{"id":1396,"date":"2023-06-22T02:22:40","date_gmt":"2023-06-22T02:22:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-chi-square-statistics-learn-it-2\/"},"modified":"2025-05-16T23:07:49","modified_gmt":"2025-05-16T23:07:49","slug":"introduction-to-chi-square-statistics-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-chi-square-statistics-learn-it-2\/","title":{"raw":"Introduction to Chi-Square Statistics \u2013 Learn It 2","rendered":"Introduction to Chi-Square Statistics \u2013 Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Write a null and alternative hypothesis for a chi-square test<\/li>\r\n\t<li>Calculate and interpret the value of a chi-square statistics in context of a real-world problem<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>We use the chi-square hypothesis test to determine whether the data \"fit\" a particular distribution or not. The chi-square statistic compares the size of any differences between the expected counts and the actual observed counts.<\/p>\r\n<h2>Chi-Square Test Statistic ([latex]\\chi^2[\/latex])<\/h2>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2324[\/ohm2_question]<\/section>\r\n<p>You just calculated the value of the chi-square (pronounced \u201ckai-square\u201d) test statistic for this problem.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>[latex]\\chi^2[\/latex] test statistic<\/h3>\r\n<p>The [latex]\\chi^2[\/latex] test statistic measures the overall distance between observed and expected counts.<\/p>\r\n<p>The greater the chi-square test statistic, the further the observed counts are from what we expected.<\/p>\r\n<p>Here is the formula for the chi-square test statistic:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\chi^2=\\sum\\dfrac{(\\text{Observed}-\\text{Expected})^2}{\\text{Expected}}[\/latex]<\/p>\r\n<\/section>\r\n<p>This formula shows what we did in the question above \u2014 we added up (the large sigma [latex]\\sum[\/latex] represents summation) the [latex]\\dfrac{(O-E)^2}{E}[\/latex] for each quarter of the year (each category).<\/p>\r\n<p>It\u2019s important to remember the intuition behind this formula\u2014we get the differences, square them to get rid of the negative values, and then scale them by dividing the squared differences by the expected counts. In this way, we get a robust measure of the overall difference between the observed and expected counts for a categorical variable.<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Write a null and alternative hypothesis for a chi-square test<\/li>\n<li>Calculate and interpret the value of a chi-square statistics in context of a real-world problem<\/li>\n<\/ul>\n<\/section>\n<p>We use the chi-square hypothesis test to determine whether the data &#8220;fit&#8221; a particular distribution or not. The chi-square statistic compares the size of any differences between the expected counts and the actual observed counts.<\/p>\n<h2>Chi-Square Test Statistic ([latex]\\chi^2[\/latex])<\/h2>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2324\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2324&theme=lumen&iframe_resize_id=ohm2324&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>You just calculated the value of the chi-square (pronounced \u201ckai-square\u201d) test statistic for this problem.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>[latex]\\chi^2[\/latex] test statistic<\/h3>\n<p>The [latex]\\chi^2[\/latex] test statistic measures the overall distance between observed and expected counts.<\/p>\n<p>The greater the chi-square test statistic, the further the observed counts are from what we expected.<\/p>\n<p>Here is the formula for the chi-square test statistic:<\/p>\n<p style=\"text-align: center;\">[latex]\\chi^2=\\sum\\dfrac{(\\text{Observed}-\\text{Expected})^2}{\\text{Expected}}[\/latex]<\/p>\n<\/section>\n<p>This formula shows what we did in the question above \u2014 we added up (the large sigma [latex]\\sum[\/latex] represents summation) the [latex]\\dfrac{(O-E)^2}{E}[\/latex] for each quarter of the year (each category).<\/p>\n<p>It\u2019s important to remember the intuition behind this formula\u2014we get the differences, square them to get rid of the negative values, and then scale them by dividing the squared differences by the expected counts. In this way, we get a robust measure of the overall difference between the observed and expected counts for a categorical variable.<\/p>\n","protected":false},"author":8,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1388,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1396"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1396\/revisions"}],"predecessor-version":[{"id":6861,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1396\/revisions\/6861"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1388"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1396\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1396"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1396"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1396"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1396"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}