I'm very happy to see "The Breeder's Equation" on Hacker News. Wow. I spent my graduate career studying mathematical biology and working with biologists.
It's a fascinating equation and I spent some time on it converting it from a discrete equation into a continuous equation along with explaining the use cases and motivations behind it here:
http://fromdata.org/2013/10/27/the-continuous-breeders-equat...
This of course, does not take into account mutation, transduction, conjugation and other sources of genetic variation.
Edit: Please take these approximate expectation equations with a grain of salt. They only apply to larger, discrete-reproducing, controlled populations. Applying results like these on spatially segregated or small populations does not really work. Also this equation does break down a bit when the phenotypic trait is a result of very complicated interactions between genetics and the environment.
From a didactic perspective, I think he goes off the rails by immediately focusing on something as abstract and hard to measure as intelligence, and where there's a long history of people talking about whether it's even a single trait. In terms of just explaining the breeder's equation, I think he could have been more effective talking about, IDK, tall Dutch people.
The other thing that's annoying here is that although this is an 'equation', as I understand it it's typically not predictive because that heritability term is so fuzzy. It's not that you know the heritability term in advance and predict the response to selection; measuring the response to selection also updates your information about the heritability.
His examples have a cherry-picking feel for this reason. Take the "national merit finalists" thing he mentions multiple times. Note that he doesn't mention the IQ distribution of the college professors (surely it's logistically impractical to get any large number of busy faculty to spend time taking such tests) or of the college towns (which might not be the same as the overall population mean), or the degree to which being a professor is correlated with IQ or the degree to which being a national merit finalists is correlated to IQ. Had the data come out differently (and I suspect he mentions _finalist_ multiple times because there wasn't as distinct a signal among _winners_), this perspective leaves plenty of room to declare that (a) maybe heritability of intelligence is lower than initially thought or assortative mating along IQ lines is weaker than expected (b) maybe professor-ness is less correlated to IQ than assumed or (c) maybe national merit finalist status isn't especially correlated to IQ (semifinalists have to submit a bunch of info other than test results).
> (surely it's logistically impractical to get any large number of busy faculty to spend time taking such tests)
You can estimate it decently by looking at SAT scores, which most professors will have taken at some point. Your error bars will be larger than if you administered IQ tests to everyone, but it's a lot easier.
This kind of quibbling always comes up. He's just using professors and national merit finalists as examples.
The truth is, you could probably take the top 10 percent highest-earning car salesmen (EDIT: and women), drop them on a desert island and, after two generations, get the highest-IQ society the world has ever known. But that's less obvious, so why use such an example?
But it’s not clear at all that intelligence is a single trait. It is incredibly messy and this means any analysis of it has to be very careful of confounders. In contrast, height is by many definitions a single trait, we know exactly what confounds it’s measurement (and those things are easy to measure!) and we have a pretty good estimate of it’s heritability.
Does it matter if intelligence is a "single trait"? IQ, at least, is a single trait and it's empirically heritable. There has been a great deal of work to address the confounders (e.g. twin and adoption studies) and shore up the construct validity of IQ. Nobody thinks IQ = intelligence, but it's a measurable trait that says a lot about intelligence.
IQ isn't on the same level of definiteness as height, but I don't think you have a bright line of "trait validity" with height on one side and IQ on the other.
Remapping intelligence to certain cognitive performance parameters, many of which can be measured in a very metrical and objective way, still tends to produce correlations. See Figure 3 of this article: [0] - there is a mix of simple perceptual tasks and more complex reasoning ones, which on the surface appear independent. The paper admits at the end that intelligence isn't one kind of task, but instead many kinds of systems used together. That still doesn't prevent their ability to construct a version of g-factor.
And very simple metrics, taken alone, seem to inter-correlate well with IQ scores, like reaction time.[1] This correlation increases with age, suggesting a biological factor in preserving brain integrity that would nullify educational effects of intelligence.
If your version of intelligence has nothing to do with cognitive activity I don't know what to tell you.
It's true that effectiveness in society might not be correlated with intelligence. That says more about what society rewards and the kind of labor required to get it. And I don't mean to say that society should reward people based off of test scores, that's dumb. Nor should we assortate people based off of pure test scores, also dumb.
But I see no real conceptual problems regarding intelligence that doesn't try to conflate intelligence entirely with general effectiveness. There are IQ realists like Steven Hsu that don't do this, placing it on equal importance with conscientiousness [4]. As with height, there are innate and behavioral/developmental components of general effectiveness (I could dope my child with growth hormone to make him taller, or mal-nourish them to go the other way). As with emotional intelligence, being a bastion of full-nurture theorists for behavior that look for non-academic modes of measuring effectiveness, we can get heritability correlates of 0.2-0.4[2], and for psychopathy up to 0.5[3], mediated by biological factors like inherent oxytocin concentrations.
It's hard to say you can succeed without without any ability to learn, but there are high paying jobs that involve mainly physical labor (while taking on some mortality risk which can be managed). And there are people who are smart assholes, and other people who are dumb yet compassionate. So I have no problem with accepting the cognitive version of intelligence as with the right attitude there is no anti-social conclusion to risk. Society takes all sorts of people to exist, even if to some extent, the more well-rounded the better.
[0] "Fractionating human intelligence", www.owenlab.uwo.ca/pdf/2012%20-%20Hampshire%20-%20Neuron.pdf
Even more so: every time IQ and regression to the mean is explained, it is always in the “more-to-less direction”, whereas it works in both and both are relevant.
The author is playing fast and loose with definitions here. Other sources[0][1] (that I would consider more reputable) define the breeder's equation in terms of populations, but this author focuses on arbitrary and ill-defined subsets of populations.
Crucially, the author defines R as "the response to selection", but he omits the second part: "from one generation to the next". Source [0] defines it even more clearly: "the change in the mean [of the population] over one complete generation".
Similarly, S is the difference in the measured trait among the entire population and the population that reproduces. So when the author considers "a set of parents with IQs of 120", it only fits the correct usage of the equation if we take that to mean that mean IQ of all parents in this generation is 120. If that's his argument, how is he defining the population of parents?
In my opinion, this is a wishy-washy argument that seems like a subtle way of advancing eugenics (or something similarly distasteful).
I'd like to understand what this comment is saying, but I can't follow. Could someone please explain more clearly what exact intellectual mistake the author allegedly made? The exposition in the OP seems very simple and clear, and I cannot seem to relate the definitional issues raised here to what is said in the article.
In the article's first thought experiment, we take a population with IQ 100, pull out the subset with IQ 120, have that IQ 120 subset breed with one another. According to the breeder's equation, we get children with average IQ 110. What is "arbitrary", "ill-defined", "fast and loose", etc. about this? It seems to be a standard application of the equation, no different than how one might breed cows for milk or tomatoes for size. It's quite unclear how the "crucial" clause "from one generation to the next" undermines any of this.
And could we perhaps focus on understanding what was said before muddying the waters with unsubstantiated accusations of eugenics? HN rules say that HN is for learning and understanding, not ideological warfare.
The breeder's equation described the expected change in a _population_ from one generation to the next due to selection pressures (natural or artificial).
For example, say a cattle rancher with a large herd may want to increase the average weight. So if he only breeds those animals in the top half of weight, what is the expected change in weight from one generation to the next?
For another example, say a particular species of lizard is hunted by a species of bird. The faster lizards are more likely to escape under a rock than the slower lizards. What is the expected change in average speed from one generation to the next?
Both of these examples have a well-defined _overall_ population and _reproducing_ population. The value of S can be calculated. In the first example, it is the difference between the mean weight of the top half of cattle and the mean weight of the entire herd. In the second example, it is difference in the mean speed of the lizards that are able to reproduce before being eaten and the entire population.
What are the analogous groups in the author's example? He doesn't define what distinguishes the population of 120 IQ parents from the population as a whole. In one reasonable reading, you could even think he means just two people when he says "a set of parents". That is what I mean about being fast and loose with terms: how are we defining the entire population and how are we defining the reproductive population?
Further, he says that 120 IQ parents having children with mean IQ of 110 is an example of regression to the mean. I would say the exact opposite: the 110 IQ children _define the mean_ of the next generation (in the correct usage of the equation). The expectation is that if there continues to be positive pressure on IQ, then future generations will continue to have increasing IQs.
With regards to "ideological warfare", the author himself explicitly introduced eugenics into the conversation with his analogy about the desert island populated by National Merit finalists (he literally described it as eugenics). His Wikipedia page [0] describes him as an anthropologist "who argues that cultural innovation resulted in new and constantly shifting selection pressures for genetic change, thereby accelerating human evolution and divergence between human races". I don't think it's unfair to say there are some unpleasant undertones to his work.
> He doesn't define what distinguishes the population of 120 IQ parents from the population as a whole.
...Surely what distinguishes them is that they have IQs of 120, whereas the parent population has average IQ of 100?
> Further, he says that 120 IQ parents having children with mean IQ of 110 is an example of regression to the mean. I would say the exact opposite: the 110 IQ children _define the mean_ of the next generation (in the correct usage of the equation).
Surely you understand there is not only one mean in play here? There is also the mean of the population from which the IQ 120 subset was drawn. It is their mean that "regression to the mean" normally refers to, and has referred to since Galton invented the concept in 1886.
> With regards to "ideological warfare", the author himself explicitly introduced eugenics into the conversation with his analogy about the desert island populated by National Merit finalists
Not all things "eugenics" are inherently evil. Genetic screening of embryos to avoid infant suffering from horrible genetic diseases is also eugenics. Should mothers not have a choice to save their children from horrible genetic diseases?
It's ideological warfare to paint all eugenics with the same moral brush, as all having "unpleasant undertones". There is no moral flaw, per se, in seeking to understand how desirable traits might be increased in the human species.
I can't speak on whatever research the author has done on "human races", but it does not seem relevant to the discussion of the breeder's equation or to eugenics in the broad, perfectly benign sense of increasing desirable traits in a population.
"And of course the breeder’s equation explains how average IQ potential is declining today, because of low fertility among highly educated women."
In the Heinlein universe, the Howard Foundation runs a voluntary eugenics program by paying people with particularly long lived ancestors to mate with each other. Protagonist Lazarus Long is a product of this.
The author is saying that this would work as expected, with offspring gaining around 50% of their parents' extra longevity.
I'm a firm opponent of any kind of coercive eugenics. But it seems like it would be a good use of a billionaire's fortune to establish such a foundation for the promotion of intelligence, by paying very smart people to make babies together.
But in Heinlein, the "Howards" became reviled and were forced off of the planet. If we do create a particularly smart population by eugenics, perhaps their primary task should be to find a way to protect themselves from us normals, a treatment for tall poppy syndrome. Of course, the first resort is secrecy. So we wouldn't know if they already exist.
An IQ is just a number coming out of a test. Whether or not that correlates to something useful for humanity, or even the individual being tested, is a different question. Surely the majority of useful discoveries are made by people with above-average IQs, but would juicing the numbers through eugenics really make much of a difference?
How many extra Isaac Newtons would it take to make controlled nuclear fusion work? Because if we don't have an estimate for that, why would we expect an intelligence-focused eugenics program to be useful at all?
>why would we expect an intelligence-focused eugenics program to be useful at all?
Are there no uses of a having a higher population average intelligence besides "does it immediately produce nuclear fusion"? Is higher average intelligence good for many many many other things besides producing fusion? Of course it is, this is an absurd premise.
Also, where did nuclear fission get invented? Are there correlated differences in IQ of people from this region? Did nuclear fission get invented in areas with low average IQ?
"An IQ is a just a number coming out of a test". Oh, but it does correlate with other group-wide achievements that aren't just numbers coming out of a test, does it not?
>Surely the majority of useful discoveries are made by people with above-average IQs
How it is not obvious then that increasing the average would also increase the level of the outliers and their ceiling?
Not to mention the regression to the mean issue discussed in the article, and the slight problem of attracting women aware that they have a very high IQ to trade career success or other personal preferences for being the baby making machine for some billionaire's experiment. It's not like they're currently disproportionately likely to choose not to have kids because they can't afford them
>to trade career success or other personal preferences for being the baby making machine for some billionaire's experiment
Why is it assumed that this isn't a more attractive proposition compared the alternative. Certainly there would be many, if not most or all, that would be willing to make this choice.
The point you are implying is that "career success and personal goals minus children" > "making smart babies and being a mother"...where ">" can be more enjoyable or fulfilling. It's not clear that this is true. Part of "baby making" means leveraging that IQ in a productive way to raise strong, intelligent children, so it would not be going to waste.
I’m a woman, and none of my smartest friends are interested in having more than 2 children. A few aren’t interested in having any. I’m the weirdo who wants 3, but I don’t see why I should be part of someone’s experiments when I can find a high quality mate on my own anyway.
Yeah, it's interesting, humans already HAVE an incredibly complex sexual selection mechanism. There's all kinds of things that you are capable of selecting for without necessarily even realizing what you're doing. That selection pressure likely already includes the relevant things such a billionaire might care about, and many other things they can't even quantify, which likely includes genetic compatibility. The system we have has worked pretty well for a long time. It's hard to optimize such a complex and fuzzy thing.
So maybe the billionaire should enact his horrific eugenics experiment by creating a dating app to help women like you find high-quality men who want lots of kids.
He'd even make money, because... well, be honest, how much would you pay for that if it really worked?
> Why is it assumed that this isn't a more attractive proposition compared the alternative. Certainly there would be many, if not most or all, that would be willing to make this choice.
I'm not sure why you're asking me this when your opening post in this exchange starts by noting low fertility amongst highly educated women. Smart women generally aren't short of offers from smart men to make babies with them (and the smart men are usually pretty wealthy). If "many, if not most or all" wanted to have lots and lots of kids instead of being academics or lawyers or rocket scientists, they'd already be doing it.
I don't think a creepy eugenics foundation that's obsessed with the IQ of their kids makes the proposition more attractive even if they're paying enough for a slightly bigger mansion and the husband to afford to retire too.
The little secret is that the highly successful do have many children. Like, Elon Musk has six kids. It's only a class of upper-middle-class workers who, like high-ranking court eunuchs, valorize work over family.
Some highly successful people have lots of children.
But Elon Musk is a little bit too rich to make his decisions on partners and kids based on whether some poorer billionaire offers a stipend. My point is upper middle classes intellectuals won't be interested trading those decisions for the money either
When affluent and educated societies worldwide show declining birthrates and increasing age at reproduction, it can be taken as a strong indicator of women's and family's preferences for fulfilling lives at demographic scale.
> The author is saying that this would work as expected, with offspring gaining around 50% of their parents' extra longevity.
Not for lifespan. What the author didn't mention is that this equation works for traits affected by a large number of genetic variants, each with small, roughly additive effects. Most heritable traits are like this, and if you plot the probability distribution for one, it looks like a normal distribution. Lifespan, on the other hand, is not at all normally distributed and you shouldn't expect it to work the same way as things like height.
Life span can be easily translated into standardized units more convenient to talk about which the breeder's equation would apply to, the same way binary or categorical or polytomous traits can often be treated as a normal latent with thresholds. And people calculate heritability of lifespan all the time. (It's low, but that's not because lifespan happens to be Gompertz-distributed.)
This wouldn't work for "longevity" because "longevity" isn't a bell curve.
If "longevity" were a bell curve, we would see some humans at the 150 years old mark. We do not. Human longevity has a "cliff" that it won't go past.
Selective breeding could get you arbitrarily close to whatever that cliff actually is, but no further.
To get further than that cliff, you would have to fix all the issues that cause it. There is no guarantee (and practically an anti-guarantee) that selective breeding would ever accomplish that.
> If "longevity" were a bell curve, we would see some humans at the 150 years old mark. We do not.
I agree with everything else (including that its not a bell curve), but this particular reasoning seems off — it could still follow a bell curve even if we never see the rarer items… because we’re only talking about approximating a bell curve. Just as height’s bell curve suggests a 30ft human could eventually occur, but we’d never actually expect it to actually happen in reality (because confounding factors will start to trigger) - but we still say height follows a bell curve.
No, the bell curve for height does not suggest a 30ft human would ever occur. That would be somewhere around 90+ standard deviations out--that's so wildly improbable as to be effectively 0. A "mere" 7 standard deviations is effectively 1 in a billion. And that's somewhere around a 7 1/2 foot tall human (who we do see occasionally).
"Standard deviation of lifespan" (a somewhat erroneous idea since we're not actually on a bell curve) is somewhere around 10 years. 150 years old is 7 standard deviations--improbable but not impossible. If we're following a bell curve, 1 in a billion people should be 140 years+ and 1 in 10,000 should be 120 years+.
The fact that we don't see these probabilities is extremely strong evidence we're not on a bell curve. And not being on a bell curve immediately throws out huge chunks of statistical "rules".
I’d expect if we find the kids of professors to be more likely to be National Merit Finalists that we’d find a non-zero upbringing/educational correlation. Sure there’s a genetic component, but I’d be fairly shocked if “being good on academic testing” has no correlation with “being raised by academics”.
There are studies of twins raised separately that directly address this. We know a great deal about the genetic heritability of IQ, please have a look at this for more info: https://en.wikipedia.org/wiki/Heritability_of_IQ
Extreme skepticism of social science findings is warranted for some subfields, but when it comes to intelligence, humanity has actually learned a lot.
Of course it would be correlated. Genes and environment and culture and family are all just one big interrelated mess. Is it even possible to tease the contributions apart, even in theory?
I guess I could imagine a study, something like: look at kids of professors, in particular identical twins vs fraternal. Not all children will be National Merit Finalists, but among those that are, do you find identical twin pairs more often? Even that is subject to environmental effects, though, because potentially identical twins are raised differently!
I agree with your last, which is very unfortunate in that if we could figure a way via research to shift upward the effectiveness of cognition (for which IQ is a proxy measure), I think peoples lives overall would be improved.
Some of these are interventions that we’ve already found to prevent wasting of ability (as you cite in another comment); others might be nutritional or environmental changes we could make to better support development in utero or as children develop.
Giving everyone 5% “better brains” would be a huge win for the world, but the sociological factors you mention preclude widespread scientific study.
You can't boost IQ, but you can stave off its decline with exercise. The reason why there's no incentive to do this is that no one wants smarter voters.
I agree with you but not the article which claims (without introducing evidence): “ By the way, when we say "environmental,” we mean “something other than additive genetics.” It doesn’t look as if the usual suspects—the way in which you raise your kids, or the school they attend—contribute much to this "environmental" variance, at least for adult IQ.”
Perhaps studies of early adopted children would another pathway for study, but there seems to be little appetite for IQ study in general.
I remember, without having googled anything additionally, studies where Malaria infections and lead exposition had a negative effect on final IQ, whereas deworming treatments and longer school attendance had a positive effect.
Wouldn't the 120 IQ subpopulation also have some people whose IQs are "genetically" more like 130 but they've gotten unlucky with environmental factors, pushing the genetic average a bit towards 120? It would be smaller than the number of 110 people who got lucky, but non-zero. Is this baked in to h^2?
> It also explains why the professors’ kids are a disproportionate fraction of the National Merit Finalists in a college town. […] But those kids, although smarter than average, usually aren’t as smart as their *fathers*: partly because their *mothers* typically aren’t theoretical physicists, partly because of regression towards the mean.
Wow, this guy is … not concerned with political correctness/not considering whether or not the college professors worth mentioning are guaranteed to be male.
That’s not right, either, since he is speaking in the present tense and comparing his current local high school’s attempts to emulate what Los Alamos high currently does:
> Reminds me of the fact that Los Alamos High School has the highest test scores in New Mexico. Our local high school tried copying their schedule, in search of the secret. Didn’t work. I know of an approach that would, but it takes about fifteen years.
I think you're simply reading too much into his imprecise usage of grammatical tense.
"Our local high school" probably refers to when he was in high school himself decades ago, being born in 1953, that would put those years right around the time period when children of Manhattan Project physicists have demonstrated their academic talent.
Given the fact that Los Alamos High School is not academically renowned in the current day, but was so half a century back, this is the only context in which his series of statements make any sense.
If you select for a single, measurable trait, perhaps. We do this all the time, with limited success, in breeding lab test animals, dog breeds and such.
But for survivability eugenics is hard: there are multiple dice rolls each cycle, not just the chromosomal mixing, but environment too, and that adds not just epigemetic change but different environments for which various traits might be temporarily valuable.
You can see this demonstrated by the inbreeding of the royal houses of Europe. They have recognized this too as they have been widening their gene pool for decades.
Even the magic island experiment described from which I extracted that sentence would suffer under that too.
This is an understatement, just considering SNPs, there is at least several hundred thousand statistically independent coin flips when two parents breed. It is difficult to overstate how powerful an evolutionary tool this is, and the implications it has for breeding.
This is a pretty poor quality article. I say this as someone who works in a population genetics research group. If you like some actually thoughtful reading on genetics and eugenics I would point you towards gwern who, as far as I can tell, seems to at least get the science and math right.
Could we trouble you for a little detail about what's wrong with the article? It seems very clear and understandable for the layman, so it would be good to have an equally clear explanation of what's wrong with it.
Like most looming disasters, whether or not this is a real problem depends on quantifying the problem. Is this likely to significantly affect the population intelligence before we learn to engineer our children, say within the next 200 years?
I always find these statements interesting. Do you think many people 200 years ago based their actions on what effect they would have on those people 200 years hence?
Oh I'm not saying we will. I'm just saying that even taking the claimed effect at face value, it only has significance if it can hurt our genetic intelligence faster than engineering can improve it.
Two people with IQ's of 120 will have kids with average IQ's of 110, but if you put those kids on an island and they have kids their average IQ's will be 110. What?
In the first example, their IQ of 120 is from a mix of factors and the author estimates some percentage that is due to the environment outside genetics, and that is what is lost in reverting to the mean of the environment. In the second example the author assumes that the environment of the desert island contributes nothing to genetics and so the offspring get the full 110 (heritability is 100%).
It's odd to me though because it wasn't clear that the environment was meant to be affecting the parent and not the child too- the trick is the author treats the heritability parameter different for the parent and the child. Also, I don't think the desert island would actually have no effect on the 110 generation's IQ, or their upbringing outside of it, but I guess I can accept it as a thought-experiment-only place that does that by definition and not a literal desert island.
The parents genes are only 120 because 10 points were based on environmental "luck". Genetically they had the genes for 110 IQ +/- 10 points to accommodate the environmental luck.
The two kids had average luck, so they got 110 IQ, but they still have the same genes. So when they have kids with others with the 110 IQ gene's, they will still be 110 +/- 10.
A simpler explanation: you might be the child of geniuses but they dropped you on your head repeatedly as a baby. You might not be so smart as an adult because of it but your kids will have genius genes, not dropped on head genes.
The heritability of intelligence is overhyped, and in my opinion of no practical use.
It's extremely hard to find data where it is even possible to distinguish between genetic and environmental influence. That would be the studies of the kind "twins raised apart". And that already excludes the reality that those twins will probably grow up in a similar background, similar health and educational system etc.
The talk about heritable IQ is pointless, even dangerous, because it leads to complacency around issues around structural racism and equality in education.
>The talk about heritable IQ is pointless, even dangerous, because it leads to complacency around issues around structural racism and equality in education.
Can you say more about this? Why would it lead to complacency? If there is scientific merit to some ideas like IQ heritability, don't you think we need to find a way to incorporate those ideas safely into our formulation of society, rather than suppress them as heretical?
You never heard about the trope of African people being called genetically inferior? That one is reverberating around time and space in a million variations.
And if intelligence is genetic, well, you don't need to improve their public schools to help them achieve equal grades, do you?
My understanding of IQ is that its not fully heritable... an individual's potential is maximized by their environment, up to some limit. This alone gives a strong argument for improving public schools especially where students may be falling behind. So I am still not understanding the complacency aspect fully. I suppose a bigoted person could say "well those kids are dumb, they're expected to do bad." But the counter to that is, "they can do much better than they're doing," and the science would support it.
The usual studies for disentangling genetics and environment involve twins raised together: they get basically the same shared environment -- this is a feature, not a bug -- but identical twins are more closely related than fraternal twins.
For example, say you're trying to find out how much of adult height comes down to genetics (in a modern, non-malnourished environment). You measure the heights of a bunch of twins and compare: how much more similar are the heights of identical twins than the heights of fraternal twins? Do a bit of calculating and you've got a number for heritability.
Yes. But then people take that street number from a "twins raised together" study and quote it as heritability of intelligence overall.
My problem with this: Yes, if you put extreme limits on the environmental factors, you are mostly left with genetic factors. But that is not the reality of Human existence.
This is dangerous view IMO. Knowledge should not be suppressed just because some people might interpret it incorrectly.
If you fear the problem is "complacency around issues around structural racism and equality in education" then why not address those problems head on rather than propose to not talk about heritable IQ?
I mean both structural racism and equality in education aren't at all difficult problems. They are only made difficult because large part of population is still effectively racist, has racist role models and has approval from government officials.
It apparently is ok for people to be racist.
If you are racist today, you can happily live with friends that have the same view and tune out the rest of the world that does not agree with you.
And as long as it is ok for people to be racist they will always find some kind of excuse for their beliefs whether it is heritable IQ or something else.
It's a fascinating equation and I spent some time on it converting it from a discrete equation into a continuous equation along with explaining the use cases and motivations behind it here: http://fromdata.org/2013/10/27/the-continuous-breeders-equat...
This of course, does not take into account mutation, transduction, conjugation and other sources of genetic variation.
Edit: Please take these approximate expectation equations with a grain of salt. They only apply to larger, discrete-reproducing, controlled populations. Applying results like these on spatially segregated or small populations does not really work. Also this equation does break down a bit when the phenotypic trait is a result of very complicated interactions between genetics and the environment.