{"id":174026,"date":"2023-11-27T09:24:08","date_gmt":"2023-11-27T08:24:08","guid":{"rendered":"https:\/\/datascientest.com\/en\/?p=174026"},"modified":"2023-11-27T09:27:57","modified_gmt":"2023-11-27T08:27:57","slug":"chi-squared-test-find-out-more-about-this-essential-statistical-test","status":"publish","type":"post","link":"https:\/\/datascientest.com\/en\/chi-squared-test-find-out-more-about-this-essential-statistical-test","title":{"rendered":"Chi squared test: Find out more about this essential statistical test"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"174026\" class=\"elementor elementor-174026\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-92ebe4c e-flex e-con-boxed e-con\" data-id=\"92ebe4c\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-943523c elementor-widget elementor-widget-heading\" data-id=\"943523c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-default\">The chi squared test (or chi 2) is a statistical test for variables that take a finite number of possible values, making them categorical variables. As a reminder, a statistical test is a method used to determine whether a hypothesis, known as the null hypothesis, is consistent with the data or not.<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-cfa7b50 e-flex e-con-boxed e-con\" data-id=\"cfa7b50\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-16c08d0 elementor-widget elementor-widget-heading\" data-id=\"16c08d0\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">What is the purpose of the Chi squared test?<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5dd5e99 elementor-widget elementor-widget-text-editor\" data-id=\"5dd5e99\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p>The advantage of the <strong>Chi squared test<\/strong> is its wide range of applications:<\/p><ul><li><strong>Test of goodness<\/strong> of fit to a predefined law or family of laws, for example: Does the size of a population follow a normal distribution?<\/li><li><strong>Test of independence,<\/strong> for example: Is hair color independent of gender?<\/li><li><strong>Homogeneity test:<\/strong> <a href=\"https:\/\/datascientest.com\/en\/datasets-top-5-places-to-find-quality-datasets\">Are two sets of data identically distributed?<\/a><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-8249584 e-flex e-con-boxed e-con\" data-id=\"8249584\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-866325f elementor-widget elementor-widget-heading\" data-id=\"866325f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">How does the Chi squared test work?<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d6155c3 elementor-widget elementor-widget-text-editor\" data-id=\"d6155c3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Its principle is to compare the proximity or divergence between the distribution of the sample and a theoretical distribution using the Pearson statistic <strong><span class=\"katex-eq\" data-katex-display=\"false\"><\/span> \\chi_{Pearson} [\\latex],<\/strong> which is based on the <strong>chi-squared distance.<\/strong><\/p><p><strong>First problem:<\/strong> Since we have only a limited amount of data, we cannot perfectly know the distribution of the sample, but only an approximation of it, the empirical measure.<\/p><p>The empirical measure <span class=\"katex-eq\" data-katex-display=\"false\"><\/span> \\widehat{\\mathbb{P}}_{n,X} [\\latex] represents the frequency of different observed values:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eec9db7 elementor-widget elementor-widget-shortcode\" data-id=\"eec9db7\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\forall x \\in \\mathbb{X} \\quad   \\widehat{\\mathbb{P}}_{n,X} (x) = \\frac{1}{n} \\sum_{k=1}^{n} 1_{X_{k} =x}<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-59a045e elementor-widget elementor-widget-text-editor\" data-id=\"59a045e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p style=\"text-align: center;\">Empirical measurement formula<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2c57ae6 elementor-widget elementor-widget-text-editor\" data-id=\"2c57ae6\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p style=\"text-align: center;\">with<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b42ff38 elementor-widget elementor-widget-shortcode\" data-id=\"b42ff38\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> X_{1},... ,{X_n}<\/span> = the sample<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8c3c6e2 elementor-widget elementor-widget-shortcode\" data-id=\"8c3c6e2\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> {\\mathbb{X}} <\/span> = all possible values<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b199b11 elementor-widget elementor-widget-text-editor\" data-id=\"b199b11\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>The Pearson statistic is defined as :<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ca5cc9e elementor-widget elementor-widget-shortcode\" data-id=\"ca5cc9e\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\chi_{Pearson} = n \\times \\chi_{2}(\\widehat{\\mathbb{P}}_{n,X}, P_{theorique} ) = n \\times \\sum_{x \\in  \\mathbb{X}} \\frac{(\\widehat{\\mathbb{P}}_{n,X} (x)- P_{theorique}(x))^{2}}{P_{theorique}(x)}<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b52caf9 elementor-widget elementor-widget-text-editor\" data-id=\"b52caf9\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p style=\"text-align: center;\">Pearson&#8217;s statistical formula<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d039d75 elementor-widget elementor-widget-text-editor\" data-id=\"d039d75\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p style=\"text-align: left;\">Under the null hypothesis, which means that there is equality between the distribution of the sample and the theoretical distribution, this Pearson statistic will converge to the <strong>chi-squared<\/strong> distribution with d degrees of freedom.<\/p><p style=\"text-align: left;\">The number of degrees of freedom, d, depends on the dimensions of the problem and is generally equal to the number of possible values minus 1.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-97134be e-flex e-con-boxed e-con\" data-id=\"97134be\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t<div class=\"elementor-element elementor-element-acf81bd e-con-full e-flex e-con\" data-id=\"acf81bd\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;full&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-741dd09 elementor-widget elementor-widget-text-editor\" data-id=\"741dd09\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>As a reminder, the chi-2 law with d degrees of freedom<\/p><p>centred reduced independent.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-66f96d7 e-con-full e-flex e-con\" data-id=\"66f96d7\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;full&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-1395ac0 elementor-widget elementor-widget-shortcode\" data-id=\"1395ac0\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\chi^{2}_{loi}(d)\n<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-b3bdaf8 e-con-full e-flex e-con\" data-id=\"b3bdaf8\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;full&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-166dc76 elementor-widget elementor-widget-text-editor\" data-id=\"166dc76\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>is that of a sum of squares of d Gaussians<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0dc7c15 elementor-widget elementor-widget-shortcode\" data-id=\"0dc7c15\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> chi^{2}_{loi}(d) := \\sum_{k=1}^{d} X_{k} \\quad avec \\quad X_{k}  \\sim  \\mathbb{N}(0,1)<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-621d66f elementor-widget elementor-widget-text-editor\" data-id=\"621d66f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p style=\"text-align: left;\">Otherwise, this statistic will diverge to infinity, reflecting the distance between empirical and theoretical distributions.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d8ffbf6 elementor-widget elementor-widget-shortcode\" data-id=\"d8ffbf6\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> Sous \\quad H_{0}  \\quad  \\lim_{n\\rightarrow  \\infty } \\chi_{Pearson} = \\chi^{2}_{loi}(d). \\\\\nSous \\quad H_{1}  \\quad  \\lim_{n\\rightarrow  \\infty } \\chi_{Pearson} = \\infty\n<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7ab7478 elementor-widget elementor-widget-text-editor\" data-id=\"7ab7478\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p style=\"text-align: center;\">Limit formula<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f2c3177 e-flex e-con-boxed e-con\" data-id=\"f2c3177\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2f446fb elementor-widget elementor-widget-heading\" data-id=\"2f446fb\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">What are the benefits of the Chi squared test?<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-61499eb elementor-widget elementor-widget-text-editor\" data-id=\"61499eb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><strong>So, we have a simple decision rule:<\/strong> if the Pearson statistic exceeds a certain threshold, we reject the initial hypothesis (the theoretical distribution does not fit the data), otherwise, we accept it.<\/p><p>The advantage of the <strong>chi-squared test<\/strong> is that this threshold depends only on the chi-squared distribution and the confidence level alpha, so it is independent of the distribution of the sample.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f66eee6 e-flex e-con-boxed e-con\" data-id=\"f66eee6\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-bf8635c elementor-widget elementor-widget-heading\" data-id=\"bf8635c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The test of independence:<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0f522ba elementor-widget elementor-widget-text-editor\" data-id=\"0f522ba\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Let&#8217;s take an example to illustrate this test: we want to determine if the genders of the first two children, X and Y, in a couple are independent?<\/p><p>We have gathered the <a href=\"https:\/\/datascientest.com\/en\/what-is-a-dataset-how-do-i-work-with-it\">data in a contingency table:<\/a><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-224e592 elementor-widget elementor-widget-shortcode\" data-id=\"224e592\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{|c|c|c|c|}\n\t\\hline X \/ Y &amp; Child 2 : son &amp; Child 2: daughter &amp; Total \\\\\n\t\\hline Child 1: son &amp; 857  &amp;  801 &amp; 1658 \\\\\n\t\\hline Child 1: daughter  &amp; 813 &amp; 828 &amp; 1641\\\\\n\t\\hline Total  &amp; 1670 &amp; 1629 &amp; 3299\n\\end{array}\n<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e7189f9 elementor-widget elementor-widget-text-editor\" data-id=\"e7189f9\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>The<strong> Pearson statistic<\/strong> will determine if the empirical measure of the joint distribution (X, Y) is equal to the product of the empirical marginal measures, which<strong> characterizes independence:<\/strong><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-07da9b8 elementor-widget elementor-widget-shortcode\" data-id=\"07da9b8\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\chi_{Pearson} = n \\times \\chi^2\n(\\widehat{\\mathbb{P}}_{X \\times Y}, \\widehat{\\mathbb{P}}_{X} \\times \\widehat{\\mathbb{P}}_{Y}) = \n\\sum_{x \\in \\{daughter, son\\}, y\\in \\{daughter, son\\}} \\frac{(Observation_{x,y} - Theory_{x,y})^{2}}{Theory_{x,y}}\n<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-434fbf4 elementor-widget elementor-widget-text-editor\" data-id=\"434fbf4\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Here, Observation(x,y) represents the frequency of the value (x, y):<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-be9fe3f elementor-widget elementor-widget-shortcode\" data-id=\"be9fe3f\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\forall x, y \\in \\{daughter, son\\} \\quad\nObservation_{x,y} = \\frac{1}{n} \\sum_{k=1}^{n} 1_{(X_{k},Y_{k}) = (x, y)}\n<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-429eed7 elementor-widget elementor-widget-text-editor\" data-id=\"429eed7\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>for exemple:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0766e1f elementor-widget elementor-widget-shortcode\" data-id=\"0766e1f\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> Observation(daughter, daughter) = \\frac{828}{3299} = 0.251\n<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e0e3f14 elementor-widget elementor-widget-text-editor\" data-id=\"e0e3f14\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>For <strong>Theory(x, y), X and Y are assumed<\/strong> to be independent, so the theoretical distribution should be the product of the <strong>marginal distributions:<\/strong><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b7df02a elementor-widget elementor-widget-shortcode\" data-id=\"b7df02a\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\forall x, y \\in \\{daughter, son\\} \\quad\nTheory_{x,y} = Observation^{X} \\times Observation^{Y} = \\sum_{y\\in\\{daughter, son\\}} Observation_{x,y} \\times\n\\sum_{x\\in\\{daughter, son\\}} Observation_{x,y}\n<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1e1e909 elementor-widget elementor-widget-text-editor\" data-id=\"1e1e909\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Thus, the theoretical probability for (son, son) is:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ee70c88 elementor-widget elementor-widget-shortcode\" data-id=\"ee70c88\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> Theory(son, son) = \\frac{857+801}{3299} \\times \\frac{857+813}{3299} = \\frac{1658 \\times 1670}{3299^{2}} = 0.254<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0dcefbf elementor-widget elementor-widget-text-editor\" data-id=\"0dcefbf\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>Let&#8217;s calculate the test statistic using the following Python code:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-de14227 elementor-widget elementor-widget-shortcode\" data-id=\"de14227\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><script src=\"https:\/\/gist.github.com\/robinTdatascientest\/8e3a8d20848edc75719ef94ee58dd804.js\"><\/script><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1ce6d99 elementor-widget elementor-widget-text-editor\" data-id=\"1ce6d99\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>In our case, the variables X and Y have only 2 possible values: daughters or sons, so the dimension of the problem is (2-1)(2-1), which is 1.<\/p><p>Therefore, we compare the test statistic to the<strong> chi-squared quantile<\/strong> with 1 degree of freedom using the chi2.ppf function from <a href=\"https:\/\/datascientest.com\/en\/scipy-all-about-the-python-machine-learning-library\">scipy.stats.<\/a><\/p><p>If the test statistic is lower than the quantile and the p-value is greater than the significance level of 0.05, we cannot reject the null hypothesis with 95% confidence.<\/p><p>Thus, we conclude that the gender of the first two children is independent.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-12e9fd4 e-flex e-con-boxed e-con\" data-id=\"12e9fd4\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b8f1989 elementor-widget elementor-widget-heading\" data-id=\"b8f1989\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">What are its limits?<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-84ffa7a elementor-widget elementor-widget-text-editor\" data-id=\"84ffa7a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>While the <strong>chi squared test<\/strong> is very practical, it does have limitations. It can only detect the existence of correlations but does not measure their strength or causality.<\/p><p>It relies on the approximation of the chi-squared distribution with the Pearson statistic, which is only valid if you have a sufficient amount of data. In practice, the validity condition is as follows:<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-98b6974 elementor-widget elementor-widget-shortcode\" data-id=\"98b6974\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\forall x \\in \\mathbb{X} \\quad n \\times P_{theoretical}(x) (1- P_{theoretical}(x)) \\geq 5<\/span><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-42cbd0b elementor-widget elementor-widget-text-editor\" data-id=\"42cbd0b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p>The Fisher exact test can address this limitation but requires significant computational power. In practice, it is often limited to 2&#215;2 contingency tables.<\/p><p>Statistical tests are crucial in <a href=\"https:\/\/datascientest.com\/en\/data-science-in-education-how-data-is-transforming-schools\">Data Science to assess the relevance of explanatory variables<\/a> and validate modeling assumptions. You can find more information about the chi-squared test and other statistical tests in our module 104 &#8211; Exploratory Statistics.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f15fbe5 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"f15fbe5\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/formation-data-scientist\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Discover our Data Scientist training<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-347ff48 e-flex e-con-boxed e-con\" data-id=\"347ff48\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b56a939 elementor-widget elementor-widget-heading\" data-id=\"b56a939\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">References:<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c276a0b elementor-widget elementor-widget-text-editor\" data-id=\"c276a0b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><a href=\"https:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.stats.chi2.html\">https:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.stats.chi2.html<\/a><\/p><p><a href=\"https:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.stats.chi2_contingency.html\">https:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.stats.chi2_contingency.html<\/a><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-2c79819 e-flex e-con-boxed e-con\" data-id=\"2c79819\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d6e2c22 elementor-widget elementor-widget-html\" data-id=\"d6e2c22\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"FAQPage\",\n  \"mainEntity\": [{\n    \"@type\": \"Question\",\n    \"name\": \"How does the test work?\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Its principle is to compare the closeness or distance between the sample law and a theoretical law, with the so-called Pearson statistic <span class=\"katex-eq\" data-katex-display=\"false\"><\/span> \\chi_{Pearson} [\\latex], based on the chi-2 distance. First problem: Since we only have a limited amount of data, we can't know the sample law perfectly, but only an approximation to it, the empirical measure.\"\n    }\n  },{\n    \"@type\": \"Question\",\n    \"name\": \"What are its advantages?\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"So we have a simple decision rule: if the Pearson statistic exceeds a certain threshold, we reject the initial hypothesis (the theoretical distribution doesn't fit the data), otherwise we accept it. The advantage of the Chi-2 test is that this threshold depends only on the distribution of the Chi-2 and the alpha confidence level, and is therefore independent of the distribution of the sample.\"\n    }\n  },{\n    \"@type\": \"Question\",\n    \"name\": \"What are its limits?\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Although it seems very practical, the chi-2 test does have its limitations: it only detects the existence of correlations, but neither their intensity nor causality. It relies on the approximation of the chi2 law with Pearson's statistic, verified only if a sufficient number of data is available.\"\n    }\n  }]\n}\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The chi squared test (or chi 2) is a statistical test for variables that take a finite number of possible values, making them categorical variables. As a reminder, a statistical test is a method used to determine whether a hypothesis, known as the null hypothesis, is consistent with the data or not. What is the [&hellip;]<\/p>\n","protected":false},"author":67,"featured_media":174027,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"content-type":"","inline_featured_image":false},"categories":[2425],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Chi squared test: Find out more about this essential statistical test<\/title>\n<meta name=\"description\" content=\"The Chi squared test is a valuable statistical tool for analyzing relationships between categorical variables. 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