Mysteries of the Bell Curve Revealed

Behold the famous Bell Curve (aka 'the normal distribution'), loved by some, loathed by many, but indispensable to social sciences like psychology. The Bell Curve is what you get when you graph the distribution of things like people's height, weight, mood, IQ, extroversion, exam scores, affinity for chocolate, willingness to vote Labor, or indeed almost anything in which people vary. Again and again the same pattern appears: some people are on one extreme (e.g., extremely tall), some people are on the other extreme (e.g., extremely short), but most people are relatively average.

Why should this be? Why should such a disparate range of variables all distribute in this way? Is this some sort of divine signal? Or perhaps a government conspiracy? How bizarre that exactly the same shape should come up again and again!

Well, there's actually a good reason for the ubiquity of the Bell Curve. I only realised this a couple of years ago when I saw a nifty little exhibit at the Questacon National Science and Technology Center in Canberra. The exhibit was quite simple. Mounted on a wall, inside a glass case, were a series of pegs arranged vertically in a triangle. Visitors to Questacon were asked to drop a ball into a small shoot at the top of the case, just above the top most peg. The ball would hit the peg and bounce either left or right, then fall and hit another peg and bounce either left or right, and so on, ricocheting all the way to the bottom.

The pegs were carefully arranged so that at each level the ball had a 50/50 chance of falling either to the left or to the right. And so, each time a ball was dropped, it would take a different path through the pyramid of pegs. When the ball reached the bottom, it would fall into one of several slots lined up along the bottom; sensors in these slots, wired up to a computer, recorded the end point of each ball's journey.

The computer kept track of the outcomes. A running tally of the number of balls that had fallen into each slot was presented on a little screen in the form of a bar graph: the greater the tally for a given slot, the higher its bar.

What kind of shape do you think this graph showed after tens of thousands of ball drops? That's right, a bell curve!


Slots 1 and 9 had the smallest tallies, slots 2 and 8 had slightly more, 3 and 7 more again, 4 and 6 had even more, but slot 5 had he largest tally of all. And from this exhibit it's not dificult to see why.

For the ball to make it to slot 9 everything has to go right...literally! (On each peg the ball has to fall to the right). Thus, there's only one path to slot 9. And similarly, for the ball to reach slot 1 everything has to go left, so there's only one path to slot 1. But for slots closer to the center there are multiple routes that the ball can take. In fact, the closer a slot is to the center, the greater the number of possible routes, and the more frequently the ball reaches it. The consequence of this is a lovely bell curve.

So how does this relate to other variables?

Consider the case of an exam. How well a given student does on the exam depends on many different factors. For example:

- How much effort was put into studying
- How intelligent the student is
- How confident the student is on the day
- How much sleep the student got the night before
- Whether the student gets to the exam on time
- Whether the student is sitting next to someone who will distract them in the exam

(etc.)

For a student to get the highest possible score on the exam everything has to go right. That is, all of the factors that influence exam score have to go favorably: the student has studied hard, is intelligent, is confident, arrived on time, etc. And conversely, for a given student to get the worst possible score everything has to go wrong. For most students, however, some things will go right and some things will go wrong (in various combinations). Thus, most students will obtain a relatively average score.

So with exams, as with the pegs, we can see that there are more paths to being average than to being extreme, resulting in a bell curve. The same is true for other variables. For example, there are many different things that influence height (genes, nutrition, etc.) and so there are many more paths to being of average height than there are to being either extremely tall or short. Similarly, there are many different influences on people's love of chocolate (past experiences, advertising etc.) and so ratings of chocolate admiration should also distribute as a bell curve.

In other words, the bell curve is the distribution you get when there are multiple independent influences. And so the ubiquity of the Bell Curve in social science is not really that mysterious after all. The bell curve, my friends, is simply the signature of complexity. No wonder it pops up everywhere.

Homework

Here's a wee experiment you can do at home. Take 5 coins and toss them all together. Count up the number of heads and write it down. Repeat this 30 or more times, each time writing down the number of heads that come up.

When you're done, count up the number of times you got 5 out of 5, 4 out of 5, 3 out of 5, 2 out of 5, and 1 out of 5. Now draw a bar graph. What does the shape of the graph look like?

If you're brave, try doing this with 10, 20, or 30, coins at a time. If you're smart, just do it in Excel.

The Teenager Song

How teaching changes me

This year I've been handballed a small teaching gig at the university where i'm doing my PhD. It's a weekly 2nd year cognitive psychology tutorial/lab. I've taught 3 classes so far, and so far so good.

What I find interesting is how radically the teaching role transforms my personality for the duration of the 2 hour class. It seems to have two effects:

1) I turn into a temporary extravert. Whereas I'm usually a bit socially inhibited and/or anti-social in the company of strangers, as a teacher I'm Captain Social (tm), introducing myself to students before the class, spraying good-cheer on all in my path. And as I bounce around the class, flashing my pedagogical grin, I'm thinking "What's happening to me? Who is this guy?" I suspect that it's my competitive drive to do better in the end of semester student evaluations than the real academics that is dominating over my shyness/introversion/anti-socialness.

2) I start channeling Carl Sagan, with the slow talking and drammmmmmmm-atic emphasis of key words. What's that about?

I'm not sure what to make of this bizzare phenomenon. In its involuntariness, it's a transformation more akin to a werewolf's than a superhero's. But then unlike a werewolf, I don't bite...so perhaps more of a pedagogical werepuppy than anything else.

Steven Pinker on the psychology of language: veiled threats and requests

Attempting to spark some inspiration for a presentation I'm preparing, I spent the afternoon watching more of the TED Talks videos, and I came across yet another fantastic talk from Steven Pinker. This one is on his specialty topic - psycholinguistics. Very interesting, very accessible. (And at no point in the talk does he mistake anyone's gender). Take a look:

My return to console games: Assassin's Creed review

It's been a long time since I've played a console game. In the late 80s I played a lot of Super Mario Bros. 3. But since then, with the exception of brief dalliances with The Sims and World of Warcraft, I've largely abandoned gaming.

"One day," I told myself, "games will be just like the Holodeck from Star Trek...I'll save my money until then."

But impatient with the lack of progress in Holodeck technology, I recently gave in and decided to see what 21st century gaming has to offer. In a moment of madness I bought an Xbox 360 and a copy of the hit new game from Ubisoft -- Assassin's Creed. Having just 'clocked' the game this week, I'd like to give you my review (warning: spoilers may follow).

First, a quick summary. In Assassin's Creed, you, the player, assume the identity of Desmond Miles, an average American, who, as chance would have it, is descended from a long line of professional assassins, dating back to 1191 AD. Desmond has just been kidnapped by a secret group of power hungry scientists, who, via the magic of science, have created a machine that can extract from his DNA the memories of his assassin ancestors; apparently Desmond's DNA contains an ancient secret that will allow them to rule the world. However, to get to this secret Desmond has to relive a series of his ancestors assassinations , and this requires being strapped into a special virtual reality bed called an 'Animus'. So, via the Animus, we're transported to The Holy Land, circa 1191 AD, to live the life of an assassin: pickpocketing, interrogating, rescuing, and assassinating.

As far as the graphics go, Assassin's Creed is uber impressive. The four cities that you visit are intricately designed and beautifully rendered. But what makes the world of Assassin's Creed particularly imersive are the interactive crowds and climbable structures. The inclusion of thousands of virtual people in the streets of Jerusalem, Acre, Damascus and Masyaf is a real innovation. This landscape of citizens for you to help, hinder, cajole, and kill adds a stimulating new dimension to game play. The climbable structures are similarly innovative: you can climb almost any surface, allowing you to scale buildings and towers, and giving you access to the rooftops for quick access around the cities. Jumping from building to building is great fun.

The combat is less impressive. The sword fighting is enjoyable at first, but you soon realise that you can defeat most opponents with mindless button mashing. It's also strange how easy it is for you -- one assassin -- to beat a group of two dozen soliders time and time again. Despite having you surrounded, AI soldiers seem quite happy for you to pick them off one by one; it's all bit easy. Moreover, the missions you must complete to get a tad repetitive. But given the acclaim Assassin's Creed has received, am I to take it that this is standard for console games?

Where the game really disappointed me was the narrative. A personal pet peev of mine is when science-fiction finds it necessary to bombard you with techno-babble and pseudo-science to convince you of the plausibility of their fantastical premise. What happened to the idea of suspension of disbelief? I would have preferred it if Desmond could just have arrived in 1191 via a wardrobe or rabbit hole or something. I certainly could have done without the crappy lecture about the magic of genetics.

The narrative within the 1191 scenes didn't do much for me either. You're forced to sit through incredibly long and wordy cut scenes again and again, with no option to skip over them. The dialogue is clichéd, melodramatic, and doesn't further the story much anyway. In particular, the game seems very unsure of itself with respect to walking to religion tightrope. If you pay attention you'll eventually work out that the bad guys are atheists, who have discovered that although the miracles described in the bible are literally true, their origin is not divine, but extraterrestrial (umm, ok!). Your 1191 character also goes on a fair bit about "letting man believe what he wants to believe", a strangely liberal attitude for a crusader. But all this religious stuff is buried in the dialog so as not to offend; instead, it's just confusing.

In summary, although promising, I think that Assassin's Creed could have been a lot better than it was. Perhaps they should take another go at it. Hire better writers and spend some time adding variety to the missions. Oh, and get rid of the cheesy sci-fi crap.