily we've got the frequency table as well. The responses seem to be spaced out rather well generally, although it does seem like the lower and higher numbers might be more popular than the middle numbers.
99 and 100 are by far the most popular. 100 was to be expected, as it is a round number in the decimal system, and the highest available whole number when treating the interval as <11, 101>. 99 is also quite close to the end of the interval, and might also be appealing because it features twice the same digit.
Two other high-scoring numbers were 13 and 77. 77 isn't too hard to explain: it's got the 7, which is popular for reasons aforementioned, and it also features twice the same digit, like 99.
13 is often associated with bad luck, but it is two away from the end of the interval, like 99.
If I had a bit more data I might've been able to describe more patterns, but at this point some things are too likely to be caused by chance.

In this interval, 0 was again the most popular number. However, it isn't the middle number here.
But like I mentioned, I do have another theory with regards to 0's popularity. You could look at it this way: in this interval, there are three types of numbers. Positive, negative, and neutral numbers. 113 numbers submitted were positive, 39 numbers submitted were negative, and only 8 numbers suggested were neutral. So neutral numbers aren't so popular after all. However, there is only one neutral number, while there are 43 negative and 51 positive numbers. I think that's why 0 is seemingly so popular: it's one of a kind.
Two patterns noticed before are quite clear here too: negative numbers are less popular than positive ones: on average, every negative number was chosen 0.9 times, while every positive number, on average, was chosen 2.2 times. Quite the difference, once again.
Also, the numbers one away from the end of the interval, in this case -42 and 50, are relatively popular.

Onto the last chart:

0 and 1 are surprisingly close together. In fact, most of the times I checked the chart, they had an equal frequency. Now, there are 47 1s and 46 0s, but if I had kept it open 0 might've taken over 1 again.
When starting this form, I was really curious as to which would be more popular, as I had no idea. And as it turns out, me having no idea was a good hypothesis, as there doesn't seem to be a general preference.
Now, this is the only chart where the beam for non-integer valid numbers is higher than any of the other beams. This is, as mentioned before, caused by many people not understanding 0 and 1 being valid answers. As such, quite some people went for non-integer decimal numbers, while in all other intervals, they were quite rare.
This interval also features more invalid numbers than any of the others, as relatively many people chose numbers outside of the interval.


In the end, I think this experiment was quite interesting! I'm glad I went through with it, and didn't brush it off as just a weird idea to forget about.
Is this useful in any way? I'm not sure.
I mean, it might help you next time somebody asks you to guess the number they have in mind… unless they've also read this, of course.
I think this emphasises another point, though: humans are bad at randomness. Even when trying to be random, there are always certain tendencies at play.

November 6th, 2016 (1960)


PS: Thanks to everyone who filled out the form!