{"id":128,"date":"2023-12-05T16:04:28","date_gmt":"2023-12-05T21:04:28","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=128"},"modified":"2024-04-18T04:22:11","modified_gmt":"2024-04-18T08:22:11","slug":"measures-of-variance","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/measures-of-variance\/","title":{"raw":"The Standard Deviation","rendered":"The Standard Deviation"},"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 standard deviations.\r\n\r\n<\/div>\r\n<\/div>\r\nThe Standard Deviation (St. Dev.):\r\n<ul>\r\n \t<li>Is the average distance that the numbers are away from the mean.<\/li>\r\n \t<li>I.e., the average of [latex](x_i-\\bar{x})[\/latex].<\/li>\r\n \t<li>It takes into account ALL the numbers when calculating the 'spread' of the data.<\/li>\r\n \t<li>Is a good measure of dispersion of data.<\/li>\r\n<\/ul>\r\nThe SAMPLE standard deviation is defined as:\r\n\r\n\\[s = \\sqrt{\\frac{\\Sigma(x_i-\\bar{x})^2}{n-1}}\\]\r\n\r\nThe POPULATION standard deviation (indicated by \u03c3<em>, <\/em>the LOWER case SIGMA) is given by:\r\n\r\n\\[\\sigma= \\sqrt{\\frac{\\Sigma(x_i-\\mu)^2}{N}}\\]\r\n<h1>Example 6.1.1 - Sunita's Grades<\/h1>\r\nLet us go back to look at Sunita's and Sanjay's grades from a couple of sections back:\r\n<div>\r\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 68.4408%;height: 124px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 36px\">\r\n<th class=\"shaded\" style=\"width: 7.31908%;height: 36px;text-align: center;vertical-align: middle\">Name<\/th>\r\n<th class=\"shaded\" style=\"width: 7.81683%;height: 36px;text-align: center;vertical-align: middle\">Homework<\/th>\r\n<th class=\"shaded\" style=\"width: 12.6027%;height: 36px;text-align: center;vertical-align: middle\">Term Test#1<\/th>\r\n<th class=\"shaded\" style=\"width: 12.7488%;height: 36px;text-align: center;vertical-align: middle\">Term Test#2<\/th>\r\n<th class=\"shaded\" style=\"width: 12.8627%;height: 36px;text-align: center;vertical-align: middle\">Term Test#3<\/th>\r\n<th class=\"shaded\" style=\"width: 12.6032%;height: 36px;text-align: center;vertical-align: middle\">Concept Quizzes<\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 7.31908%;height: 18px;text-align: center;vertical-align: middle\">Sunita<\/td>\r\n<td style=\"width: 7.81683%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td style=\"width: 12.6027%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td style=\"width: 12.7488%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td style=\"width: 12.8627%;height: 18px;text-align: center;vertical-align: middle\">82<\/td>\r\n<td style=\"width: 12.6032%;height: 18px;text-align: center;vertical-align: middle\">83<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 7.31908%;height: 18px;text-align: center;vertical-align: middle\">Sanjay<\/td>\r\n<td style=\"width: 7.81683%;height: 18px;text-align: center;vertical-align: middle\">48<\/td>\r\n<td style=\"width: 12.6027%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td style=\"width: 12.7488%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td style=\"width: 12.8627%;height: 18px;text-align: center;vertical-align: middle\">98<\/td>\r\n<td style=\"width: 12.6032%;height: 18px;text-align: center;vertical-align: middle\">99<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div>\r\n<ul>\r\n \t<li><span style=\"text-align: initial\">Remember, Sunita's average grade was 405\/5 = 81 and her median grade was 80.<\/span><\/li>\r\n \t<li>The range for Sunita's grades = [latex]83 - 80 = 3[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<ul>\r\n \t<li>Sunita's range is small because ALL the numbers are close to the mean of 81.<\/li>\r\n \t<li>We will regard this data as a sample, and therefore we will talk about [latex]\\bar{x} [\/latex]and [latex]s[\/latex].<\/li>\r\n<\/ul>\r\nLet us work Sunita's standard deviation of her grades. It is easiest if we put it in the following table:\r\n<table class=\"grid aligncenter\" style=\"width: 403px\">\r\n<tbody>\r\n<tr>\r\n<th class=\"shaded\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">[latex]x_i[\/latex]<\/th>\r\n<th class=\"shaded\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">[latex]\\bar{x}[\/latex]<\/th>\r\n<th class=\"shaded\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">[latex](x_i -\\bar{x})[\/latex]<\/th>\r\n<th class=\"shaded\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">[latex](x_i-\\bar{x})^2[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\r\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">-1<\/td>\r\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\r\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">-1<\/td>\r\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">80<\/td>\r\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\r\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">-1<\/td>\r\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">82<\/td>\r\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\r\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">1<\/td>\r\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">83<\/td>\r\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\r\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">2<\/td>\r\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\"><\/td>\r\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\"><\/td>\r\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">SUM<\/td>\r\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">[latex]\\Sigma = 8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis gives the following standard deviation for Sunita's grades:\r\n\r\n\\[s = \\sqrt{\\frac{\\Sigma (x_i-\\bar{x})^2}{n-1}} = \\sqrt{\\frac{8}{5-1}}=\\sqrt{2}=1.41\\]\r\n<h1>Example 6.1.2 - Sanjay's Vs. Sunita's Grades (Excel)<\/h1>\r\nNow, let us go back to look at Sanjay's grades:\r\n<ul>\r\n \t<li>For Sanjay, the range= [latex]99 - 48 = 51[\/latex].<\/li>\r\n \t<li>Sanjay's range is large because SOME of the numbers are not so very close to the mean of 81.<\/li>\r\n<\/ul>\r\nLet us calculate Sanjay's standard deviation in Excel and compare it to Sunita's standard deviation:\r\n\r\n<img class=\"alignnone wp-image-579\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-300x54.jpg\" alt=\"\" width=\"711\" height=\"128\" \/>\r\n<ul>\r\n \t<li>We see that the standard deviation of Sanjay's grades is much higher.<\/li>\r\n \t<li>Ie: Sanjay's grades have a larger deviation, on average, from the average grade of 81.<\/li>\r\n \t<li><a style=\"text-align: initial\" href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/6-2-1.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 Standard Deviation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>When the standard deviation is small, the numbers are close to the mean.<\/li>\r\n \t<li>When the standard deviation is large, some of the numbers are not close to the mean.<\/li>\r\n \t<li>It measures the average distance that the numbers are away from the mean.<\/li>\r\n \t<li>There are different formulas for the sample and population standard deviation.<\/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 standard deviations.<\/p>\n<\/div>\n<\/div>\n<p>The Standard Deviation (St. Dev.):<\/p>\n<ul>\n<li>Is the average distance that the numbers are away from the mean.<\/li>\n<li>I.e., the average of [latex](x_i-\\bar{x})[\/latex].<\/li>\n<li>It takes into account ALL the numbers when calculating the &#8216;spread&#8217; of the data.<\/li>\n<li>Is a good measure of dispersion of data.<\/li>\n<\/ul>\n<p>The SAMPLE standard deviation is defined as:<\/p>\n<p>\\[s = \\sqrt{\\frac{\\Sigma(x_i-\\bar{x})^2}{n-1}}\\]<\/p>\n<p>The POPULATION standard deviation (indicated by \u03c3<em>, <\/em>the LOWER case SIGMA) is given by:<\/p>\n<p>\\[\\sigma= \\sqrt{\\frac{\\Sigma(x_i-\\mu)^2}{N}}\\]<\/p>\n<h1>Example 6.1.1 &#8211; Sunita&#8217;s Grades<\/h1>\n<p>Let us go back to look at Sunita&#8217;s and Sanjay&#8217;s grades from a couple of sections back:<\/p>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 68.4408%;height: 124px\">\n<tbody>\n<tr style=\"height: 36px\">\n<th class=\"shaded\" style=\"width: 7.31908%;height: 36px;text-align: center;vertical-align: middle\">Name<\/th>\n<th class=\"shaded\" style=\"width: 7.81683%;height: 36px;text-align: center;vertical-align: middle\">Homework<\/th>\n<th class=\"shaded\" style=\"width: 12.6027%;height: 36px;text-align: center;vertical-align: middle\">Term Test#1<\/th>\n<th class=\"shaded\" style=\"width: 12.7488%;height: 36px;text-align: center;vertical-align: middle\">Term Test#2<\/th>\n<th class=\"shaded\" style=\"width: 12.8627%;height: 36px;text-align: center;vertical-align: middle\">Term Test#3<\/th>\n<th class=\"shaded\" style=\"width: 12.6032%;height: 36px;text-align: center;vertical-align: middle\">Concept Quizzes<\/th>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 7.31908%;height: 18px;text-align: center;vertical-align: middle\">Sunita<\/td>\n<td style=\"width: 7.81683%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\n<td style=\"width: 12.6027%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\n<td style=\"width: 12.7488%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\n<td style=\"width: 12.8627%;height: 18px;text-align: center;vertical-align: middle\">82<\/td>\n<td style=\"width: 12.6032%;height: 18px;text-align: center;vertical-align: middle\">83<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 7.31908%;height: 18px;text-align: center;vertical-align: middle\">Sanjay<\/td>\n<td style=\"width: 7.81683%;height: 18px;text-align: center;vertical-align: middle\">48<\/td>\n<td style=\"width: 12.6027%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\n<td style=\"width: 12.7488%;height: 18px;text-align: center;vertical-align: middle\">80<\/td>\n<td style=\"width: 12.8627%;height: 18px;text-align: center;vertical-align: middle\">98<\/td>\n<td style=\"width: 12.6032%;height: 18px;text-align: center;vertical-align: middle\">99<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div>\n<ul>\n<li><span style=\"text-align: initial\">Remember, Sunita&#8217;s average grade was 405\/5 = 81 and her median grade was 80.<\/span><\/li>\n<li>The range for Sunita&#8217;s grades = [latex]83 - 80 = 3[\/latex].<\/li>\n<\/ul>\n<\/div>\n<ul>\n<li>Sunita&#8217;s range is small because ALL the numbers are close to the mean of 81.<\/li>\n<li>We will regard this data as a sample, and therefore we will talk about [latex]\\bar{x}[\/latex]and [latex]s[\/latex].<\/li>\n<\/ul>\n<p>Let us work Sunita&#8217;s standard deviation of her grades. It is easiest if we put it in the following table:<\/p>\n<table class=\"grid aligncenter\" style=\"width: 403px\">\n<tbody>\n<tr>\n<th class=\"shaded\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">[latex]x_i[\/latex]<\/th>\n<th class=\"shaded\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">[latex]\\bar{x}[\/latex]<\/th>\n<th class=\"shaded\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">[latex](x_i -\\bar{x})[\/latex]<\/th>\n<th class=\"shaded\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">[latex](x_i-\\bar{x})^2[\/latex]<\/th>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">80<\/td>\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">-1<\/td>\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">80<\/td>\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">-1<\/td>\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">80<\/td>\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">-1<\/td>\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">82<\/td>\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">1<\/td>\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\">83<\/td>\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\">81<\/td>\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">2<\/td>\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">4<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 118.181px;text-align: center;vertical-align: middle\"><\/td>\n<td class=\"border\" style=\"width: 97.75px;text-align: center;vertical-align: middle\"><\/td>\n<td class=\"border\" style=\"width: 66.3889px;text-align: center;vertical-align: middle\">SUM<\/td>\n<td class=\"border\" style=\"width: 126.347px;text-align: center;vertical-align: middle\">[latex]\\Sigma = 8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This gives the following standard deviation for Sunita&#8217;s grades:<\/p>\n<p>\\[s = \\sqrt{\\frac{\\Sigma (x_i-\\bar{x})^2}{n-1}} = \\sqrt{\\frac{8}{5-1}}=\\sqrt{2}=1.41\\]<\/p>\n<h1>Example 6.1.2 &#8211; Sanjay&#8217;s Vs. Sunita&#8217;s Grades (Excel)<\/h1>\n<p>Now, let us go back to look at Sanjay&#8217;s grades:<\/p>\n<ul>\n<li>For Sanjay, the range= [latex]99 - 48 = 51[\/latex].<\/li>\n<li>Sanjay&#8217;s range is large because SOME of the numbers are not so very close to the mean of 81.<\/li>\n<\/ul>\n<p>Let us calculate Sanjay&#8217;s standard deviation in Excel and compare it to Sunita&#8217;s standard deviation:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-579\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-300x54.jpg\" alt=\"\" width=\"711\" height=\"128\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-300x54.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-1024x185.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-768x138.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-65x12.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-225x41.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1-350x63.jpg 350w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Example6-1-2_Image1-1.jpg 1431w\" sizes=\"auto, (max-width: 711px) 100vw, 711px\" \/><\/p>\n<ul>\n<li>We see that the standard deviation of Sanjay&#8217;s grades is much higher.<\/li>\n<li>Ie: Sanjay&#8217;s grades have a larger deviation, on average, from the average grade of 81.<\/li>\n<li><a style=\"text-align: initial\" href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/6-2-1.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 Standard Deviation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>When the standard deviation is small, the numbers are close to the mean.<\/li>\n<li>When the standard deviation is large, some of the numbers are not close to the mean.<\/li>\n<li>It measures the average distance that the numbers are away from the mean.<\/li>\n<li>There are different formulas for the sample and population standard deviation.<\/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":883,"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-128","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/128","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\/883"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/128\/revisions"}],"predecessor-version":[{"id":827,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/128\/revisions\/827"}],"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\/128\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=128"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=128"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=128"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}