That is not how statistics works. This is how you reverse a conditional probability (Bayes's Rule):

  P(B|A) = P(A|B) * P(B) / P(A)

If A is "female" and ~A is "male", and B is "good engineer", then what we know , including ancestor post's assumption, is this:

  P(A|B) = 0.4 (the population of known-good engineers is 40% female)
  P(A) = 0.5 (the population as a whole is 50% female)

We can further assume, arbitrarily, that P(B) = 0.02 (one in every fifty people make good engineers). So the likelihood that a person is a good engineer given that they are female is P(B|A).

  P(B|A) = 0.4 * 0.02 / 0.5 = 0.016
  P(B|~A) = 0.6 * 0.02 / 0.5 = 0.024

What we learn from this is that sex is not a good screening criterion for finding good engineers. You realize an advantage of only 0.8%, not 60%.

This also isn't how statistics works.

> We can further assume, arbitrarily, that P(B) = 0.02 (one in every fifty people make good engineers). So the likelihood that a person is a good engineer given that they are female is P(B|A).

  > P(B|A) = 0.4 * 0.02 / 0.5 = 0.016
  > P(B|~A) = 0.6 * 0.02 / 0.5 = 0.024

> What we learn from this is that sex is not a good screening criterion for finding good engineers. You realize an advantage of only 0.8%, not 60%.

if you multiply two numbers (here 0.4 and 0.6) by the same constant (here 0.02 / 0.5) you can't then say "You realize an advantage of only 0.8%, not 60%"... 0.016/0.024 == 40/60.

Re: OP's point ("When you pick out an individual there's still a 60% chance than one is better than the other given no other information than their sex"), I agree that's wrong, but not for the reason given. He's not reversing a conditional probability, but rather asking a question about the unconditional probability (the base rate), so the 60/40 conditional information is not relevant.

If you pick two random individuals, one male and one female, and the IQ average of the two groups is equal (roughly true, from the article), the chance that one has a higher IQ than the other is 50%.

  julia> sum([randn() > (randn() * 10) for i=1:1_000_000])
  499884

IE assuming that men have 10x the IQ variance, the variance makes literally no difference when comparing two random people.

Your post looks like a non-sequitur to me. But for the sake of math...

For an arbitrary variance multiplier of 10 (std. dev. sqrt(10)), there are no non-imaginary solutions for a high-pass threshold where you can achieve a 60:40 split between two populations that have the same median.

You can get about a 58:42 split, putting your threshold a bit below the common median. But to achieve a 60:40 split, you need one population to have 25 times the variance, or for the populations to have different medians. Above that, you have two possible real solutions for the threshold value.

You can try it yourself, and hopefully my math is correct for this one:

  60/40 * erfc( x / sqrt(2))/2 = erfc( sqrt( variance_ratio ) * x / sqrt(2))/2

Since the threshold value equates to a different standard deviation for each population, you can say with some certainty that given the knowledge of which population a person is in, there is a different probability that person is above the threshold value, which works out to be 60% and 40%, respectively. But that's just begging the question, since those are the same numbers you used to work out the threshold value, and you still had to assume values for the medians and variances of both populations.

I'm super confused by this reply. The article says that the stddev of the distribution of male IQ is larger than the stddev of the distribution for female IQ, and also seems to state the mean and median are equal. GGP threw out 60/40 as their believed split of men to women being good engineers. I took this to mean that GGP was saying he believed engineering talent to be independent of gender when IQ was present, and that engineering required a higher than average IQ. More formally:

  P(good engineer | IQ) * P(IQ | gender) == P(good engineer | gender)

In that model the 60/40 split comes from a conditional distribution of gender given an IQ above some threshold. So something like:

  julia> women = sum([randn() > 1 for i=1:10_000_000])
  julia> men = sum([(randn() * 2) > 1 for i=1:10_000_000])
  julia> women / (women + men)
  0.3395449535000696

Which isn't exactly 60/40, but is fairly close. It says that [iff the 60/40 split is true], the stddev difference needed is less than 2x.

If you prefer exact math, the ratio above is

  erfc(1/sqrt(2)) / (erfc(1 / (2 * sqrt(2))) + erfc(1 / sqrt(2))) =~ 0.339593

and the exact solution for a 60/40 split is

  1 / (sqrt(2) * inverseerfc((3/2) * erfc(1/sqrt(2)))) =~ 1.4

Though FWIW I find the simulation version more intuitive (likely because of a misspent youth programming rather than a misspent youth mathing :p).

Can you explain the model you were using to motivate the mean vs median calculation? I feel like I'm missing something interesting.

Easy to explain. I made a mistake in my math. The equation I needed to solve was

  40/60 * erfc( x / sqrt(2))/2 = erfc( sqrt( variance_ratio ) * x / sqrt(2))/2

Thanks for presenting this so clearly. However, note that, as you show in this scenario, males are, on a relative basis, 50% more likely than females to be good engineers, knowing nothing else about them other than gender (P(B|~A) = 1.5 * P(B|A)). While the absolute probabilities (and difference in probabilities) are small, the relative improvement in classification error rate attributable to knowing the person's gender could very well mean that gender is still a useful feature to include (among many others! it's not so useful on its own, as you note) in a classifier for good engineers.

If you were writing an automated Bayesian classifier to find good engineers, males would get one measly additional point towards the refer-to-a-human threshold.

In contrast, there are many other hypothetical rules that would be better predictors. Liking Star Wars, 4 points. Star Trek, 9 points. Knowing how much mana a Lightning Bolt costs, 6 points. Owning a 3-wolf shirt, 2 points. Being left-handed, 15 points. Reading HN, 40 points.

Anything you can glean statistically from a population of known-good engineers can be transformed into Bayesian classifier rules, in exactly the same way you can predict an e-mail is spam if it has certain strings in it.

But even including the one point for knowing the sex could be considered sex discrimination, even though it would be totally supported by the math. But since you likely want your threshold value to be high enough to weed out false positives, while still low enough to avoid false negatives, that one point rule is practically a waste of time. The cases where that one point makes a difference will be just those people who barely meet the threshold. If you have two people who are only just barely good enough to be considered good engineers, and otherwise exactly the same, the statistical argument says to prefer the male.

I have never met anyone that is content to hire engineers who are likely to be only barely adequate in preference to those who are likely much better. That hypothetical person is the only one who might care about male or female. Everyone else will be looking for the highest point totals, gained from criteria that are better predictors.

This assumes it's independent of other data.

If your classifier also knows that an applicant has a CS degree and hypothetically a higher percentage of females with CS degrees than males with CS degrees are talented. Then the relationship might reverse.

The point is stats about populations often fail to carry over to sub populations.

PS: This flip is actually fairly common in the face of discrimination. Basically, you would expect the first hundred female fighter pilots to be way above average.

There are three hypotheses:

1. The selection criteria are the same for different populations, and the there are significant statistical differences in the populations that may lead to the observed result.

2. The selection criteria are the same for different populations, and random noise produced the observed result.

3. The selection criteria are different for different populations.

You cannot assert one without ruling out the others.

In the fighter pilot case, I imagine that it would be easy to disprove #1 and #2. Not all cases of discrimination are so obvious that there are no non-imaginary solutions to the equations that conform to the assumptions of the hypothesis.

It makes some people uncomfortable to face the idea that in order to prove discrimination is occurring, they must first employ accurate statistics for partitioned populations, which may be unflattering to one or more of the partitioned groups. You can't really say "because race" or "because sex" or "because disability" with certainty over "because selection-relevant attribute" using a statistical argument without showing the statistics. But those who attempt to gather accurate statistics are often accused of something-ism and vilified just because some people don't like the results.

Some people just want to work without constantly watching their backs and covering their asses. And that's why we assume independence without having ironclad proof that is also easily understood by the general public. If you're going to suggest that there is something intrinsic to femaleness that causally links to less engineering skill, you had better have a flawless data set with mathematically perfect analysis, and then also invent a psychological "out" which absolves anyone of guilt for relying on the results.

This is why all HR departments suck. Most of their job is covering someone else's ass, and if they choose to use statistics, they must be both factually correct and politically correct.

I don't understand why you are taking the advantage between the two options to be the difference between their probabilities, rather than the ratio. If I am 2% successful at X and you are only 1% successful, I am 0.02/0.01 = twice as successful as you, not 0.02-0.01 = 1% more successful.

As dlss has explained, it doesn't matter what constant you multiply by, the ratio is the same.

You are taking the ratio of P(B|~A) over P(B|A):

  P(A) = P(~A) = 0.5
  P(B|~A) / P(B|A) = ( P(~A|B) * P(B) / P(~A)) / ( P(A|B) * P(B) / P(A))
  = P(~A|B) / P(A|B)
  = ( 1 - P(A|B)) / P(A|B)

I don't understand why you consider this to be a meaningful number. It isn't even a probability any more; it's just a dimensionless ratio.

In your example, your result is 2. That alone tells me it cannot be a probability, as those numbers are restricted to the domain from zero to one.

If you partition the population by sex, the probability of randomly selecting a good engineer from the all-male group is only 0.8% higher than picking from the all-female group. That is the advantage you realize by selecting that criterion for your partition. If you are taking a ratio of conditional probabilities, you are no longer partitioning the probability space by that condition.

Yup, "insignificant" is the wrong word, since it has a specific meaning in statistics, and for that meaning, my paragraph is definitely wrong. How about if I change that word: "But at an individual level, that difference becomes trivial."

I get what you mean, regardless of wording. People should be seen as individuals. I don't know how we can be blind to aggregate differences enough to not be perturbed when we have unequal outcomes but not so cognizant of them that we demotivate people.

But as I write this, I'm arriving at an answer: we need to recognize that people are making choices based upon who they are and what they like.

As I have read this thread, I keep coming back to how we measure competence and ability.

For example, at one point in history our IQ tests included questions based on baseball rules. So someone taking the test that didn't know the rules of baseball couldn't do well – regardless of their mathematical or logical ability, i.e. what the question was supposedly measuring.

Along those lines, I would strongly:

- recommend listening to the Hanselminutes episode on Women in Technology in the Muslim World (http://hanselminutes.com/203/women-in-technology-in-the-musl...)

- remind people that the first programmer was Ada Lovelace, and computers used to be actual people and mostly female (https://en.wikipedia.org/wiki/Human_computer) & suggest reading some of the reasons this changed (http://www.smithsonianmag.com/smart-news/computer-programmin...)

In other words, culture can strongly impact both perceptions of ability and measurements of ability, as well as impacting the equitable training and opportunity that people are afforded. I suspect that most differences in most fields are not actual sex differences (i.e. biological), but in fact are social differences based on gender (i.e. cultural based on a person's gender identity).

I recently took an artificial intelligence class that required writing an agent to answer non-verbal IQ test questions. One of the greatest challenges was that some of the questions had ambiguous answers. For example, an axial symmetry in the example frame could be reproduced with either a rotation or a mirror operation in the output frame, and both options were available as candidate solutions.

Naturally, the question came up: how do we define the "right" answer when there are multiple valid choices? As it turns out, the folks who write the test determine the correct answer to these types of question based on the most common response from test takers who get the highest scores on non-ambiguous questions. The thinking goes that if folks who get most of the other questions right largely agree about a preference for one answer over the other, that must be the most correct answer.

The most interesting part of that discussion to me was when a classmate observed that we use human intelligence as a baseline for measuring human intelligence, and then marvel at the extent of our own intelligence; if a machine used more sophisticated logic than a human was capable of to choose a different answer, we would look down on the machine for being less intelligent than man. It reminded me very much of Plato and the shadows in the cave.

You’re trying to redefine words out of context.

Significant is often used in science to mean large enough to be concerned with. So, a 1% chance of death tomorrow is a significant chance of death tomorrow.