{"id":509,"date":"2024-04-09T01:08:26","date_gmt":"2024-04-09T05:08:26","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=509"},"modified":"2024-04-18T04:22:02","modified_gmt":"2024-04-18T08:22:02","slug":"the-variance","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/the-variance\/","title":{"raw":"The Variance","rendered":"The Variance"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCalculate and understand the sample and population variances.\r\n\r\n<\/div>\r\n<\/div>\r\nThe Variance (Var):\r\n<ul>\r\n \t<li>The Variance is the square of the Standard Deviation.<\/li>\r\n \t<li>It is rarely used in an introductory course in Statistics.<\/li>\r\n<\/ul>\r\nThe SAMPLE variance is defined as:\r\n<p style=\"padding-left: 40px\">[latex] s^2= \\frac{\\Sigma(x_i-\\bar{x})^2}{n-1} [\/latex]<\/p>\r\nThe POPULATION variance is defined as:\r\n<p style=\"padding-left: 40px\">[latex] \\sigma ^2= \\frac{ \\Sigma (x_i - \\bar{x} ) ^2 }{ n-1} [\/latex]<\/p>\r\n\r\n<h1>Example 7.1 - Variances of Sunita and Sanjay's Grades (Excel)<\/h1>\r\nLet us revisit Sunita and Sanjay's grades. Let us now calculate the sample variance for each in Excel:\r\n\r\n<img class=\"alignnone wp-image-601\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-300x56.jpg\" alt=\"\" width=\"616\" height=\"115\" \/>\r\n<ul>\r\n \t<li>Again, we see that Sanjay's grades are much more 'dispersed' than Sunita's<\/li>\r\n \t<li>The variance for Sanjay's grades is much larger than Sunita's variance for her grades.<\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/7-1.xlsx\">Click here to download the spreadsheet shown above.<\/a><\/li>\r\n<\/ul>\r\n<h1>Key Takeaways<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: The Variance<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>The variance is the squared standard deviation.<\/li>\r\n \t<li>The variance is not often used in this course.<\/li>\r\n \t<li>There are different formulas for the sample and population variance.<\/li>\r\n \t<li>Often, we assume that we have a sample (if nothing is stated).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Calculate and understand the sample and population variances.<\/p>\n<\/div>\n<\/div>\n<p>The Variance (Var):<\/p>\n<ul>\n<li>The Variance is the square of the Standard Deviation.<\/li>\n<li>It is rarely used in an introductory course in Statistics.<\/li>\n<\/ul>\n<p>The SAMPLE variance is defined as:<\/p>\n<p style=\"padding-left: 40px\">[latex]s^2= \\frac{\\Sigma(x_i-\\bar{x})^2}{n-1}[\/latex]<\/p>\n<p>The POPULATION variance is defined as:<\/p>\n<p style=\"padding-left: 40px\">[latex]\\sigma ^2= \\frac{ \\Sigma (x_i - \\bar{x} ) ^2 }{ n-1}[\/latex]<\/p>\n<h1>Example 7.1 &#8211; Variances of Sunita and Sanjay&#8217;s Grades (Excel)<\/h1>\n<p>Let us revisit Sunita and Sanjay&#8217;s grades. Let us now calculate the sample variance for each in Excel:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-601\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-300x56.jpg\" alt=\"\" width=\"616\" height=\"115\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-300x56.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-1024x190.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-768x143.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-65x12.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-225x42.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1-350x65.jpg 350w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/Example7-1_Image1.jpg 1383w\" sizes=\"auto, (max-width: 616px) 100vw, 616px\" \/><\/p>\n<ul>\n<li>Again, we see that Sanjay&#8217;s grades are much more &#8216;dispersed&#8217; than Sunita&#8217;s<\/li>\n<li>The variance for Sanjay&#8217;s grades is much larger than Sunita&#8217;s variance for her grades.<\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/04\/7-1.xlsx\">Click here to download the spreadsheet shown above.<\/a><\/li>\n<\/ul>\n<h1>Key Takeaways<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: The Variance<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>The variance is the squared standard deviation.<\/li>\n<li>The variance is not often used in this course.<\/li>\n<li>There are different formulas for the sample and population variance.<\/li>\n<li>Often, we assume that we have a sample (if nothing is stated).<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Your Own Notes<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n","protected":false},"author":865,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-509","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/509","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/865"}],"version-history":[{"count":26,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/509\/revisions"}],"predecessor-version":[{"id":826,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/509\/revisions\/826"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/509\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=509"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=509"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=509"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=509"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}