{"id":5,"date":"2023-11-28T16:54:51","date_gmt":"2023-11-28T21:54:51","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?p=5"},"modified":"2024-04-18T04:23:10","modified_gmt":"2024-04-18T08:23:10","slug":"chapter-1","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/chapter-1\/","title":{"raw":"The Mean","rendered":"The Mean"},"content":{"raw":"<div class=\"The Mean\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Outcomes<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Understand and calculate means (averages) both by hand and in Excel.<\/div>\r\n<\/div>\r\n<img class=\" wp-image-428 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-300x78.jpg\" alt=\"\" width=\"208\" height=\"54\" \/>\r\n<ul>\r\n \t<li>The sum of the numbers divided by the number of numbers.<\/li>\r\n \t<li>Also called the 'Arithmetic Mean' or Average<\/li>\r\n \t<li>The formula is: [latex] \\bar{x} = \\frac{\\sum_{i=1}^{n} X_i }{n}[\/latex]<\/li>\r\n<\/ul>\r\n<h1>Understanding Summation Notation (\u03a3)<\/h1>\r\n<ul>\r\n \t<li>\u03a3 means the summation (to add up all of the x terms).<\/li>\r\n \t<li>It is often used in Statistics<\/li>\r\n \t<li>Many metrics require us to sum up the values or squared values of a data set<\/li>\r\n \t<li>And it is quicker to use the 'summation notation' to express that sum.<\/li>\r\n<\/ul>\r\n<h2>EXAMPLE 1.1 \u2014 USING SUMMATION NOTATION<\/h2>\r\nIf the data set 'X' contains the following data:\r\n<p style=\"padding-left: 40px\">x<sub>1\u00a0<\/sub>= 55, x<sub>2\u00a0<\/sub>= 62, x<sub>3\u00a0<\/sub>= 67, x<sub>4\u00a0<\/sub>= 60, x<sub>5\u00a0<\/sub>= 71, x<sub>6\u00a0<\/sub>= 66<\/p>\r\nthen:\r\n<p style=\"padding-left: 40px\">\u03a3 x<sub>i<\/sub> = x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> + x<sub>4<\/sub> + x<sub>5<\/sub>\u00a0+ x<sub>6<\/sub><\/p>\r\n<p style=\"padding-left: 80px\">= 55 + 62 + 67 + 60 + 71 + 66<\/p>\r\n<p style=\"padding-left: 80px\">= 381<\/p>\r\n\r\n<h1>Calculating Mean Values by Hand (Exercise)<\/h1>\r\nTo calculate an average (mean), we simply do the following:\r\n<ol>\r\n \t<li>Sum up all of the values (\u03a3 x<sub>i<\/sub>)<\/li>\r\n \t<li>Count the number of values in the data set (n)<\/li>\r\n \t<li>Divide the sum of all the terms by the number of data in the data set\u00a0(\u03a3 x<sub>i <\/sub>\/ n)<\/li>\r\n<\/ol>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Knowledge Check 1.1.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div>Try the following exercise below to practice what you've learned:<\/div>\r\n<div><\/div>\r\n<div>[h5p id=\"1\"]<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div>Conclusion: The average income per resident is roughly $63,500.<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Calculating the MEAN USing Excel (Video)<\/h1>\r\nWe will use the <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/calculate-the-average-of-a-group-of-numbers-e158ef61-421c-4839-8290-34d7b1e68283#:~:text=Click%20a%20cell%20below%2C%20or,Average%2C%20and%20then%20press%20Enter.\">average call<\/a> in Excel:\r\n<ul>\r\n \t<li>Type: ' = average()'<\/li>\r\n \t<li>Select all of the elements across the row or column you would like to average<\/li>\r\n \t<li>Hit enter<\/li>\r\n<\/ul>\r\n<h2>EXAMPLE 1.2 \u2014 USING EXCEL TO FIND AN AVERAGE<\/h2>\r\n<ul>\r\n \t<li>Let us again, average out the incomes from the apartment residents,<\/li>\r\n \t<li>But this time, we will use Excel (see the video below).<\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/1-1-2.xlsx\">Click here to download the file shown in the video below<\/a>:<\/li>\r\n<\/ul>\r\n[embed]https:\/\/youtu.be\/MJhsU4ocWCo[\/embed]\r\n<h1>Sample vs population Notation<\/h1>\r\n<h3>Populations<\/h3>\r\n<ul>\r\n \t<li>A population is the entire group that you want to draw conclusions about<\/li>\r\n \t<li>The population mean is indicated by the symbol \u03bc (mu)<\/li>\r\n \t<li>Ex: All students currently registered in the BCIT School of Business make up the population<\/li>\r\n<\/ul>\r\n<h3>Samples<\/h3>\r\n<ul>\r\n \t<li>A sample is the specific group that you will collect data from that will not include all members of the population<\/li>\r\n \t<li>The sample mean is indicated by the symbol x\u0304 (x-bar)<\/li>\r\n \t<li>Ex: A randomly selected group of students from BCIT whose last name begins with the letter \u201dB\u201d.<\/li>\r\n<\/ul>\r\n<h1>Key Takeaways (exercise)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: The Mean<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"2\"]\r\n\r\n[h5p id=\"3\"]\r\n\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\"]\r\n\r\n<\/div>","rendered":"<div class=\"The Mean\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Outcomes<\/p>\n<\/header>\n<div class=\"textbox__content\">Understand and calculate means (averages) both by hand and in Excel.<\/div>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-428 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-300x78.jpg\" alt=\"\" width=\"208\" height=\"54\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-300x78.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-1024x266.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-768x199.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-1536x399.jpg 1536w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-2048x531.jpg 2048w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-65x17.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-225x58.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/11\/Mean-350x91.jpg 350w\" sizes=\"auto, (max-width: 208px) 100vw, 208px\" \/><\/p>\n<ul>\n<li>The sum of the numbers divided by the number of numbers.<\/li>\n<li>Also called the &#8216;Arithmetic Mean&#8217; or Average<\/li>\n<li>The formula is: [latex]\\bar{x} = \\frac{\\sum_{i=1}^{n} X_i }{n}[\/latex]<\/li>\n<\/ul>\n<h1>Understanding Summation Notation (\u03a3)<\/h1>\n<ul>\n<li>\u03a3 means the summation (to add up all of the x terms).<\/li>\n<li>It is often used in Statistics<\/li>\n<li>Many metrics require us to sum up the values or squared values of a data set<\/li>\n<li>And it is quicker to use the &#8216;summation notation&#8217; to express that sum.<\/li>\n<\/ul>\n<h2>EXAMPLE 1.1 \u2014 USING SUMMATION NOTATION<\/h2>\n<p>If the data set &#8216;X&#8217; contains the following data:<\/p>\n<p style=\"padding-left: 40px\">x<sub>1\u00a0<\/sub>= 55, x<sub>2\u00a0<\/sub>= 62, x<sub>3\u00a0<\/sub>= 67, x<sub>4\u00a0<\/sub>= 60, x<sub>5\u00a0<\/sub>= 71, x<sub>6\u00a0<\/sub>= 66<\/p>\n<p>then:<\/p>\n<p style=\"padding-left: 40px\">\u03a3 x<sub>i<\/sub> = x<sub>1<\/sub> + x<sub>2<\/sub> + x<sub>3<\/sub> + x<sub>4<\/sub> + x<sub>5<\/sub>\u00a0+ x<sub>6<\/sub><\/p>\n<p style=\"padding-left: 80px\">= 55 + 62 + 67 + 60 + 71 + 66<\/p>\n<p style=\"padding-left: 80px\">= 381<\/p>\n<h1>Calculating Mean Values by Hand (Exercise)<\/h1>\n<p>To calculate an average (mean), we simply do the following:<\/p>\n<ol>\n<li>Sum up all of the values (\u03a3 x<sub>i<\/sub>)<\/li>\n<li>Count the number of values in the data set (n)<\/li>\n<li>Divide the sum of all the terms by the number of data in the data set\u00a0(\u03a3 x<sub>i <\/sub>\/ n)<\/li>\n<\/ol>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Knowledge Check 1.1.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div>Try the following exercise below to practice what you&#8217;ve learned:<\/div>\n<div><\/div>\n<div>\n<div id=\"h5p-1\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-1\" class=\"h5p-iframe\" data-content-id=\"1\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"OPMT 1130 - 1.1.2 - Finding the Mean\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div><\/div>\n<div>Conclusion: The average income per resident is roughly $63,500.<\/div>\n<\/div>\n<\/div>\n<h1>Calculating the MEAN USing Excel (Video)<\/h1>\n<p>We will use the <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/calculate-the-average-of-a-group-of-numbers-e158ef61-421c-4839-8290-34d7b1e68283#:~:text=Click%20a%20cell%20below%2C%20or,Average%2C%20and%20then%20press%20Enter.\">average call<\/a> in Excel:<\/p>\n<ul>\n<li>Type: &#8216; = average()&#8217;<\/li>\n<li>Select all of the elements across the row or column you would like to average<\/li>\n<li>Hit enter<\/li>\n<\/ul>\n<h2>EXAMPLE 1.2 \u2014 USING EXCEL TO FIND AN AVERAGE<\/h2>\n<ul>\n<li>Let us again, average out the incomes from the apartment residents,<\/li>\n<li>But this time, we will use Excel (see the video below).<\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/1-1-2.xlsx\">Click here to download the file shown in the video below<\/a>:<\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Finding an average (mean) using Excel\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/MJhsU4ocWCo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>Sample vs population Notation<\/h1>\n<h3>Populations<\/h3>\n<ul>\n<li>A population is the entire group that you want to draw conclusions about<\/li>\n<li>The population mean is indicated by the symbol \u03bc (mu)<\/li>\n<li>Ex: All students currently registered in the BCIT School of Business make up the population<\/li>\n<\/ul>\n<h3>Samples<\/h3>\n<ul>\n<li>A sample is the specific group that you will collect data from that will not include all members of the population<\/li>\n<li>The sample mean is indicated by the symbol x\u0304 (x-bar)<\/li>\n<li>Ex: A randomly selected group of students from BCIT whose last name begins with the letter \u201dB\u201d.<\/li>\n<\/ul>\n<h1>Key Takeaways (exercise)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: The Mean<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-2\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-2\" class=\"h5p-iframe\" data-content-id=\"2\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"1.1 Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-3\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-3\" class=\"h5p-iframe\" data-content-id=\"3\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"1.1 Accordion\"><\/iframe><\/div>\n<\/div>\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<\/div>\n","protected":false},"author":865,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[48],"contributor":[],"license":[],"class_list":["post-5","chapter","type-chapter","status-publish","hentry","chapter-type-standard"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/5","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":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/5\/revisions"}],"predecessor-version":[{"id":832,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/5\/revisions\/832"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/5\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=5"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=5"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=5"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}