There has been much made of the apparent pay discrepancies between men and women. At the recent Miss America Pageant, Nene Leakes posed the question,
A recent report shows that in 40% of American families with children women are the primary earners, yet they continue to earn less than men. What does this say about society?”
Miss Utah took a lot of flak for her poor response to the question. But sometimes we need to look a bit deeper at these questions, and the answers aren’t always so straightforward as to quote a simple statistic. I’ve been reading Superfreakonomics by Stephen Dubner and Steve Levitt, and as usual they make some interesting points that are often against conventional wisdom. In analyzing the pay gap among men and women, they find some reasons that aren’t necessarily sexist. Among women with an MBA, the pay gap can be explained by the following reasons.
-
“Women have slightly lower GPAs than men and, perhaps more important, they take fewer finance courses. All else being equal, there is a strong correlation between finance background and career earnings.
-
Over the first fifteen years of their careers, women work fewer hours than men, 52 hours per week versus 58. Over fifteen years, that six-hour difference adds up to six months’ less experience.
-
Women take more career interruptions than men. After ten years in the workforce, only 10 percent of male MBAs went for six months or longer without working, compared to 40 percent of female MBAs.
The big issue seems to be that many women, even those with MBAs, love kids. The average female MBA with no children works only 3 percent fewer hours than the average male MBA. But female MBAs with children work 24 percent less. “The pecuniary penalties from shorter hours and any job discontinuity are enormous,” the three economists write. “It appears that many MBA mothers, especially those with well-off spouses, decided to slow down within a few years following their first birth.”
…
There’s one more angle to consider when examining the male-female wage gap. Rather than interpreting women’s lower wages as a failure, perhaps it should be seen as a sign that a higher wage isn’t as meaningful an incentive for women as it is for men. Could it be that men have a weakness for money just as women have a weakness for children?
So, in answer to Leakes question to Miss Utah, rather than the “obvious” answer that society discriminates against women, perhaps it says that society rewards people who don’t take time off work.
Here’s an example that I use when I teach introductory statistics classes to show that statistics can be paradoxical. A university offers 2 degree programs: electrical engineering and English. Admission is competitive, and women expect discrimination. Is it fair to conclude discrimination?
We can see that more males apply, but when we look at the rates of admission, it appears that males are admitted at a higher rate.
|
Male |
Female |
|
| Admit |
35 |
20 |
| Deny |
45 |
40 |
| Totals |
80 |
60 |
| Percent Admitted |
43.8% |
33.3% |
It looks like discrimination is going on. But is this a fair test? At this point I will illustrate to students how to perform a Chi-Square test, and when we are through with out calculations, the Chi-Square test shows that there is no difference in admission between men and women. It is at this point that I like to quote Mark Twain: “There are liars, damn liars, and statisticians.”
The simpler calculation seems to show that there is discrimination, but the more complex calculation shows that there isn’t. So which one do we believe? Well, perhaps there is discrimination going on in one of the departments, so let’s looks at each department and see if we can figure out where the discrimination is going on.
| Engineering |
Male |
Female |
| Admit |
30 |
10 |
| Deny |
30 |
10 |
| Total |
60 |
20 |
In the Engineering Dept, we can see that the admission rate is 50% for both genders. Now let’s check the English Dept.
| English |
Male |
Female |
| Admit |
5 |
10 |
| Deny |
15 |
30 |
| Total |
20 |
40 |
Well, there’s no discrimination between genders here either. Men and women are admitted at a rate of 25%.
This is an example of what we call Simpson’s Paradox. We can see that there is no discrimination going on in either department. So why did there appear to be discrimination in the first table shown? The reason for this paradox is that the 2 groups are unequally sized. Because the number of men is larger than women (especially in the Engineering department), those numbers overwhelm the smaller numbers in the English Department, giving us a false impression of discrimination. Therefore the Chi-Square test is the better test than the simple admission calculation. There is no discrimination going on.
This example is based on a real-life lawsuit filed at the University of California-Berkeley. (If you would like to learn how to perform the calculation using a free stats package called R, here is a tutorial.) Women cited as fact that they were being admitted at a lower rate than men. However, when they looked at each department to determine where the discrimination was actually happening, they found that women were actually admitted at a HIGHER rate than men. Once again, it was a trick of the numbers that there were more male applicants than female.
So contrary to the charge that I have a feminist agenda, I just want data to back up assertions. Sometimes discrimination isn’t quite so clear cut, but sometimes it really is discrimination. I wish Simpson was here to cut through all the paradoxes that can be found in statistics, but we shouldn’t be quite so cynical as Mark Twain was. Sometimes we have to do a more complex statistical test to know the real answers, or we have to cut through erroneous assumptions. Comments?
Wheat & Tares 
Well, that’s some interesting info you have there.
But I have a very dear friend who owns his own company and flat out admits he’s been to many seminars over the years where he was taught to hire women–they work harder for less pay than men. So he has, and they do.
It’s never been fair for women.
Women still do not have equality……in pay or anything else. That 100% statistic is absolute truth.
Chi square does not tell you that there is no difference in the observed numerical difference. It tells you the probability of the difference you observe occurring by chance. You are dealing with probability, not absolutes. Something always causes a difference in numbers. What you are saying, in this particular case, is that you attribute the difference, based upon a statistical test, to chance. You have not shown that women are not discriminated against.
Parker, yes you are being more precise in your wording than I was. However in my example, I have shown that women are not discriminated against, and in fact are being treated identical to men with regards to admission rates. Since the admission rates are identical in the 2 departments, you can’t credibly claim that women are discriminated against in this situation.
The interesting thing about Simpson’s Paradox is that sometimes the associations become reversed. In the Cal-Berkeley case, when you break it down by department, women were actually given preference in admission. (By department, it could be argued that men were discriminated against because they had lower admission rates, though the Chi-Square test argues that the differences are so small that they are not statistically significantly different.)
Rockies Gma, to be clear I am not arguing that discrimination does not exist, but I am arguing that sometimes we need to look deeper at the numbers. It may not be so clear-cut as first impressions imply.
You haven’t shown that they weren’t discriminated against. You have shown that the difference in the observed numbers has a probability (which you haven’t listed) of occurring by chance, or in other words random variation. That is all you can say. Even if you had reported a statistically significant difference, you couldn’t have said it was due to discrimination, without considerable other data. You really have no basis for interpreting the numbers of your particular set, other than the variation is likely due to chance..
Parker, here’s a quick google definition of Discrimination:
The unjust or prejudicial treatment of different categories of people or things, esp. on the grounds of race, age, or sex.
Table 1 above showed that men were admitted at a higher rate than women. Based on this table alone, then it could be argued that men and women are treated unequally.
Tables 2 and 3 showed that men and women had equal rates of admission in the English and engineering depts. With regards to admission rates, The data show that men and women have equal rates of admission.
Based on Table 1, we asked, “Is it fair to claim discrimination?”
Based on Tables 2 and 3, clearly it is not fair to claim discrimination, regardless of whatever the Chi-Square test results were. (But for the record, the Chi-Square test show a pvalue of 0.2117.)
So, if admission rates are used as a proxy on which to judge if discrimination is taking place, then clearly there is not enough evidence to conclude that there is discrimination in admission rates. If you want to propose another definition of discrimination, then fine. But based on the data provided so far, the evidence seems to indicate that there is equal treatment in both the engineering and English departments with regards to admission rates. If there is some other discrimination going on, then you are welcome to speculate, but admission rates should not be used as a bludgeon to indicate discrimination. It is simply an invalid measure of discrimination on this data set.
Another way of putting it, the null hypothesis of no discrimination cannot be rejected with this data. Since ‘no discrimination’ is your starting point in the statistical test, and you can’t reject it, than the correct conclusion is that this data show no evidence of discrimination. Of course, that does not mean discrimination is not happening, just that this data do not show it.
There would have been discrimination in admissions if the women were admitted at the same rate as the men BUT had higher demonstrated aptitude. That would be discrimination, although difficult to prove, and MH is right that there is no evidence here to prove it.
This is an interesting point: “society rewards people who don’t take time off work,” and in my experience, it is true overall, or rather the reverse is true. Society (and circumstance) punishes those who take time off work, whether men or women, with fewer opportunities and employers often view those people as unreliable, lower contributors. Which they basically are from the employer’s standpoint. Even where the law protects them from outright consequences for lengthy leaves of absence, it is difficult for someone taking frequent long leaves of absence to contribute on par with someone who is there all the time. Out of sight, out of mind. They will lack the connections to get things done in the network. This is more true in some fields than others, though.
There is also a subtle form of sexism at play when girls are routinely discouraged from the hard sciences or from participating in fields like business that typically pay well. The problem is that many women & girls buy into the idea that “math is hard” or that they will be held accountable disproportionately for child rearing to the point that they cannot pursue a field that pays well or is demanding. This is the whole point of Sandberg’s Lean In book. I would love to see more women decide to buck this idea. Entire fields that are woman-dominated are traditionally paid less in the US: nursing, teaching, child care. Women continue to be herded into these fields, and so the status quo never changes, even though many men are very suited to do this type of work. Because men are taught they must be a provider, they shy away from lower paying jobs that they might be more suited to do. The sexism cuts both ways.
Rockies Gma: I have never heard anyone say that openly, but I’ve always worked for very large companies that wouldn’t do such a thing for fear of being sued. However, I have observed that women are often easier to work with on issues of pay. If I compare two of my employees, a man and a woman, I have often found the woman works harder, complains less, is paid less, and is very grateful for pay increases. I have often found men to feel entitled to raises and demand them or pout about them if they don’t get them, to resist correction or otherwise lack social skills – using stubbornness as a tactic to negotiate higher pay, and to be more likely to quit and change jobs. Should women start acting like men or should men start acting like women in these examples? I reward the right behaviors and let the chips fall.
How would Hawkgirls comment be received if it read like this?
If I compare two of my employees, a man and a woman, I have often found the man works harder, complains less, is paid less, and is very grateful for pay increases. I have often found women to feel entitled to raises and demand them or pout about them if they don’t get them, to resist correction or otherwise lack social skills – using stubbornness as a tactic to negotiate higher pay, and to be more likely to quit and change jobs.
“But based on the data provided so far, the evidence seems to indicate that there is equal treatment in both the engineering and English departments with regards to admission rates.”
That is the conclusion your data allows you to make in this one instance. I just don’t understand why you think you can now make absolute statements about discrimination, which you have failed to operationally define, based upon one small data set, with non non-parametric statistic.
I’m not saying you are wrong in your apparent conviction that women are discriminated against. I’m saying your data doesn’t allow the conclusions you’ve drawn.
oops–“aren’t discriminated against.” #9
Parker, please read my previous post on this topic. You seem to be viewing this post with blinders on. See http://www.wheatandtares.org/12379/how-racism-sexism-and-other-stereotypes-hurt-our-performance/ Perhaps that will help you appreciate better what I’m trying to say.
Phil, to clarify, I was referring to 2 specific employees in that case, although there is some pattern I have observed that can be generalized. I have often inherited a team of employees of both sexes and found some of the highest performing to be the lowest paid (whether make or female) and vice versa. Here’s another way to phrase that. Those who have been overlooked are grateful for crumbs. Those who have been given extras expect and demand more extras. Why did my assessment of their skills differ from their prior leaders? Maybe the treatment they got drove the behavior I saw. Maybe different leadership styles yielded different results. Hard to say.
I’m not sure about blinders, but I am wearing glasses. Let’s try one more time. There are many people who argue that LDS women are discriminated against in the Church. So let’s construct a 2 cell matrix. In one cell we will put the number of all the males you have been chosen as bishop in all the wards of a particular stake for a 25 year period. In the other cell we will put the number of females chosen as bishop. Then we will run a chi square. Without even running it, I know that we will find a significant statistical difference. Can we now conclude that indeed the Church does discriminate against women? If not, how does that differ from you case in your OP?
Parker – I don’t know why you are bringing the church into this. The OP is about society at large (emphasis on US culture). The only potential church connection is that Miss Utah was a complete dunderhead.
Why wouldn’t I use data derived from something we all know about as an example of what you can and can’t do with a particular statistical test? There was no criticism expressed, and I didn’t attack anyone cherished beliefs. I certainly apologize if I stepped on your toes in any way.
I actually like Parker’s take, and in fact I did ask on my previous post for some ideas on how to demonstrate discrimination within the church. The whole women and priesthood thing is a bit like shooting fish in a barrel, because as demonstrated, Parker nailed a proposed experiment (though I will mention that the Chi-square test doesn’t perform well when 0 is in one of the cells, or if 25% of cells are less than 5.) Still that’s a very obvious example. A harder example would be to show that women are natural nurterers while men aren’t. I’d love to see someone design an experiment like that; the Radio West guys said that while women have the “advantage” of breast feeding, it is really no more natural for women to be nurturers than men. It is a learned skill. (Frankly, my wife doesn’t consider herself a nurturer, but she does spend more time with the kids, so she is better than me by a long shot.)
But this post was to point out that sometimes common wisdom isn’t necessarily common. It is nice to have data to back up assertions, rather than making a claim that women are natural nurturers. Statistics can help remove (though not eliminate) some of the subjectivity of certain claims.
This might be an oversimplification, but, if you could hire an equally good woman for 75 cents on the dollar, who would be stupid enough to ever hire a man???
I am in a profession that has historically been male dominated. In 50% of the facilities in which I have worked there have been at times zero women in the operations department(s). Currently there is one woman out of 80 total. In the closely related disciplines of engineering, maintenance and supervision & management there are also few women in the facilities with few women operators.
In larger facilities, there are more women in operations, ~10-20%, and significantly more women in the closely related disciplines. The percentages would be close to the college graduation percentages in the targeted majors at the larger facilities.
Is there overt discrimination against the few women at the smaller facilities? Not that I have seen. I think it is more likely that women self select to areas where they feel more comfortable. There are far more women at our corporate office where they work around less environmental hazards and extremes and do not have a core area with no women working.
I say all of this as a background for the town where I work. Our facility probably has the largest payroll of any private employer in town. The average starting pay for an operator which requires only a HS diploma is higher than the starting pay of the local school district. The disparity increases after 5 and 10 years on the job. Does our town discriminate against women? How about our facility? In both cases, I do not think so. The outcomes look like discrimination, but the details point to a free market and self selection.
Is it discrimination? Not when self selection is at play, at least not legally. But do people self select because the environment is make dominated and doesn’t feel welcoming to women as a result? Maybe. The other side of the self selection coin is that women are encouraged to take the easy road career wise. When one sex gets extra credit for being ambitious but dinged for not being nurturing and the other sex gets extra credit if they change a single diaper or wash one dish but they get dinged if they are insufficiently ambitious or don’t earn enough then we have a system that is less efficient and less fulfilling than it could be.
Parker, I shouldn’t have been so quickly dismissive of your point since you were talking stats. I just didn’t see the post being very church focused and it is tough to quantify the sexism in the church in terms of discrimination. It is definitely discriminatory but it is not legally discrimination because churches are exempt. Here is the stat to look at. Compare the percent of seminary teachers who are female who are volunteers (unpaid) vs those who are paid employees of CES. CES uses ministerial privilege to discriminate against women with children. It’s not illegal but that doesn’t mean it’s right. Of course those jobs pay crap. But they pay even less (zero) when it’s a calling.
Superfreakonomics said that men will enter traditionally female fields, if the money is right. For example, they cited TItle IX, the federal law that requires that male and female athletics be on a more equal footing in college athletics. Before Title IX, nearly all coaches of female sports teams were women. After TItle IX, more money flowed into female sports departments, and now only 40% of coaches are female.
They also did a very entertaining chapter on prostitution, a traditionally female dominated occupation. At the turn of the century, most of the pimps were female. As the money started getting better, most of the pimps became male.
Now in my previous post I mentioned that just 15% of editors at Wikipedia are female, despite the fact that women are more active in social media and many other internet technologies. The authors of the study theorized that men are more competitive (and that’s where the matriarchal and patriarchal studies came in to see if culture played a role in competition) and women aren’t comfortable with competition in patriarchal societies, but they are equally competitive in matriarchal societies (of which there are but a few.)
Certainly, as nursing has become more lucrative, more men have entered that job market. But it could be that $$$ doesn’t have the same allure for women as it does for men.
Well, my daughter interviewed with a firm in Salt Lake. Every year they advertise that they are looking for a couple of engineers, and intimate that one slot is reserved for a female hire for diversity purposes.
She talked to the secretaries. They have never hired a woman other than as a secretary or a janitor. The sole purpose of the diversity/female hire information is so that the guys who do not get the job re-direct their ire towards female engineers and not towards the company.
Turns out that such a practice is not that uncommon. It does lead to increased hostility in the field towards women in it, resulting in fewer women.
“s nursing has become more lucrative, more men have entered that job market” — very true and it is interesting to compare male CRNAs to female ones, for example in terms of job placement, treatment on the job, etc.
As for CES and seminary, even when it is a “calling” there is some reimbursement, it is just very little.
MH, I like your description of Simpson’s Paradox and the example dataset you show. I do think, though, that the fact that the chi square test is nonsignificant is not a good indicator that the data need to be disaggregated into their groups. It’s just an issue of statistical power. If you multiplied all your counts by 10, the chi square test would become significant (chi square (1) = 15.60, p < .001), even though the pattern in the data would be exactly the same. I think the only way to see that the data need to be disaggregated into groups is to actually do the disaggregation and look at the pattern in the groups.
Ziff, that’s an excellent point. In my classes we talk about how too large of a sample can lead to statistical significance but not practical (or clinical) significance. I often use the example that you may be able to collect data on traffic patterns and get a large sample easily (like 10,000 cars.) Because the sample size is so large, you will probably find a statistically significant difference (p<0.001 for example) that shows that the commute time increased/decreased by 30 seconds over a 45 minute commute. So it's important to pay attention to sample size and not look only at statistical significance because in the scheme of things, even if 30 seconds is statistically significant, from a practical point of view, nobody is going to notice that their commute time is 30 seconds less. Practical significance isn't the same as statistical significance.
The other concept that I like to bring up is whether a study is generalizable. Parker above seemed to misunderstand that in using the data above, I was only applying it to this data set. It's easy to want to apply it to other data sets (and of course we hope that our studies are generalizable.) One famous example that I like to use in my classes is that someone did a study and found that by adding a 3rd brake light in the top center of a car, rear-end collisions were reduced by 45%. Congress thought this was such a great way to reduce rear-end collisions that they mandated all car manufacturers install the 3rd brake light that is now standard today.
However, it turns out that accidents decreased by just 5%. The problem was that in the study, the 3rd brake light was such a new, novel thing that people noticed them. Once they became routine, people didn't notice them. Yes they did reduce accidents, but not nearly as much as in the original study, so the results weren't generalizable. It's the same problem with drug studies. In the studies, when doctors are monitoring patients and have strict inclusion criteria, drug studies are very promising. But in the real world, people forget to take the medication, or may have different characteristics than the study patients, so the drug isn't as effective as the study would have you believe.
It's one reason why randomized controlled trials aren't always as useful as observational studies (though I know many think a randomized trial is the gold standard for proving anything.)