1.0Probably Overthinking Ithttps://www.allendowney.com/blogAllenDowneyhttps://www.allendowney.com/blog/author/allendowney_6dbrc4/rich600338<blockquote class="wp-embedded-content" data-secret="UQasZuDpng"><a href="https://www.allendowney.com/blog/2025/09/25/the-poincare-problem/">The Poincaré Problem</a></blockquote><iframe sandbox="allow-scripts" security="restricted" src="https://www.allendowney.com/blog/2025/09/25/the-poincare-problem/embed/#?secret=UQasZuDpng" width="600" height="338" title="“The Poincaré Problem” — Probably Overthinking It" data-secret="UQasZuDpng" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" class="wp-embedded-content"></iframe><script> /*! This file is auto-generated */ !function(d,l){"use strict";l.querySelector&&d.addEventListener&&"undefined"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!/[^a-zA-Z0-9]/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret="'+t.secret+'"]'),o=l.querySelectorAll('blockquote[data-secret="'+t.secret+'"]'),c=new RegExp("^https?:$","i"),i=0;i<o.length;i++)o[i].style.display="none";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute("style"),"height"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):"link"===t.message&&(r=new URL(s.getAttribute("src")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener("message",d.wp.receiveEmbedMessage,!1),l.addEventListener("DOMContentLoaded",function(){for(var e,t,s=l.querySelectorAll("iframe.wp-embedded-content"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute("data-secret"))||(t=Math.random().toString(36).substring(2,12),e.src+="#?secret="+t,e.setAttribute("data-secret",t)),e.contentWindow.postMessage({message:"ready",secret:t},"*")},!1)))}(window,document); //# sourceURL=https://www.allendowney.com/blog/wp-includes/js/wp-embed.min.js </script> Selection bias is the hardest problem in statistics because it’s almost unavoidable in practice, and once the data have been collected, it’s usually not possible to quantify the effect of selection or recover an unbiased estimate of what you are trying to measure. And because the effect is systematic, not random, it doesn’t help to collect more data. In fact, larger sample sizes make the problem worse, because they give the false impression of precision. But sometimes, if we are... Read More Read Morehttps://www.allendowney.com/blog/wp-content/uploads/2025/09/35cfe517511163559ff87ce035853e291ea5625f0b9eba3d13b81592c0427cca.png442292