{"id":1156,"date":"2021-06-23T11:12:05","date_gmt":"2021-06-23T15:12:05","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=1156"},"modified":"2024-02-08T14:34:25","modified_gmt":"2024-02-08T19:34:25","slug":"11-2-chi-square-distribution","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/11-2-chi-square-distribution\/","title":{"raw":"11.2 Chi-Square Distribution","rendered":"11.2 Chi-Square Distribution"},"content":{"raw":"Just as z-tests are based on the normal distribution and t-tests are based on the t-distribution, chi-square tests are based on the chi-square distribution.\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Facts: Chi-Square Distribution<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>If [latex]Z \\sim N(0, 1)[\/latex], then [latex]Z^2 \\sim \\chi^2[\/latex] distribution with degrees of freedom [latex]df=1[\/latex].<\/li>\r\n \t<li>If [latex]Z_1, Z_2, \\dots, Z_n[\/latex] are independent and follow a standard normal distribution [latex]N(0, 1)[\/latex], then [latex]Z_1^2 + Z_2^2 + \\dots + Z_n^2 \\sim \\chi^2 [\/latex] distribution with [latex]df = 1 + 1 + \\dots + 1 = n[\/latex].<\/li>\r\n \t<li>If [latex]W \\sim \\chi^2[\/latex] with [latex]df = p[\/latex], [latex]V \\sim \\chi^2[\/latex] with [latex]df = q[\/latex] and they are independent, then [latex]W + V \\sim \\chi^2 [\/latex] with [latex]df = p + q[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nLike the <em>t<\/em> distribution, the chi-square distribution is determined by one parameter, the degrees of freedom. The figure below shows the density curves of chi-square distributions with [latex]df = 1, 3, 5, 9, 15[\/latex].<a id=\"retfig11.1\"><\/a>\r\n\r\n[caption id=\"attachment_1159\" align=\"aligncenter\" width=\"445\"]<a class=\"alignnone size-medium wp-image-1159\" href=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions-300x284.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-1159\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions.png\" alt=\"Several chi-square density curves are shown with different degrees of freedom. Image description available.\" width=\"445\" height=\"421\" \/><\/a> <strong>Figure 11.1<\/strong>: Chi-Square Density Curves. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig11.1\">Image Description (See Appendix D Figure 11.1)<\/a>][\/caption]The properties of the chi-square density curve are as follows:\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Facts: Properties of Chi-Square Density Curve<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>The total area under the curve is 1.<\/li>\r\n \t<li>It is right skewed.<\/li>\r\n \t<li>As the degrees of freedom increase, the chi-square curves appear more symmetric.<\/li>\r\n \t<li>Every chi-square random variable is non-negative with possible values between 0 and [latex]\\infty[\/latex].<\/li>\r\n \t<li>The mean of a chi-square is equal to its degrees of freedom and the standard deviation is the square root of twice the degrees of freedom. Suppose the degrees of freedom of a chi-square distribution is [latex]\\gamma[\/latex], then [latex]\\mu=\\gamma, \\sigma=\\sqrt{2\\gamma}[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nLike the <em><a href=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/08\/t_table_more_crop.png\" target=\"_blank\" rel=\"noopener\">t score table<\/a><\/em>, the [latex]\\chi^2[\/latex] table (Table V) gives critical values [latex]\\chi_{\\alpha}^2[\/latex]. Each critical value [latex]\\chi_{\\alpha}^2[\/latex] has an area of [latex]\\alpha[\/latex] to its right under the curve of the chi-square distribution with a certain degrees of freedom <em>df<\/em>.\r\n<p style=\"text-align: center;\"><strong>Table 11.1<\/strong>: Part of Chi-Square Table\u00a0 (Table V)<a id=\"rettab11.1\"><\/a><\/p>\r\n\r\n[caption id=\"attachment_3093\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-3093 size-large\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-1024x402.png\" alt=\"Part of the chi-square table is shown. Image description available.\" width=\"1024\" height=\"402\" \/> [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#tab11.1\">Image Description (See Appendix D Table 11.1)<\/a>][\/caption]\r\n<div align=\"center\">\r\n\r\n&nbsp;\r\n\r\n<\/div>","rendered":"<p>Just as z-tests are based on the normal distribution and t-tests are based on the t-distribution, chi-square tests are based on the chi-square distribution.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Facts: Chi-Square Distribution<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>If [latex]Z \\sim N(0, 1)[\/latex], then [latex]Z^2 \\sim \\chi^2[\/latex] distribution with degrees of freedom [latex]df=1[\/latex].<\/li>\n<li>If [latex]Z_1, Z_2, \\dots, Z_n[\/latex] are independent and follow a standard normal distribution [latex]N(0, 1)[\/latex], then [latex]Z_1^2 + Z_2^2 + \\dots + Z_n^2 \\sim \\chi^2[\/latex] distribution with [latex]df = 1 + 1 + \\dots + 1 = n[\/latex].<\/li>\n<li>If [latex]W \\sim \\chi^2[\/latex] with [latex]df = p[\/latex], [latex]V \\sim \\chi^2[\/latex] with [latex]df = q[\/latex] and they are independent, then [latex]W + V \\sim \\chi^2[\/latex] with [latex]df = p + q[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>Like the <em>t<\/em> distribution, the chi-square distribution is determined by one parameter, the degrees of freedom. The figure below shows the density curves of chi-square distributions with [latex]df = 1, 3, 5, 9, 15[\/latex].<a id=\"retfig11.1\"><\/a><\/p>\n<figure id=\"attachment_1159\" aria-describedby=\"caption-attachment-1159\" style=\"width: 445px\" class=\"wp-caption aligncenter\"><a class=\"alignnone size-medium wp-image-1159\" href=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions-300x284.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1159\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions.png\" alt=\"Several chi-square density curves are shown with different degrees of freedom. Image description available.\" width=\"445\" height=\"421\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions.png 611w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions-300x284.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions-65x61.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions-225x213.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2021\/06\/m11_DentsityCurves_Chi-squareDistributions-350x331.png 350w\" sizes=\"auto, (max-width: 445px) 100vw, 445px\" \/><\/a><figcaption id=\"caption-attachment-1159\" class=\"wp-caption-text\"><strong>Figure 11.1<\/strong>: Chi-Square Density Curves. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig11.1\">Image Description (See Appendix D Figure 11.1)<\/a>]<\/figcaption><\/figure>\n<p>The properties of the chi-square density curve are as follows:<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Facts: Properties of Chi-Square Density Curve<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>The total area under the curve is 1.<\/li>\n<li>It is right skewed.<\/li>\n<li>As the degrees of freedom increase, the chi-square curves appear more symmetric.<\/li>\n<li>Every chi-square random variable is non-negative with possible values between 0 and [latex]\\infty[\/latex].<\/li>\n<li>The mean of a chi-square is equal to its degrees of freedom and the standard deviation is the square root of twice the degrees of freedom. Suppose the degrees of freedom of a chi-square distribution is [latex]\\gamma[\/latex], then [latex]\\mu=\\gamma, \\sigma=\\sqrt{2\\gamma}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Like the <em><a href=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/08\/t_table_more_crop.png\" target=\"_blank\" rel=\"noopener\">t score table<\/a><\/em>, the [latex]\\chi^2[\/latex] table (Table V) gives critical values [latex]\\chi_{\\alpha}^2[\/latex]. Each critical value [latex]\\chi_{\\alpha}^2[\/latex] has an area of [latex]\\alpha[\/latex] to its right under the curve of the chi-square distribution with a certain degrees of freedom <em>df<\/em>.<\/p>\n<p style=\"text-align: center;\"><strong>Table 11.1<\/strong>: Part of Chi-Square Table\u00a0 (Table V)<a id=\"rettab11.1\"><\/a><\/p>\n<figure id=\"attachment_3093\" aria-describedby=\"caption-attachment-3093\" style=\"width: 1024px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3093 size-large\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-1024x402.png\" alt=\"Part of the chi-square table is shown. Image description available.\" width=\"1024\" height=\"402\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-1024x402.png 1024w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-300x118.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-768x302.png 768w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-1536x603.png 1536w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-2048x805.png 2048w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-65x26.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-225x88.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/06\/chisq_table-350x138.png 350w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption id=\"caption-attachment-3093\" class=\"wp-caption-text\">[<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#tab11.1\">Image Description (See Appendix D Table 11.1)<\/a>]<\/figcaption><\/figure>\n<div style=\"margin: auto;\">\n<p>&nbsp;<\/p>\n<\/div>\n","protected":false},"author":19,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1156","chapter","type-chapter","status-publish","hentry"],"part":1148,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1156","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":34,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1156\/revisions"}],"predecessor-version":[{"id":5314,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1156\/revisions\/5314"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/1148"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1156\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=1156"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=1156"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=1156"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=1156"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}