{"id":90,"date":"2021-01-12T22:19:45","date_gmt":"2021-01-12T22:19:45","guid":{"rendered":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/facts-about-the-chi-square-distribution\/"},"modified":"2023-04-19T19:57:35","modified_gmt":"2023-04-19T19:57:35","slug":"facts-about-the-chi-square-distribution","status":"publish","type":"chapter","link":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/facts-about-the-chi-square-distribution\/","title":{"rendered":"Facts About the Chi-Square Distribution"},"content":{"raw":"\r\n[latexpage]\r\n<\/span>\r\n
\r\n

The notation for the chi-square distribution<\/span> is:<\/p>\r\n$$\\chi \\sim \\chi_{df}^2$$\r\n\r\nwhere df<\/em> = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use df<\/em> = n<\/em> - 1. The degrees of freedom for the three major uses are each calculated differently.)\r\n

For the \u03c72<\/sup><\/em> distribution, the population mean is \u03bc<\/em> = df<\/em> and the population standard deviation is $\\sigma = \\sqrt{2(df)}$<\/p>\r\n

The random variable is shown as \u03c72<\/sup><\/em>, but may be any upper case letter.<\/p>\r\n

The random variable for a chi-square distribution with k<\/em> degrees of freedom is the sum of k<\/em> independent, squared standard normal variables.<\/p>\r\n

\u03c7<\/em>2<\/sup> = (Z<\/em>1<\/sub>)2<\/sup> + (Z<\/em>2<\/sub>)2<\/sup> + ... + (Z<\/em>k<\/sub>)2<\/sup><\/p>\r\n\r\n

    \r\n \t
  1. The curve is non-symmetrical and skewed to the right.<\/li>\r\n \t
  2. There is a different chi-square curve for each df<\/em>.\r\n
    \r\n
    \r\n\"Part\r\n<\/span><\/figure>\r\n
    Figure <\/span>11.2<\/span><\/div>\r\n<\/div><\/li>\r\n \t
  3. The test statistic for any test is always greater than or equal to zero.<\/li>\r\n \t
  4. When df<\/em> > 90, the chi-square curve approximates the normal distribution. For $\\chi \\sim \\chi_{1000}^2$ the mean, \u03bc<\/em> = df<\/em> = 1,000 and the standard deviation, $\\sigma = \\sqrt{2(1000)}=44.7$. Therefore, $X\\sim N(1,000, 44.7)$, approximately.<\/li>\r\n \t
  5. The mean, \u03bc<\/em>, is located just to the right of the peak.\r\n
    \r\n
    \r\n\"This\r\n<\/span><\/figure>\r\n
    Figure <\/span>11.3<\/span><\/div>\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>","rendered":"


    \n[latexpage]
    \n<\/span><\/p>\n

    \n

    The notation for the chi-square distribution<\/span> is:<\/p>\n

    $$\\chi \\sim \\chi_{df}^2$$<\/p>\n

    where df<\/em> = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use df<\/em> = n<\/em> – 1. The degrees of freedom for the three major uses are each calculated differently.)<\/p>\n

    For the \u03c72<\/sup><\/em> distribution, the population mean is \u03bc<\/em> = df<\/em> and the population standard deviation is $\\sigma = \\sqrt{2(df)}$<\/p>\n

    The random variable is shown as \u03c72<\/sup><\/em>, but may be any upper case letter.<\/p>\n

    The random variable for a chi-square distribution with k<\/em> degrees of freedom is the sum of k<\/em> independent, squared standard normal variables.<\/p>\n

    \u03c7<\/em>2<\/sup> = (Z<\/em>1<\/sub>)2<\/sup> + (Z<\/em>2<\/sub>)2<\/sup> + … + (Z<\/em>k<\/sub>)2<\/sup><\/p>\n

      \n
    1. The curve is non-symmetrical and skewed to the right.<\/li>\n
    2. There is a different chi-square curve for each df<\/em>.\n
      \n

      \n\"Part
      \n<\/span><\/figure>\n
      Figure <\/span>11.2<\/span><\/div>\n<\/div>\n<\/li>\n
    3. The test statistic for any test is always greater than or equal to zero.<\/li>\n
    4. When df<\/em> > 90, the chi-square curve approximates the normal distribution. For $\\chi \\sim \\chi_{1000}^2$ the mean, \u03bc<\/em> = df<\/em> = 1,000 and the standard deviation, $\\sigma = \\sqrt{2(1000)}=44.7$. Therefore, $X\\sim N(1,000, 44.7)$, approximately.<\/li>\n
    5. The mean, \u03bc<\/em>, is located just to the right of the peak.\n
      \n

      \n\"This
      \n<\/span><\/figure>\n
      Figure <\/span>11.3<\/span><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":1,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-90","chapter","type-chapter","status-publish","hentry"],"part":89,"_links":{"self":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/90","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":9,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/90\/revisions"}],"predecessor-version":[{"id":558,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/90\/revisions\/558"}],"part":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/parts\/89"}],"metadata":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/90\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/media?parent=90"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=90"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/contributor?post=90"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/license?post=90"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}