[{"id":247043,"date":"2026-05-26T12:59:53","date_gmt":"2026-05-26T17:59:53","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247043"},"modified":"2026-05-26T12:59:53","modified_gmt":"2026-05-26T17:59:53","slug":"calculating-expected-normal-range","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/05\/26\/calculating-expected-normal-range\/","title":{"rendered":"Calculating the expected range of normal samples"},"content":{"rendered":"

The previous post<\/a> looked at the expected IQ range in a jury of 12. This post will look more generally at computing the expected range of\u00a0n<\/em> samples from a\u00a0N<\/em>(0, 1) random variable. This will give the expected range in units of \u03c3, i.e. multiply the results by \u03c3 if your \u03c3 isn’t 1.<\/p>\n

As mentioned in the previous post, the expected range is given by<\/p>\n

\"d_n<\/p>\n

where \u03c6 and \u03a6 are the PDF and CDF of a standard normal. The integral can be calculated in closed form for\u00a0n<\/em> \u2264 5, but in general it requires numerical integration [1].<\/p>\n

The following Python code can compute d<\/em>n<\/em><\/sub>.<\/p>\n

\r\nfrom scipy.stats import norm\r\nfrom scipy.integrate import quad\r\nimport numpy as np\r\n\r\ndef d(n):\r\n    integrand = lambda x: x*norm.pdf(x)*norm.cdf(x)**(n-1)\r\n    res, info = quad(integrand, -np.inf, np.inf)\r\n    return 2*n*res\r\n<\/pre>\n
For large n<\/em> we have the asymptotic approximation<\/p>\n
d_n = 2 \\Phi^{-1}\\left( \\frac{n \\,\u2013\\, 0.375}{ n + 0.25} \\right)<\/p>\n
which we could implement in Python by<\/p>\n
\r\ndef approx(n):\r\n    return 2*norm.ppf((n - 0.375)\/(n + 0.25))\r\n<\/pre>\n
For very large n<\/em> the asymptotic expression may be more accurate than the integral due to numerical integration error. <\/p>\n
Here are a few example values.<\/p>\n
\r\n|-----+-------|\r\n|   n |   d_n |\r\n|-----+-------|\r\n|   2 | 1.128 |\r\n|   3 | 1.693 |\r\n|   5 | 2.326 |\r\n|  10 | 3.078 |\r\n|  12 | 3.258 |\r\n|  23 | 3.858 |\r\n|  50 | 4.498 |\r\n| 100 | 5.015 |\r\n|-----+-------|\r\n<\/pre>\n
[1] Order Statistics by H. A. David. John Wiley & Sons. 1970.<\/p>\n","protected":false},"excerpt":{"rendered":"
The previous post looked at the expected IQ range in a jury of 12. This post will look more generally at computing the expected range of\u00a0n samples from a\u00a0N(0, 1) random variable. This will give the expected range in units of \u03c3, i.e. multiply the results by \u03c3 if your \u03c3 isn’t 1. As mentioned […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[5],"tags":[105],"class_list":["post-247043","post","type-post","status-publish","format-standard","hentry","category-computing","tag-probability"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247043","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=247043"}],"version-history":[{"count":0,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/247043\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=247043"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=247043"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=247043"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":247041,"date":"2026-05-26T08:50:51","date_gmt":"2026-05-26T13:50:51","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247041"},"modified":"2026-05-26T13:01:40","modified_gmt":"2026-05-26T18:01:40","slug":"expected-iq-spread-on-a-jury","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/05\/26\/expected-iq-spread-on-a-jury\/","title":{"rendered":"Expected IQ spread on a jury"},"content":{"rendered":"
There’s been some discussion online lately about how a large difference in IQ makes it difficult for two people to communicate. There have been studies that confirm this effect. The difficulty is not insurmountable, but it takes deliberate effort to overcome.<\/p>\n
Someone dismissed this communication difficulty by pointing out that the expected difference in IQ between two individuals is around 17, suggesting that most communication is between people who differ by more than one standard deviation in IQ. But this calculation assumes people are chosen at random, which they usually are not. People tend to live around and work around others of similar intelligence.<\/p>\n
However, a jury\u00a0is<\/em> a random sample. It’s not a perfect random sample. For one thing, it starts with a random sample of people who are registered to vote, or who have a drivers license, not all individuals. Furthermore, the pool of potential jurors is reduced to a jury through the process of voir dire<\/em>, which is not random.<\/p>\n
For this post I will make the simplifying assumption that a jury is a random sample from a population with normally distributed IQ with standard deviation \u03c3 = 15. The mean doesn’t matter here, but you could assume it’s 100 if you’d like.<\/p>\n
By symmetry, the expected range of n<\/em> samples from a normal random variable is twice the maximum. For\u00a0n<\/em> = 12 the range is about 3.26\u03c3, which corresponds to nearly 50 IQ points<\/strong>.<\/p>\n
This suggests there’s usually a big spread of IQ on a jury. Even if IQ doesn’t measure intelligence, it measures\u00a0something<\/em>, and that something varies a lot over 12 people chosen at random [1].<\/p>\n
Related posts<\/h2>\n