Before reading this entire entry, please answer the poll question below.

Imagine you live in a totalitarian state and you and your partner are planning to have a child. The doctor, a geneticist, tells you that you must choose a wide or narrow intelligence range for your child. If it is a wide range, your child has a better chance of being a genius -- or being mentally handicapped. If it's a narrow range, your child is unlikely to be a genius, but also unlikely to be mentally handicapped.

Will you choose a wider intelligence range?

**37**(69.8%)

**16**(30.2%)

Early in 2005, Lawrence Summers, the former president of Harvard University, touched off a firestorm of controversy and was forced to resign when he suggested that women were not as intelligent as men.

Well, that's what the media told us.

Summers didn't say that. He didn't say anything close to that. What he said, however, was strange enough that people either by accident or design misrepresented what he said.

What Summers claimed was that his preliminary statistical analysis of IQs suggested that women have a narrower IQ *variance*. Summers was very careful to note that his analysis was fraught with possible error, but if true, could be used to explain the gender inequality in certain fields. Unfortunately, issues such as race and gender are such hot button topics that daring to suggest innate differences can easily lead to controversy. Few dispute that people of African descent are more likely to have sickle cell anemia. Ashkenazi Jews might have higher IQs than average (there is some controversy here, but I suspect that since it plays into stereotypes, there's less controversy than one might expect). Breast cancer is more likely to strike women and black, African-American men are more likely to get prostate cancer.

Since we know that many traits are closely related to race and gender, it's not terribly surprising that men and women have differences, but to suggest a difference in IQ is very, very dangerous to your careers. Summers knew this and was very careful to include plenty of caveats, but it wasn't enough.

So what did Summers really say?

Let's flip two coins. We'll think of them as two-sided dice with heads being equal to one and tails being equal to two. Assuming it's a fair coin, each side has a 50% chance of coming up and we can create a chart of possible flips as follows:

Coin 1 | Coin 2 | Total | Percent |
---|---|---|---|

1 (heads) | 1 (heads) | 2 | 25% |

1 (heads) | 2 (tails) | 3 | 25% |

2 (tails) | 1 (heads) | 3 | 25% |

2 (tails) | 1 (tails) | 4 | 25% |

As you can see, each of the four combinations has an equal chance of occuring, but the total of 3 has a 50% chance of occuring since there are two ways that it can occur. Now this is pretty simple to see from the above list, but think about a classic role playing game such as D&D where for each statistic of your character, you roll 3 six-sided dice (3d6). The lowest base stat you can get is 3 (rolling 3 ones) and the highest base stat you can get is 18 (rolling 3 sixes). As it turns out, each of those numbers has less than a .5% chance of coming up, but the mostly likely results are 10 and 11 (the mean is 10.5, but you can't roll that). In fact, those numbers occur, on average, 54 out of 216 times, or 25% of the time. In fact, here's a simple table of distributions:

Number | Occurred | Percent |
---|---|---|

3 | 1 | 0.46 |

4 | 3 | 1.39 |

5 | 6 | 2.78 |

6 | 10 | 4.63 |

7 | 15 | 6.94 |

8 | 21 | 9.72 |

9 | 25 | 11.57 |

10 | 27 | 12.50 |

11 | 27 | 12.50 |

12 | 25 | 11.57 |

13 | 21 | 9.72 |

14 | 15 | 6.94 |

15 | 10 | 4.63 |

16 | 6 | 2.78 |

17 | 3 | 1.39 |

18 | 1 | 0.46 |

So as you can see, you've likely to have an average stat and therefore an average character. If you're only allowed to roll once, a smart gamer will examine his or her stats and choose a profession (character class) which matches the stats. However, in some games, stats are decided by two ten sided dice (2d10). Let's look at those distributions:

Number | Occurred | Percent |
---|---|---|

2 | 1 | 1.00 |

3 | 2 | 2.00 |

4 | 3 | 3.00 |

5 | 4 | 4.00 |

6 | 5 | 5.00 |

7 | 6 | 6.00 |

8 | 7 | 7.00 |

9 | 8 | 8.00 |

10 | 9 | 9.00 |

11 | 10 | 10.00 |

12 | 9 | 9.00 |

13 | 8 | 8.00 |

14 | 7 | 7.00 |

15 | 6 | 6.00 |

16 | 5 | 5.00 |

17 | 4 | 4.00 |

18 | 3 | 3.00 |

19 | 2 | 2.00 |

20 | 1 | 1.00 |

If you only had one game and you were given a choice between 3d6 and 2d10, which would you choose? With the former, you're more likely to be average. With the latter, you're more likely to be exceptionally good or exceptionally bad. It all depends on your tolerance for risk.

Dice rolls form a normal distribution and are plotted as a bell curve. The most commonly occurring numbers form the rise in the center of the bell and slope down to each side. The distribution of these numbers follows what's called the 'standard deviation'. This number tells us how likely a given number is to deviate from the mean (average) value of the numbers. About 68% of numbers are one standard deviation from the mean, 95% are two standard deviations from the mean and 99.7% are three standard deviations from the mean.

For the 3d6 stats, the standard deviation turns out to be roughly 3 (2.96 with a mean of 10.5). For 2d10, the standard deviation is roughly 4 (4.06 with a mean of 11). So 95% (2 standard deviations) of 3d6 rolls will be roughly 5 to 16 (rounded off) and 95% of 2d10 rolls will be between 3 and 19 (also rounded off).

What does this mean (no pun intended) for IQ? Well, the average IQ is 100 and the standard deviation is 15 points. This means that 68% of all IQs are expected to be between 85 and 115. 95% are between 70 and 130 and 99.7% of all IQs are between 55 and 145.

Now Summers stated that his results implied that "the difference in implied standard deviations" is about 20%. I took that quote from the excellent book Super Crunchers, but it wasn't terribly clear what Summers thought the deviations might be per gender, so I'll take the simplest (and probably incorrect) assumption that he meant men have a standard deviation of 15 while women are at 13. This means that 99.7% of women (3 standard deviations) will have IQs between 61 and 139 instead of the men's 55 to 145. Men are 2d10 and women are 3d6.

Of course, IQ is often considered to be a rather dubious metric. Tests sometimes reflect cultural bias, they may not reflect a fixed quantity (there's evidence that diet and exercise have a large role) and, at its core, IQ may not even reflect intelligence. An overly emotional individual with a high IQ may be more error prone in some situations than a calm person with a lower IQ. There's also the thorny question of whether IQ is more a function of genetics or culture.

That being said, let's finish off with an interesting quote from the Super Crunchers book I mentioned earlier.

You are told you can choose the range of possible IQs that your child will have but this range must be centered on an IQ of 100. Any IQ withing that range is equally likely to occur. What range would you choose -- 95 to 105, or would you roll the dice on a wider range of, say, 60 to 140? When I asked this question to a group of fourth and sixth graders, they invariably chose ranges that were incredibly small (nothing wider than 95-105).

So these children, at least, clearly felt that the female's alleged IQ variance was a better choice, but obviously the journalists hounding for Summer's job didn't ask them. Frankly, I find this a bit frustrating. Some say that we simply shouldn't be allowed to ask "disturbing" questions because they don't gain us anything, but that's wrong. In science, we constantly find that asking questions leads to unexpected answers and while we don't always get the answers we want, we often learn new things which can help us. Constraining us beforehand merely means that we're less likely to get the unpleasant answers, but we're also less likely to get the pleasant ones. Sound familiar?