Gender Differences and Statistical Distributions
With respect to the recent Google memo, intelligent discussion of the topic may be impeded by bad math being employed by both "sides" of this debate.
First of all, and probably more prevalent, are those who misunderstand the memo author's claim, and are assuming that he's using the average male and average female as proxies for all males and females. He's clearly not, and his memo shows this in notional graph form.
As a first approximation, the memo goes on to argue that such statistical distributions may explain differing rates of representation among Google's workforce for different genders.
For example, from this link, we see the following data, which has been qualitatively consistent for many years.
Looking specifically at Google, their share of technical jobs held by women stands at 19% (2016), while approximately 32% of those jobs are held by Asian workers, and 57% by white workers. As non-Hispanic whites make up approximately 62% of the US population, this means that whites are slightly underrepresented at Google, still ignoring the math skill results. If we assume that Google gender ratios (.81 : .19) hold across ethnicities, for the sake of argument, that would mean that white men are overrepresented at Google (46%), while white women are underrepresented (11%). White men only make up 31% of the US population, so 46% is considered overrepresentation. However, this effect is much greater for Asian men, who have approximately 26% of tech jobs at Google, compared to 3% of the US population. Likewise, Asian women hold 6% of Google tech jobs, compared to 3% of the population. So, Asian men are employed in Google IT jobs at a level more than 800% of their expected level, based on population alone. Asian women are employed at 200% of their expected level. White men are employed at 150% of expectations, and white women at 35% of expectation.
Qualitatively, these 4 groups' basic results are consistent with the ranking of each of the groups' averages in the SAT math scores table. This is essentially one of the memo's major points - that skewed ethnic or gender representation at Google should be expected.
Limitations
There's multiple problems with the memo's content, as well as the simplified example I provided above. One problem is that the SAT math scores, and the notional graph of Trait distributions that the memo showed, are both indications only of gross population features. The published results of the tests may simply be averages (as shown in the table above), or perhaps an average and standard deviation, for a normal distribution. But, these two numbers and a distribution type (e.g. "normal") are only a rough approximation of the entire distribution. In reality, distributions of traits can have shapes unlike the ones shown in the notional memo graph above. For some analyses, this may be irrelevant, and a normal distribution may adequately describe the results.
There's reason to suspect, however, that Google's IT workforce may be an exception here. IT staff in general are not drawn from an entire population. In terms of a trait like math skill, they will generally only be from the upper echelons of ability, likely all above the average. For Google, specifically, though, the staff are likely to come exclusively from the very highest performers in our measured traits. Consider these new notional distributions.
In the above graph, I've represented male (blue) and female (pink) math scores with overlapping distributions. The male graph has a higher average math score. For most minimum values of "mathiness" one could pick, there is a greater area under the blue curve than under the pink curve, suggesting that we should expect more men than women to meet that minimum criterion. However, recall that Google isn't sourcing employees from near the average. The "upper tail" of a distribution (aka "right tail") is the part of the curve at the highest values on the X (Mathiness) axis. I have shown a possible zoomed-in view of that portion of the curves in a second plot:
This is completely hypothetical. However, what it attempts to illustrate is that since Google is an elite employer, they are interested in an area of the distributions that may not be well-represented by the overall male-female distributions shown first. It's entirely possible, for example, that the male distribution is not strictly normal. It may not be symmetric to the left and right of the average point (and I have drawn it to be asymmetric). If this is the case, while most parts of the male mathiness curve exist at higher values of mathiness than the female curve, the curves may cross near their upper tails. If so, it may be expected that for very high minimum values of mathiness (i.e. Google's elite standards), we would see more women than men meeting that criterion. This is depicted in my second (zoomed) graph.
Is this the case? That's a harder question to answer than the one about entire male and female populations. Many studies have quantified overall gender traits, but most aren't oriented toward the top 1%, or top 0.1%, of the population. That highest echelon may not be well known, based on existing research. If the second graph is representative, then the Google memo's conclusions are actually completely wrong. For that reason, caution should be used in applying general results to such an elite company as Google.
Other Data
Do I believe the Google memo's main premise is wrong? No, I don't. But, I'm also not confident. In the absence of better data, I'm willing to use the general population math data as a starting point. But, disclaimers about its applicability should be made, and the Google memo did not do that.
However, one reason I think the upper tail data would likely still show an expected male advantage in math is that we do have more sophisticated statistics available for IQ. Whether math scores or IQ are better predictors of ability at a software company is another question. However, IQ results do suggest that while male and female average IQs are similar, the distribution of male IQ is such that at very high levels of IQ, males are disproportionately well-represented. If math or other software skills mimic IQ, then the memo's premise may hold.
As one more simple data point, I tallied the entire graduating class of Caltech for 2017. Caltech has the highest SAT scores among US colleges, and is almost exclusively STEM majors. Therefore, it may help better characterize the "upper tails". I used the commencement program to count students with relevant degrees. For this, I counted any of the majors with "comput" in their name, and also minors, so long as the student's major didn't indicate another likely career choice. For example, a "physics major, computer science minor" was counted, while a "biology major, computational neural systems minor" was not.
Among the students with majors likely desired by a software company, women comprised approximately 27% of the graduates. This is compared to about 40% of the graduating class overall, suggesting women may be choosing, or excel at, other STEM majors more than computer fields. The Caltech class at 27% female, however, is higher than current US averages of 18-20% female, in "computer science" majors. This could mean that in fact, although more men would still be expected near the upper tails, the distributions are skewed at those upper tails to narrow the gender gap shown among the broader population (where women make up less than 20% of computer science majors).
One problem with Caltech is its extreme small size, so to be useful, this same tally should be performed for several years running. Nevertheless, without considering upper tails, Google's current ratio of female IT staff of 19% looks almost entirely expected. If the Caltech result is more representative of the upper tail, though, then perhaps we should expect closer to 27% female IT workers at an elite company like Google.
There are many other limitations to results in the Google memo, and my presentation here. I've only attempted to identify a couple strictly related to statistics. Neither are proper scientific presentations, but merely attempts to further a delicate conversation with the addition of some data.
