{"id":520,"date":"2023-05-09T08:42:27","date_gmt":"2023-05-09T08:42:27","guid":{"rendered":"https:\/\/milkyeggs.com\/?p=520"},"modified":"2023-10-30T22:53:48","modified_gmt":"2023-10-30T22:53:48","slug":"estimated-iq-distribution-of-children-given-iq-of-parents","status":"publish","type":"post","link":"https:\/\/milkyeggs.com\/biology\/estimated-iq-distribution-of-children-given-iq-of-parents\/","title":{"rendered":"Estimated IQ distribution of children given IQ of parents"},"content":{"rendered":"\n<p>Let&#8217;s try to estimate the distribution of the &#8220;genetic IQ&#8221; of some offspring assuming that we know the parental &#8220;genetic IQ.&#8221; Following Hsu (2014), <a href=\"https:\/\/arxiv.org\/abs\/1408.3421\">On the genetic architecture of intelligence and other quantitative traits<\/a>, let us start with the following assumptions:<\/p>\n\n\n\n<ul>\n<li>A number of independent, additive alleles determine your IQ<\/li>\n\n\n\n<li>The &#8216;major&#8217; (+, higher probability) version of each allele increases your IQ, whereas the &#8216;minor&#8217; (-, lower probability) version lowers your IQ<\/li>\n\n\n\n<li>Each (+) or (-) effect is equal in magnitude<\/li>\n\n\n\n<li>The frequency of the minor allele is approximately <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-201682e11dcd0ed88ee679cd124048c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#32;&#61;&#32;&#48;&#46;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"65\" style=\"vertical-align: -4px;\"\/><\/li>\n\n\n\n<li>Each standard deviation of IQ difference corresponds to differences in about 40 SNPs<\/li>\n\n\n\n<li>Therefore, we can calculate that approximately 10,000 alleles control IQ<\/li>\n<\/ul>\n\n\n\n<p>These are inexact but plausible, &#8220;good enough&#8221; assumptions; see the paper for a more detailed discussion.<\/p>\n\n\n\n<p>Note that this ignores environmental contribution, does not account for regression to the mean, and so on; it is purely an attempt to work through the math, in a &#8220;bottom-up&#8221; approach, deriving the distribution of genetic outcomes expected in the offspring given what we know about the additive genetic architecture of intelligence.<\/p>\n\n\n\n<p>A person with exactly average IQ would have 8.5k (+) alleles and 1.5k (-) alleles. Similarly, a person with 1 SD higher IQ would have 40 more (+) alleles and 40 fewer (-) alleles.<\/p>\n\n\n\n<p>Suppose that two parents have IQs of <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/> standard deviations away from the mean. Then:<\/p>\n\n\n\n<ul>\n<li>Parent 1 has <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-07bfcea60b0621257120e133b3af7091_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#53;&#48;&#48;&#32;&#43;&#32;&#52;&#48;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"85\" style=\"vertical-align: -2px;\"\/> (+) alleles and <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-3afdf6dbc3fb323dbe481e02aa835c1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"84\" style=\"vertical-align: 0px;\"\/> (-) alleles<\/li>\n\n\n\n<li>Parent 2 has <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-a9ddb917a756daab45da28a49771d319_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#53;&#48;&#48;&#32;&#43;&#32;&#52;&#48;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\"\/> (+) alleles and <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-03f4109cc30a4d542b122dbbd12ad1fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -4px;\"\/> (-) alleles<\/li>\n<\/ul>\n\n\n\n<p>Our goal is to calculate the distribution of (-) alleles which a child will inherit from these parents.<\/p>\n\n\n\n<p><strong>Theoretical calculation<\/strong><\/p>\n\n\n\n<p>Assume that the genotypes of parents 1 and 2 are completely independent, and that at each allele, the child has a 50\/50 chance of inheriting each parent&#8217;s copy of that allele.<\/p>\n\n\n\n<p>At the alleles where both parents 1 and 2 are (-), the child will definitely inherit a (-) allele. The number of such positions follows a distribution<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 19px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-c02be62a909ed36cecb9b71a2240597b_l3.png\" height=\"19\" width=\"409\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#88;&#32;&#92;&#115;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#72;&#121;&#112;&#101;&#114;&#103;&#101;&#111;&#109;&#101;&#116;&#114;&#105;&#99;&#125;&#40;&#49;&#48;&#48;&#48;&#48;&#44;&#32;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#44;&#32;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#121;&#41;&#46;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>For intuition, in the case where <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-67ea41b13cc0fd38677a37a8a29e9310_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#32;&#61;&#32;&#121;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -4px;\"\/> (parents have average IQ), this distribution has mean <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-ce8fc33e6e6dc0fd37a325b47cf2275c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#117;&#95;&#88;&#32;&#61;&#32;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\"\/> and variance <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-1cfdbaa234874fa664e3f3bedddd1f51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#50;&#95;&#88;&#32;&#61;&#32;&#49;&#54;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"73\" style=\"vertical-align: -5px;\"\/>. In general, of course, we have<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 38px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-44a4acacfe5598f744654904dff731e2_l3.png\" height=\"38\" width=\"246\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#92;&#109;&#117;&#95;&#88;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#40;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#41;&#40;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#121;&#41;&#125;&#123;&#49;&#48;&#48;&#48;&#48;&#125;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>and<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 43px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-312b3b1ced28800b6fe57d15e6d92f91_l3.png\" height=\"43\" width=\"452\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#50;&#95;&#88;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#40;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#41;&#40;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#121;&#41;&#125;&#123;&#49;&#48;&#48;&#48;&#48;&#125;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#40;&#56;&#53;&#48;&#48;&#32;&#43;&#32;&#52;&#48;&#120;&#41;&#40;&#56;&#53;&#48;&#48;&#32;&#43;&#32;&#52;&#48;&#121;&#41;&#125;&#123;&#49;&#48;&#48;&#48;&#48;&#32;&#40;&#49;&#48;&#48;&#48;&#48;&#32;&#45;&#32;&#49;&#41;&#125;&#46;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>Assuming that <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> (-) alleles overlap, the rest of course do not overlap. That means that:<\/p>\n\n\n\n<ul>\n<li>Parent 1 has <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-461b4121eac9601599dd5467ff21139f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#32;&#45;&#32;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"122\" style=\"vertical-align: 0px;\"\/> (-) alleles paired with Parent 2&#8217;s (+) alleles<\/li>\n\n\n\n<li>Parent 2 has <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-b53c98499cfd24db2788bf7bfbf37c18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#121;&#32;&#45;&#32;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"121\" style=\"vertical-align: -4px;\"\/> (-) alleles paired with Parent 1&#8217;s (+) alleles<\/li>\n<\/ul>\n\n\n\n<p>At each position, we basically have a fair coin flip for the (+) or (-) allele, and here the number of (-) alleles should follow a distribution<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 19px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-ccbe112331bbe5d0f1fe58a8fa6d7fdc_l3.png\" height=\"19\" width=\"367\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#89;&#32;&#92;&#109;&#105;&#100;&#32;&#88;&#32;&#92;&#115;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#125;&#40;&#51;&#48;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#32;&#45;&#32;&#52;&#48;&#121;&#32;&#45;&#32;&#50;&#88;&#44;&#32;&#48;&#46;&#53;&#41;&#46;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>Calculating the mean is easy, because we are just taking the expectation in an additive way, and so<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 78px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-622f517505c113b960953d45290f082b_l3.png\" height=\"78\" width=\"237\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;&#92;&#109;&#117;&#95;&#89;&#32;&#38;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#32;&#45;&#32;&#52;&#48;&#121;&#32;&#45;&#32;&#50;&#92;&#109;&#117;&#95;&#88;&#125;&#123;&#50;&#125;&#92;&#92;&#38;&#61;&#32;&#49;&#50;&#55;&#53;&#32;&#45;&#32;&#49;&#52;&#120;&#32;&#45;&#32;&#49;&#52;&#121;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#121;&#125;&#123;&#50;&#53;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>For the variance, let us simplify the algebra by defining <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-def22be95a0c4a58edc1bc20ac80f35f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;&#39;&#32;&#92;&#115;&#105;&#109;&#32;&#51;&#48;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#32;&#45;&#32;&#52;&#48;&#121;&#32;&#45;&#32;&#50;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"223\" style=\"vertical-align: -4px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-c51bbf355382d3d76d8aaeb8418de6db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#32;&#61;&#32;&#49;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"\/>, so that <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-940898e3b1adc4e83545629714c5c415_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;&#32;&#92;&#109;&#105;&#100;&#32;&#89;&#39;&#32;&#92;&#115;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#125;&#40;&#89;&#39;&#44;&#32;&#112;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"190\" style=\"vertical-align: -5px;\"\/>. Then from the law of total variance,<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 150px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-c2f99f7e611d50a3f3a3e2f099268ae5_l3.png\" height=\"150\" width=\"347\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#86;&#97;&#114;&#125;&#91;&#89;&#93;&#32;&#38;&#61;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#86;&#97;&#114;&#125;&#91;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#125;&#91;&#89;&#32;&#92;&#109;&#105;&#100;&#32;&#89;&#39;&#93;&#93;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#125;&#91;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#86;&#97;&#114;&#125;&#91;&#89;&#32;&#92;&#109;&#105;&#100;&#32;&#89;&#39;&#93;&#93;&#32;&#92;&#92;&#38;&#61;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#86;&#97;&#114;&#125;&#91;&#112;&#89;&#39;&#93;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#125;&#91;&#112;&#94;&#50;&#89;&#39;&#93;&#32;&#92;&#92;&#38;&#61;&#32;&#112;&#94;&#50;&#32;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#86;&#97;&#114;&#125;&#91;&#89;&#39;&#93;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#125;&#91;&#89;&#39;&#93;&#41;&#32;&#92;&#92;&#38;&#61;&#32;&#112;&#94;&#50;&#32;&#40;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#50;&#95;&#88;&#32;&#43;&#32;&#51;&#48;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#32;&#45;&#32;&#52;&#48;&#121;&#32;&#45;&#32;&#50;&#92;&#109;&#117;&#95;&#88;&#41;&#32;&#92;&#92;&#38;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#50;&#95;&#88;&#32;&#43;&#32;&#51;&#48;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#32;&#45;&#32;&#52;&#48;&#121;&#32;&#45;&#32;&#50;&#92;&#109;&#117;&#95;&#88;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>Now, the total number of (-) alleles the child inherits follows a distribution <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-403a33a872120f764463ed8c369e597c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#90;&#32;&#92;&#115;&#105;&#109;&#32;&#88;&#32;&#43;&#32;&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"88\" style=\"vertical-align: -2px;\"\/>, with a mean<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 92px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-db926a180459677e84ba0b2621fbd6ed_l3.png\" height=\"92\" width=\"478\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;&#92;&#109;&#117;&#95;&#90;&#32;&#38;&#61;&#32;&#92;&#109;&#117;&#95;&#88;&#32;&#43;&#32;&#92;&#109;&#117;&#95;&#89;&#32;&#92;&#92;&#38;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#40;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#41;&#40;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#121;&#41;&#125;&#123;&#49;&#48;&#48;&#48;&#48;&#125;&#32;&#43;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#49;&#50;&#55;&#53;&#32;&#45;&#32;&#49;&#52;&#120;&#32;&#45;&#32;&#49;&#52;&#121;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#121;&#125;&#123;&#50;&#53;&#125;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92;&#38;&#61;&#32;&#49;&#53;&#48;&#48;&#32;&#45;&#32;&#50;&#48;&#120;&#32;&#45;&#32;&#50;&#48;&#121;&#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>and a variance<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 201px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-fc4c3279851ba842a07d0e655135f8a7_l3.png\" height=\"201\" width=\"353\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#90;&#94;&#50;&#32;&#38;&#61;&#32;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#88;&#94;&#50;&#32;&#43;&#32;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#89;&#94;&#50;&#32;&#92;&#92;&#38;&#61;&#32;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#50;&#95;&#88;&#32;&#43;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#92;&#115;&#105;&#103;&#109;&#97;&#94;&#50;&#95;&#88;&#32;&#43;&#32;&#51;&#48;&#48;&#48;&#32;&#45;&#32;&#52;&#48;&#120;&#32;&#45;&#32;&#52;&#48;&#121;&#32;&#45;&#32;&#50;&#92;&#109;&#117;&#95;&#88;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92;&#38;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#94;&#50;&#40;&#52;&#121;&#94;&#50;&#43;&#55;&#48;&#48;&#121;&#45;&#51;&#49;&#56;&#55;&#53;&#41;&#125;&#123;&#52;&#57;&#57;&#57;&#53;&#48;&#48;&#125;&#32;&#92;&#92;&#38;&#92;&#113;&#117;&#97;&#100;&#32;&#43;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#48;&#120;&#40;&#55;&#48;&#121;&#94;&#50;&#43;&#50;&#50;&#53;&#49;&#121;&#45;&#49;&#52;&#51;&#50;&#55;&#50;&#53;&#41;&#125;&#123;&#52;&#57;&#57;&#57;&#53;&#48;&#48;&#125;&#32;&#92;&#92;&#38;&#92;&#113;&#117;&#97;&#100;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#55;&#53;&#40;&#51;&#52;&#48;&#121;&#94;&#50;&#43;&#49;&#53;&#50;&#56;&#50;&#52;&#121;&#45;&#49;&#49;&#50;&#48;&#56;&#53;&#50;&#53;&#41;&#125;&#123;&#52;&#57;&#57;&#57;&#53;&#48;&#48;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>Intuitively, we see that the resulting IQ of the child is distributed with a mean that is exactly equal to the average of the parents&#8217; IQs. This is not too surprising. The more interesting component is the variance; because 40 (-) alleles correspond to one phenotypic standard deviation in the population distribution (i.e., a difference of 15 IQ points), a variance of <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-72b89e9d1d01891d8b7f0731dd7c2162_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#90;&#94;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"20\" style=\"vertical-align: -5px;\"\/> (-) alleles corresponds to a standard deviation of <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-5f4ade3f89b54524fb1fc1c87903e410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#90;&#94;&#50;&#125;&#47;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"83\" style=\"vertical-align: -11px;\"\/> IQ points.<\/p>\n\n\n\n<p><strong>Empirical results<\/strong><\/p>\n\n\n\n<p>Suppose that <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-d2a42b622186a56a48af302a8dea81be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#32;&#61;&#32;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\"\/>, so that the parents have identical IQs. Then for all reasonable choices of <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-40ca3e51a9803f260de25590c47d925e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#32;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\"\/>, the child&#8217;s IQ will vary with a standard deviation of about 10.5 to 11.5 IQ points:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image.png\" alt=\"\" class=\"wp-image-551\" width=\"436\" height=\"260\" srcset=\"https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image.png 871w, https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-300x179.png 300w, https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-768x458.png 768w\" sizes=\"(max-width: 436px) 100vw, 436px\" \/><\/figure><\/div>\n\n\n<p>It in fact looks that for <em>any<\/em> reasonable choices of <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-40ca3e51a9803f260de25590c47d925e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;&#32;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: -4px;\"\/> (not necessarily identical), the child&#8217;s IQ will vary with a standard deviation of about 10.5 to 11.5 IQ points:<br><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-1.png\" alt=\"\" class=\"wp-image-552\" width=\"376\" height=\"168\" srcset=\"https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-1.png 751w, https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-1-300x134.png 300w\" sizes=\"(max-width: 376px) 100vw, 376px\" \/><\/figure><\/div>\n\n\n<p>This is actually a pretty large range of variation: nearly as high as the population standard deviation! Furthermore, there is an environmental component to intelligence; the genetic heritability in adulthood is estimated by Vinkhuysen <em>et al.<\/em> (2012), <a href=\"https:\/\/link.springer.com\/article\/10.1007\/s10519-011-9507-9\">Reconsidering the Heritability of Intelligence in Adulthood: Taking Assortative Mating and Cultural Transmission into Account<\/a> as 0.82, raising our estimates of the standard deviation of the child&#8217;s IQ to ~12 IQ points.<\/p>\n\n\n\n<p>There are various effects here that may affect the quality of our estimate. For example, parents should have very positive genetic correlation (<em>e.g.<\/em> similar ethnic group), leading the variance quoted here to be a modest <em>over<\/em>-estimate. Parents&#8217; intelligence levels may themselves be subject to measurement error, and the parental IQ itself has an environmental component, effects which lead us to <em>under<\/em>-estimate the child variance. Ultimately, while the specific number may be not be precise, the argument here nevertheless seems to suggest that IQs of children are sampled from quite a wide distribution, with a standard deviation potentially well exceeding &gt;50% of the population distribution&#8217;s standard deviation.<\/p>\n\n\n\n<p><strong>Real-world significance<\/strong><\/p>\n\n\n\n<p>What is the significance of this result? It means that IQ variation within families can be quite high. For example, suppose that two middle-class professionals have a child. What might the IQ of the parents be? From Wolfram (2023), <a href=\"https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/S0160289623000363\">(Not just) Intelligence stratifies the occupational hierarchy: Ranking 360 professions by IQ and non-cognitive traits<\/a>:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-2-1024x714.png\" alt=\"\" class=\"wp-image-553\" width=\"512\" height=\"357\" srcset=\"https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-2-1024x714.png 1024w, https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-2-300x209.png 300w, https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-2-768x536.png 768w, https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-2-1536x1071.png 1536w, https:\/\/milkyeggs.com\/wp-content\/uploads\/2023\/05\/image-2-2048x1429.png 2048w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/><\/figure><\/div>\n\n\n<p>Let&#8217;s say that both parents are <em>above-average<\/em> &#8216;physical scientists&#8217; with IQs of 120 or so. (This category may include lab technicians and so on, but it also includes PhDs, researchers, <em>etc.<\/em>). Their children will have IQ distributed with a mean of 120 points and a standard deviation of 12 points. Assuming for simplicity that their children have normally distributed IQ (which is maybe not quite the case, but should be close enough for illustration), a full 5% of their children will have IQs below the population average of 100, and nearly a full quarter (25%) of their children will have IQs below 110. In both cases, it is relatively unlikely that these children would be able to follow in their parents&#8217; footsteps as physical scientists.<\/p>\n\n\n\n<p><strong>Regression to the mean<\/strong><\/p>\n\n\n\n<p>Several people have commented on this post noting that my model does not yield regression to the mean. This is true, but I feel that the omission is relatively minor.<\/p>\n\n\n\n<p>Let us review why regression to the mean exists. Suppose that you have two very smart parents, both at +4 SD intelligence, and assume that intelligence is equally determined by genes and environment, so that<\/p>\n\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 14px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-37ecca7bd844a3dba70e82f2e5cb2ffa_l3.png\" height=\"14\" width=\"82\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#73;&#32;&#61;&#32;&#71;&#32;&#43;&#32;&#69;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p>where the variables represent <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/>-scores.<\/p>\n\n\n\n<p>If both <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-30a79c32f18567063fe44716929e7ced_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are normally distributed, then <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-0fc37eedbe0169b46d3fc85ce49c18c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#32;&#61;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"\/> smart parents will be more likely to be, say, <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-31ea81310346a0759fe73464cf317532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#32;&#61;&#32;&#69;&#32;&#61;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"83\" style=\"vertical-align: 0px;\"\/> rather than <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-e0fed3c2e895bad84eef3332036d0ce4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#32;&#61;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"\/> and <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-0912f43b169665cfc3d617c00285a0ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#32;&#61;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\"\/> (the probability mass of a Gaussian tails off exponentially). Then supposing the parents are indeed <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-31ea81310346a0759fe73464cf317532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#32;&#61;&#32;&#69;&#32;&#61;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"83\" style=\"vertical-align: 0px;\"\/>, and assuming that the children have <em>completely independent <\/em>environmental contribution <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> as compared to their parents, then indeed the child IQ distribution will have a peak at <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-ffda130641b635f5110d1e914dd6b181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#32;&#61;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"\/>, the genetic mean of the parents, rather than at <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/milkyeggs.com\/wp-content\/ql-cache\/quicklatex.com-0fc37eedbe0169b46d3fc85ce49c18c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#32;&#61;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"\/>, the full phenotypic mean of the parents.<\/p>\n\n\n\n<p>In the modern world, which is fairly egalitarian (almost everyone has enough to eat), and where high IQ parents are probably richer and capable of supplying their children with an advantageous environment, I do not believe such independence assumptions hold, weakening the overall effect of regression to the mean.<\/p>\n\n\n\n<p>Furthermore, note that this is an attempt to reproduce the child IQ distribution from the &#8220;first principles&#8221; of the genetic architecture of intelligence, assuming perfect knowledge of the parents&#8217; genetic IQs. In that sense, it is not really accurate to say that the model fails to capture regression to the mean; it is more that it is focused on a subproblem simple enough that regression to the mean does not appear. One can of course argue that for real-world interpretability of the results, we need to take regression to the mean into account properly, but that would require a more complex model and a deeper discussion.<\/p>\n\n\n\n<p><strong>External references<\/strong><\/p>\n\n\n\n<p>I recommend these for further reading:<\/p>\n\n\n\n<ul>\n<li><a href=\"https:\/\/infoproc.blogspot.com\/2008\/10\/regression-to-mean.html\">Regression to the mean<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/infoproc.blogspot.com\/2010\/08\/iq-regression-from-terman-data.html\">IQ regression from Terman data<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/infoproc.blogspot.com\/2010\/07\/assortative-mating-regression-and-all.html\">Assortative mating, regression and all that: offspring IQ vs parental midpoint<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Let&#8217;s try to estimate the distribution of the &#8220;genetic IQ&#8221; of some offspring assuming that we know the parental &#8220;genetic IQ.&#8221;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"_links":{"self":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts\/520"}],"collection":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/comments?post=520"}],"version-history":[{"count":41,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts\/520\/revisions"}],"predecessor-version":[{"id":577,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/posts\/520\/revisions\/577"}],"wp:attachment":[{"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/media?parent=520"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/categories?post=520"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/milkyeggs.com\/wp-json\/wp\/v2\/tags?post=520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}