William M. Briggs

Statistician to the Stars!

Disparity is inevitable: a counter argument to filing discrimination lawsuits


Know a lawyer who is involved in a discrimination lawsuit? Particularly one in which the plaintiff alleges discrimination because actual disparities are found in company hiring practices? Were you aware that, just by chance, a company can be absolutely innocent of discrimination even though they actually are found to have under-hired a particular group? No? Then read on to find out how.

What are diversity and disparity?

We discussed earlier that there are (at least) two definitions of diversity: one meaning a display of dissimilar and widely varying behaviors, a philosophical position that is untenable and even ridiculous (but strangely widely desired). The second meaning is our topic today.

Diversity of the second type means parity in the following sense. Suppose men and women apply in equal numbers and have identical abilities to perform a certain job. Then suppose that a company institutes a hiring policy that results in 70% women and 30% men. It can be claimed that that company does not properly express diversity, or we might say a disparity in hiring exists. Diversity thus sometimes means obtaining parity.

Disparity is an extraordinarily popular academic topic, incidentally: scores of professors scour data to find disparities and bring them to light. Others—lawyers—notice them and, with EEOC regulations in hand that call such disparities illegal, sue.

And it’s natural, is it not, to get your dudgeon up when you see a statistic like “70% women and 30% men hired”? That has to be the result of discrimination!

Of course, it was in the past routinely true that some companies unfairly discriminated against individuals in matters that had nothing to do with their ability. Race and sex were certainly, and stupidly, among these unnecessarily examined characteristics. Again, it’s true that some companies still exhibit these irrational biases. For example, Hollywood apparently won’t hire anybody over the age of 35 to write screenplays, nor will they employ actors with IQs greater than average.

Sue ’em!

It’s lawsuits that interest us. How unusual is a statistic like “70% women and 30% men hired”? Should a man denied employment at that company sue claiming he was unfairly discriminated against? Would we expect that all companies that do not discriminate would have exactly 50% women and 50% men? This is a topic that starts out easy but gets complicated fast, so let’s take our time. We won’t be able to investigate this topic fully given that it would run to a monograph-length document. But we will be able to sketch an outline of how the problem can be attacked.

Parity depends on several things: the number of categories (men vs. women, black vs. white, black men vs. black women vs. white men vs. white women, etc.; the more subdivisions that are represented, the more categories we have to track), the proportion those categories exist in the applicant population (roughly 51% men, 49% women at job ages in the USA; we only care about the characteristics of those who apply to a job and not their rates in the population), the exact definition of parity, the number of employees the company has, and the number of companies hiring. That last one is the one everybody forgets and is the one that makes disparities inevitable. Let’s see why.

Men vs. Women

Throughout all examples we assume that companies hire blindly, that they have no idea of the category of its applicants, that all applicants and eventual hires are equally skilled; that is, that there is no discrimination in place whatsoever, but also that there is no quota system in place either. All hires are found randomly. Thus, any eventual ratio of observed categories in a company is the result of chance only, and not due to discrimination of any kind (except on ability). This is crucial to remember.

First suppose that there are in our population of applicants 51% men and 49% women.

Now suppose a company hires just one employee. What is the probability that that company will attain parity? Zero. There is (I hope this is obvious) no way the company can hire equal numbers of men and women, even with a quota system in place. Company size, then, strongly determines whether parity is possible.

To see this, suppose the company can hire two employees. What is the probability of parity? Well, what can happen: a man is hired first followed by another man, a man then a woman, a woman then a man, or a woman followed by another woman. The first and last cases represent disparity, so we need to calculate the probability of them occurring by chance. It’s just slightly over 50%.

(Incidentally, we do need to consider cases where men are discriminated against: in the past, we could just focus on cases where women were, but in the modern age of rampant tort lawyers, we have to consider all kinds of disparity lawsuits. For example, the New York Post of 12 May 2009, p. 21, writes of a a self-identified “white, African, American” student from Mozambique who is suing a New Jersey medical school for discrimination.)

Now, if a woman saw that there were two men hired, she might be inclined to sue the company for discrimination, but it’s unlikely. Why? Because most understand that with only two employees, the chance for seeming, or false discrimination is high; that is, disparity resulting by chance is pretty likely (in fact, 50%).

So let’s increase the size of our company to 1000 employees. Exact parity would give us 510 men and 490 women, right? But the probability of exact parity—given random hiring—is only 2.5%! And the larger the company the less it is likely exact parity can be reached.

Surprised? You shouldn’t be. It would be extremely difficult to hire randomly and end up with exactly 510 men and 490 women. Let’s go further. Most people, even tort lawyers, would agree that 511 men and 489 women reflects approximate parity, so we can see that our interest is not exact parity but only something close to it. How close? Would 520 men and 480 women be approximate? Probably. But 700 men and 300 women probably would not.

I have no idea of what the exact definition of exact parity is, and neither does anybody else. It is highly dependent on what a tort lawyer thinks he can get away with. The more emotional the alleged discrimination, the closer to exact parity the lawyer can go. But men vs. women is no longer as emotional as it used to be, so the lawyer cannot demand exact parity; a pretty wide discrepancy has to be in place before red flags are raised.

A company with 520 men and 480 women would reflect a disparity of two actual percentage points: 1 from the increase in men, and 1 from the decrease in women. The same would be true if the company had 500 men and 500 women. Thus, if the company had 1000 men and 0 women, the disparity would be 98 percentage points (it’s not 100 because there are more men than women in the population). Perhaps a reasonable number above which discrimination is indicated is 10 (or more) percentage points. For a company with 1000 this would be at 560 men and 440 women. Thus, if a company had 560 or more men, we might suspect something fishy.


This graph makes it easier to understand. It is the probability of seeing the number of men hired by chance (the number of men on the horizontal axis). The most likely number is 510, at just over 2.5%. Seeing at least 560 turns out to be 0.037%; in other words, a rare event, and if we did see 560 or more men we might be suspicious that discrimination has occurred.

It’s difficult to discern the individual dots, but it’s easy to tell that numbers like 511, 509, etc. are also relatively highly probable. 560 is highlighted by the dashed vertical line, a point which appears unlikely.

We are prepared to claim discrimination if we see at least 560 men, the probability of such we know is 0.037% and unlikely. If there were just one company of 1000 employees, this would be good evidence of discrimination at that company.

But there is more than one company of course (and suppose each of them also has 1000 employees). How unlikely would it be that at least one of them had—merely by chance—employed at least 560 men? The next picture shows this for the number of companies from 1 to 100 thousand.


At 1 company the probability of at least one disparity (defined as at least 560 men out of 1000) is 0.037% as we know. If there are 100 companies the probability is 3.7% of at least one disparity. With 1000 companies it jumps to 32%. At 10,000 it’s 98%. And at 100,000 it’s virtually certain that at least one company will show a disparity. About 3000 companies gives a 50% chance that at least one shows a disparity.

In the USA, there are at least 3000 companies with roughly 1000 employees—but maybe they won’t all have exactly 1000 employees. That means in order to do these types of graphs properly, we’d have to look at the number of companies and the number of employees in each company and then calculate that probability that at least one (or at least two or any number) show a disparity just by chance.

However, it is clear that (non-discriminatory) disparity somewhere is very highly probable (given this definition of disparity; if we saw 800 men and 200 women, it wouldn’t matter how many companies there were: that’s pretty good evidence of discrimination).

Sex and Race

A lot of information so far, but we’re not done. Because it’s possible to find disparities in other areas than sex. We could do the same analysis for race, for example. In the USA population there are about 60% Whites, 15% Blacks, and 25% Hispanic and others (these are only crude estimates: exact numbers can always be substituted). And again, we cannot just look at “extra” numbers of Whites; we have to consider disparities of all kinds (recall the Supreme Court discrimination case brought by white Connecticut firemen).

But we’ll skip that and move to more complicated territory. Let’s look at the mix between race and sex. Our applicant population might have about 30.6% white males, 7.7% black males, and 12.8% Hispanic and other males; the breakdown for females is 29.4% white, 7.3% black, and 12.2% Hispanic and others (all numbers approximate).

This means, in our hypothetical company of 1000 employees, exact parity would insist there were 306 white males, 77 black males, 128 Hispanic and other males, 294 white females, 73 black females, and 122 Hispanic and other females. It’s easy to get the feel that exact parity is even more difficult to attain than when just considering sex.

If we consider that a member of any of these groups will sue if their proportional representation drops by three or more percentage points, then what is the probability our company—purely by chance—with be in trouble? Three percentage points would mean, for example, the number of white men in the company would be 276 or less; for black males this would be 47 or less. The chance of this, for a company of 1000, is 3.7%. Not very big, but not impossible, either.

But what if we increase the number of companies like we did before? We get this picture:


This shows the probability that at least one company out of the Number of companies will show a disparity. After only 18 companies (of size 1000), the probability of at least one disparity is already 50%. After 50 companies, it’s about 85% likely. And anything over 100 shows it’s nearly certain to find a company that is not in parity.

In other words, since there are certainly more than 100 companies of roughly 1000 or so employees, it’s inevitable that there will be false evidence of discrimination. This suggest that the only thing that companies can do to avoid charges is to institute quotas.


The point to take away is that the finer we chop up characteristics—if we add age, sexual preference, national identity, and on and on—it becomes easier and easier for a company to be out of parity just by chance alone, even though they do nothing but hire by ability.

Since disparity it inevitable, all it takes for trouble to start is for each separate group (as in the six groups in the sex by race example) to start counting. Chances are, in some company, one of those groups will have a basis to complain.

But they shouldn’t, not until they have checked how many other companies there are, how many employees those companies have, and so on. And if they do sue, then the defense lawyers should make these calculations to at least show that, by chance alone, disparities occur and that they are not that uncommon.



Not all companies are 1000 strong, of course: there is a distribution of sizes. The techniques are easily adapted to account for this distribution (provided somebody has made the measurements). The numbers I used above to show percentage representation of applicants are all approximately correct, but they vary geographically, a point which should be seriously considered when doing these calculations for real. They could even vary by industry, and even by division in very large companies. And it is the applicant pool, and not the population pool, of characteristics that interests us. Yes, many companies are unfairly castigated for not garnering enough diverse applications, but unless the company is purposely dissuading certain individuals from applying, it is a stupid argument (it’s like faulting a university or corporate physics department for not having enough females when only a bare fraction of females apply; presuming of course the department is not actively discriminating against female applications).

The definitions I used to define disparity (10 percentage points overall, or any group being 3 percentage points away from parity) are completely arbitrary, of course, but they seem reasonable. The story remains the same if the definitions are adjusted, only the probability limits are modified.

And yes, yes, yes, some companies are filled with jerks who actually do discriminate based on characteristics other than ability. It happens all the time and always will, humans being what they are. But this fact does not mean that all disparities are the result of idiocy, as this article tries to make clear.


  1. And of course, this assumes that people of all races, sexes, religions, and national origins are identical in their abilities. That they all have equal intelligence, education, social skills, experience, work ethic, motivation, cultural and family expectations, etc.

  2. OT, but you might find these quotes re stats interesting:

    The abstract reads

    “ A cautionary tale in which previously published results are shown to be invalid due to the
    lack of statistical analyses in the original work.”

    Text from this paper includes

    “Kinsmen (1957) wrote, ‘the job of a scientist is to invent a story which accounts for a set of observations and then decide how likely the story is.’ In his 1957 work, ‘Proper and improper use of statistics in geophysics,’ he emphasized the role of the correct use of statistics in this decision. However, statistics
    continue to be misused or altogether neglected in the refereed literature, with the inevitable result of misleading or erroneous conclusions.”

    From Pielke Sr

  3. Despite it being well known and well documented that certain professions attract more of one sex than the other (engineering: predominantly male for example), we must have ‘parity’. Some of the sexually-differentiated attractions are byproducts of brain structure, brain chemistry and plain old hormones as much as culture or upbringing. Of what sense parity in this reality? Which professions shall we degrade because of the delusion that the mind is a blank slate?

    But it did please one day when I got out of the dentist office. My wife (English degree, HR analyst) is a big fan of the parity thing. She saw the dentist and hygienist come out with me – and made the stereotypical assumption about who was who. The dentist is female, the hygienist male.

  4. More than 30 years ago the Federal Government disposed of merit-based hiring and promotions, and went to strictly racist/sexist/political affiliation hiring practices.

    The result was that any smidgen of competency that might have previously existed in the Civil Service was stuffed in a gunny sack and drowned. Today our Federal Government is operated by the most incompetent boobs imaginable. There is no diversity in levels of competency — only one kind is on display — and that is the lowest level possible, approximately negative 12.

  5. Merit of the candidate should always be the deciding factor. However, this doesn’t happen in very large companies where they have a quota system. Banks and the public sector are good examples of this.
    I wonder if you’re female and disabled whether this counts as a double whammy! Or whether they have to fill two posts for two different quotas.

    I’m picturing some unusual people working underground in the bowels of large banks that can be dragged out when the quota man comes to call.

    I’d be mortified if I was one of a quota; what an insult. We are all told that this sort of thing goes on. I prefer not to take the chance, but if I were ever rejected on the basis of my disability I would consider this a lucky escape.
    Society’s attitude has changed considerably towards disability since I left school; not for the better in my view. So many people claim disability where there is no real claim simply for the advantage it gives them. It has made lazy people richer and hard working people less ready to believe disability at all.
    We were taught that you must be better, not as good as your sighted counterparts, in order to be accepted. This is the right approach. It is up to the individual to rise to the challenge. To expect society to make allowances is only to risk disappointment. An hour out in the city in a wheelchair or with a blind fold and a white stick for every eighteen year old would do much to help society and would be far cheaper to implement, freeing up the lawcourts for more important things.
    On second thoughts, that’s a stupid idea, there’d be all sorts of claims of injury and theft resulting from such silliness.

    There’s no solution. To line the lawyers pockets is not the answer.

  6. Mike D and Joy:

    Do yourself a favor and read the following.


    “The mark of a mature mind is the ability to suspend judgment until all the evidence is in.”

    “…The result is an attack on multiculturalism and the need for a sense of psychological superiority expressed in prejudice at the affective level.”

    I am just a troll, don’t bother feeding me. I stumbled on your blog via Google, Mr. William Briggs. I won’t come back to a site that attracts ignorant people. Remember, guilty by association is no nonsense.

  7. Briggs

    May 17, 2009 at 6:26 am

    So, Ann, on the chance you do come back and demonstrate an ability to have a mature mind (and not a self-satisfied ignorant one, confident she is already right), what exactly in my calculations did you find wrong?

  8. Looks like his comments were directed at Mike D and Joy.

    Your article and the numbers are an intersting read as usual. The problem comes when the topics you choose tend to draw polarized comments and people tossing in their personal experience which only applies to a fraction of reality.

    Comments like this “Merit of the candidate should always be the deciding factor.” are good, but the next statement selectively chooses one type of instance where this doesn’t happen, while neglecting to recognize the other reasons. “However, this doesn’t happen in very large companies where they have a quota system.” It also doesn’t happen in small companies who are not subject to government quotas.

    The main problem with your analysis is the underestimation of “some companies are filled with jerks who actually do discriminate based on characteristics other than ability”.

    In NYC this number may be low, in California the quotas and opposite bias may actually shift the numbers in the other direction, but come down south and spend several months. “Disparity” is rampant. the demographics of Memphis,TN are 65% black, but outside of national chain stores, county/city jobs and universities you will either see 100% white or 100% black employees.

  9. Speaking of poor use of statistics with respect to parity …

    Some years back I worked at a major state university that was concerned about pay equity between its male and female faculty. So they did a salary comparison study in which they corrected for many factors including educational background, length of employment, years of experience, number of publications, quality of research, national reputation, etc. When the study was complete they announced that the women faculty were being paid, on average, $178 per year less that their male counterparts after all of the corrections had been applied. But instead of saying “Wow, a difference of less than a quarter of one percent is astonishingly equal”, they instituted an elaborate salary equity program in which any female faculty member could ask to have her salary compared to a group of her male peers (who she got to choose), and any disparity would be corrected.

    I think that was the moment that I lost respect for academia.

  10. Ann, Ann, Ann. Trolls are people, too. Why be ashamed of your height disparity?

    The real question is: if I was to hire you, could you do the job better than a non-troll? Or am I supposed to pay you the standard wage for your sub-standard work, just because you claim to be a member of an allegedly oppressed class?

    I prefer not to do that, due to the fact that I am in a competitive business and saddling that business with incompetents would wreck it.

    I suggest you apply for a job with the government. They have no competition and thus no incentive to do anything right, or well, or at all. They are looking for trolls; I am not.

  11. Thanks for an interesting post. You could have gone a lot further. Try hiring 50% females for plumbing / labouring work. The Chinese and Indians should be delighted by your (and our – UK) restrictive rules. Well-meaning but simple-minded legislation will drive us into poverty eventually.

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