{"id":345,"date":"2022-02-21T17:54:34","date_gmt":"2022-02-21T17:54:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=345"},"modified":"2022-05-20T16:46:53","modified_gmt":"2022-05-20T16:46:53","slug":"summary-of-interpreting-the-mean-and-median-of-a-dataset","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/summary-of-interpreting-the-mean-and-median-of-a-dataset\/","title":{"raw":"Summary of Interpreting the Mean and Median of a Data Set","rendered":"Summary of Interpreting the Mean and Median of a Data Set"},"content":{"raw":"<h3><strong>Monitoring Your Readiness<\/strong><\/h3>\r\n<div class=\"textbox\">\r\n\r\nTo effectively plan and use your time wisely, it helps to think about what you know and do not know. For each of the following, rate how confident you are that you can successfully do that skill. Use the following descriptions to rate yourself:\r\n\r\n5\u2014I am extremely confident I can do this task.\r\n\r\n4\u2014I am somewhat confident I can do this task.\r\n\r\n3\u2014I am not sure how confident I am.\r\n\r\n2\u2014I am not very confident I can do this task.\r\n\r\n1\u2014I am definitely not confident I can do this task.\r\n\r\n<\/div>\r\n<h3 style=\"text-align: center;\">Skills Needed for Interpreting the Mean and Median of a Data Set: Forming Connections<\/h3>\r\n<div align=\"center\">\r\n<table style=\"height: 107px;\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<td style=\"width: 307.984px; height: 10px; text-align: center;\"><strong>Skill or Concept: I can . . .<\/strong><\/td>\r\n<td style=\"width: 253.984px; height: 10px; text-align: center;\"><strong>Questions to check your understanding<\/strong><\/td>\r\n<td style=\"width: 113.031px; height: 10px; text-align: center;\"><strong>Rating\u00a0<\/strong><strong>from 1 to 5<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td style=\"width: 307.984px; height: 26px;\">Calculate the median of a data set.<\/td>\r\n<td style=\"width: 253.984px; height: 26px; text-align: center;\">1<\/td>\r\n<td style=\"width: 113.031px; height: 26px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 10px;\">\r\n<td style=\"width: 307.984px; height: 10px;\">Calculate the mean of a data set.<\/td>\r\n<td style=\"width: 253.984px; height: 10px; text-align: center;\">3<\/td>\r\n<td style=\"width: 113.031px; height: 10px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 31px;\">\r\n<td style=\"width: 307.984px; height: 31px;\">Interpret the median of a data set.<\/td>\r\n<td style=\"width: 253.984px; height: 31px; text-align: center;\">2<\/td>\r\n<td style=\"width: 113.031px; height: 31px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 10px;\">\r\n<td style=\"width: 307.984px; height: 10px;\">Interpret the mean of a data set.<\/td>\r\n<td style=\"width: 253.984px; height: 10px; text-align: center;\">4<\/td>\r\n<td style=\"width: 113.031px; height: 10px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"width: 307.984px; height: 20px;\">Use the terms \u201cleft skewed,\u201d \u201cright skewed,\u201d \u201csymmetric,\u201d and \u201coutlier\u201d to describe the features of a data set.<\/td>\r\n<td style=\"width: 253.984px; height: 20px; text-align: center;\">6, 10-12, 14<\/td>\r\n<td style=\"width: 113.031px; height: 20px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 307.984px;\">Make connections between the distribution of a data set and how the mean and median relate.<\/td>\r\n<td style=\"width: 253.984px; text-align: center;\">5-9<\/td>\r\n<td style=\"width: 113.031px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow use the ratings to get ready for your next in-class activity. <strong>If your rating is a 3 or below, you should get help with the material before class.<\/strong> Remember, your instructor is going to assume that you are confident with the material and will not take class time to answer questions about it.\r\n\r\nWays to get help:\r\n<ul>\r\n \t<li>See your instructor before class for help.<\/li>\r\n \t<li>Ask your instructor for on-campus resources.<\/li>\r\n \t<li>Set up a study group with classmates so you can help each other.<\/li>\r\n \t<li>Work with a tutor.<\/li>\r\n<\/ul>\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The median stays relatively fixed in a data set if one value changes by a large amount, the mean does not. This is indication that the mean is sensitive to the presence of extreme values in the data set.<\/li>\r\n \t<li>When a distribution is symmetric, the mean and median occupy the same value. Under a skew, the mean is \"pulled\" in the direction of the outliers:\r\n<ul>\r\n \t<li>Right-skewed: the mean is greater than the median.<\/li>\r\n \t<li>Left-skewed: the mean is less than the median.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The mean, under certain conditions, can be a misleading indicator of a \"typical\" observation value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Study Tips: Evidence-based strategies for learning<\/h2>\r\n<ul>\r\n \t<li>Recall the mnemonic for skewness:\u00a0<strong>L<\/strong>eft-skewed distributions have tails to the <strong>L<\/strong>eft (<strong>L<\/strong>ower values).\u00a0<strong>R<\/strong>ight-skewed distributions have tails to the\u00a0<strong>R<\/strong>ight (<strong>R<\/strong>ising values).<\/li>\r\n \t<li>Practice your skill using the tool by using it from memory (without looking at notes or directions) to create a histogram for a list of data values you either enter by hand or paste from a spreadsheet. Locate the mean and median in the tool from memory.\r\n<ul>\r\n \t<li>What do you know well about this process? What is still challenging?<\/li>\r\n \t<li>Try entering incorrect data to identify clues that could let you know that your distribution is not reasonable.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Explain out loud to a friend, real or imaginary, how the mean relates to the median in distributions that are skewed left or right. Paraphrase any technical language and include an explanation of\u00a0<em>why <\/em>this happens.<\/li>\r\n \t<li>Use a mnemonic to remember that the mean is pulled in the direction of the skew. For example, \"the median is always the middle.\"<\/li>\r\n \t<li>Re-read the\u00a0<em>Forming Connections<\/em> activity, pausing after the questions to reflect on how you could know the answers are correct. Make brief notes of understanding and identify areas of confusion that linger.<\/li>\r\n \t<li>Starting from a blank sheet of paper, without consulting the text or your notes, handwrite an explanation of how you can know when it would be inappropriate to use the mean as a measure of center to represent a \"typical\" data value in a data set.<\/li>\r\n<\/ul>\r\n<h2>Foundational Knowledge<\/h2>\r\n<ul>\r\n \t<li><a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/interpreting-the-mean-and-median-of-a-data set-corequisite-support-activity\/\">Support Activity for Interpreting the Mean and Median of a Data Set<\/a><\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572229168\" class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1170572482608\" class=\"definition\">\r\n \t<dt>left-skewed (negative skew)<\/dt>\r\n \t<dd id=\"fs-id1170572482614\">most of the data is bunched up to the right of the graph with a \"tail\" of infrequent values on the left (lower) end of the distribution.<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>\r\n<dl id=\"fs-id1170572482619\" class=\"definition\">\r\n \t<dt>resistant<\/dt>\r\n \t<dd id=\"fs-id1170572482624\">not affected by the skewness of a graph.<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>right-skewed (positive skew)<\/dt>\r\n \t<dd id=\"fs-id1170572229174\">most of the data is bunched up to the left of the graph with a \"tail\" of infrequent values on the right (upper) end of the distribution.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572229190\" class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1170572482683\" class=\"definition\">\r\n \t<dt>symmetric<\/dt>\r\n \t<dd id=\"fs-id1170572482689\">the left and right sides of the distribution (closely) mirror each other. If you drew a vertical line down the center of the distribution and folded the distribution in half, the left and right sides would closely match one another.<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1170572482683\" class=\"definition\">\r\n \t<dt>outlier<\/dt>\r\n \t<dd id=\"fs-id1170572482689\">an unusual or extreme value, given the other values in the data set.<\/dd>\r\n<\/dl>\r\n&nbsp;\r\n<h2>My Skills Checklist:<\/h2>\r\n<ul>\r\n \t<li>I can identify medians as being resistant to influence from skew and outliers.<\/li>\r\n \t<li>I can identify means as not being resistant to influence from skew and outliers.<\/li>\r\n \t<li>I can identify, in certain circumstances, when the mean is misleading.<\/li>\r\n \t<li>I can identify misleading claims made using means and suggest the most appropriate measure of center to use in different situations.<\/li>\r\n<\/ul>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/08025259\/Screen-Shot-2022-03-07-at-6.52.16-PM-225x193.png\" alt=\"Check mark list on clipboard\" \/>\r\n<p style=\"text-align: center;\"><strong>Topic Complete \u2013 now test your understanding in the Self-Check.<\/strong><\/p>\r\n\r\n\r\n<hr \/>\r\n\r\n&nbsp;","rendered":"<h3><strong>Monitoring Your Readiness<\/strong><\/h3>\n<div class=\"textbox\">\n<p>To effectively plan and use your time wisely, it helps to think about what you know and do not know. For each of the following, rate how confident you are that you can successfully do that skill. Use the following descriptions to rate yourself:<\/p>\n<p>5\u2014I am extremely confident I can do this task.<\/p>\n<p>4\u2014I am somewhat confident I can do this task.<\/p>\n<p>3\u2014I am not sure how confident I am.<\/p>\n<p>2\u2014I am not very confident I can do this task.<\/p>\n<p>1\u2014I am definitely not confident I can do this task.<\/p>\n<\/div>\n<h3 style=\"text-align: center;\">Skills Needed for Interpreting the Mean and Median of a Data Set: Forming Connections<\/h3>\n<div style=\"margin: auto;\">\n<table style=\"height: 107px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<td style=\"width: 307.984px; height: 10px; text-align: center;\"><strong>Skill or Concept: I can . . .<\/strong><\/td>\n<td style=\"width: 253.984px; height: 10px; text-align: center;\"><strong>Questions to check your understanding<\/strong><\/td>\n<td style=\"width: 113.031px; height: 10px; text-align: center;\"><strong>Rating\u00a0<\/strong><strong>from 1 to 5<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td style=\"width: 307.984px; height: 26px;\">Calculate the median of a data set.<\/td>\n<td style=\"width: 253.984px; height: 26px; text-align: center;\">1<\/td>\n<td style=\"width: 113.031px; height: 26px;\"><\/td>\n<\/tr>\n<tr style=\"height: 10px;\">\n<td style=\"width: 307.984px; height: 10px;\">Calculate the mean of a data set.<\/td>\n<td style=\"width: 253.984px; height: 10px; text-align: center;\">3<\/td>\n<td style=\"width: 113.031px; height: 10px;\"><\/td>\n<\/tr>\n<tr style=\"height: 31px;\">\n<td style=\"width: 307.984px; height: 31px;\">Interpret the median of a data set.<\/td>\n<td style=\"width: 253.984px; height: 31px; text-align: center;\">2<\/td>\n<td style=\"width: 113.031px; height: 31px;\"><\/td>\n<\/tr>\n<tr style=\"height: 10px;\">\n<td style=\"width: 307.984px; height: 10px;\">Interpret the mean of a data set.<\/td>\n<td style=\"width: 253.984px; height: 10px; text-align: center;\">4<\/td>\n<td style=\"width: 113.031px; height: 10px;\"><\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"width: 307.984px; height: 20px;\">Use the terms \u201cleft skewed,\u201d \u201cright skewed,\u201d \u201csymmetric,\u201d and \u201coutlier\u201d to describe the features of a data set.<\/td>\n<td style=\"width: 253.984px; height: 20px; text-align: center;\">6, 10-12, 14<\/td>\n<td style=\"width: 113.031px; height: 20px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 307.984px;\">Make connections between the distribution of a data set and how the mean and median relate.<\/td>\n<td style=\"width: 253.984px; text-align: center;\">5-9<\/td>\n<td style=\"width: 113.031px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Now use the ratings to get ready for your next in-class activity. <strong>If your rating is a 3 or below, you should get help with the material before class.<\/strong> Remember, your instructor is going to assume that you are confident with the material and will not take class time to answer questions about it.<\/p>\n<p>Ways to get help:<\/p>\n<ul>\n<li>See your instructor before class for help.<\/li>\n<li>Ask your instructor for on-campus resources.<\/li>\n<li>Set up a study group with classmates so you can help each other.<\/li>\n<li>Work with a tutor.<\/li>\n<\/ul>\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The median stays relatively fixed in a data set if one value changes by a large amount, the mean does not. This is indication that the mean is sensitive to the presence of extreme values in the data set.<\/li>\n<li>When a distribution is symmetric, the mean and median occupy the same value. Under a skew, the mean is &#8220;pulled&#8221; in the direction of the outliers:\n<ul>\n<li>Right-skewed: the mean is greater than the median.<\/li>\n<li>Left-skewed: the mean is less than the median.<\/li>\n<\/ul>\n<\/li>\n<li>The mean, under certain conditions, can be a misleading indicator of a &#8220;typical&#8221; observation value.<\/li>\n<\/ul>\n<\/div>\n<h2>Study Tips: Evidence-based strategies for learning<\/h2>\n<ul>\n<li>Recall the mnemonic for skewness:\u00a0<strong>L<\/strong>eft-skewed distributions have tails to the <strong>L<\/strong>eft (<strong>L<\/strong>ower values).\u00a0<strong>R<\/strong>ight-skewed distributions have tails to the\u00a0<strong>R<\/strong>ight (<strong>R<\/strong>ising values).<\/li>\n<li>Practice your skill using the tool by using it from memory (without looking at notes or directions) to create a histogram for a list of data values you either enter by hand or paste from a spreadsheet. Locate the mean and median in the tool from memory.\n<ul>\n<li>What do you know well about this process? What is still challenging?<\/li>\n<li>Try entering incorrect data to identify clues that could let you know that your distribution is not reasonable.<\/li>\n<\/ul>\n<\/li>\n<li>Explain out loud to a friend, real or imaginary, how the mean relates to the median in distributions that are skewed left or right. Paraphrase any technical language and include an explanation of\u00a0<em>why <\/em>this happens.<\/li>\n<li>Use a mnemonic to remember that the mean is pulled in the direction of the skew. For example, &#8220;the median is always the middle.&#8221;<\/li>\n<li>Re-read the\u00a0<em>Forming Connections<\/em> activity, pausing after the questions to reflect on how you could know the answers are correct. Make brief notes of understanding and identify areas of confusion that linger.<\/li>\n<li>Starting from a blank sheet of paper, without consulting the text or your notes, handwrite an explanation of how you can know when it would be inappropriate to use the mean as a measure of center to represent a &#8220;typical&#8221; data value in a data set.<\/li>\n<\/ul>\n<h2>Foundational Knowledge<\/h2>\n<ul>\n<li><a href=\"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/interpreting-the-mean-and-median-of-a-data set-corequisite-support-activity\/\">Support Activity for Interpreting the Mean and Median of a Data Set<\/a><\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572229168\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>left-skewed (negative skew)<\/dt>\n<dd id=\"fs-id1170572482614\">most of the data is bunched up to the right of the graph with a &#8220;tail&#8221; of infrequent values on the left (lower) end of the distribution.<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572482619\" class=\"definition\">\n<dt>resistant<\/dt>\n<dd id=\"fs-id1170572482624\">not affected by the skewness of a graph.<\/dd>\n<\/dl>\n<p> \tright-skewed (positive skew)<br \/>\n \tmost of the data is bunched up to the left of the graph with a &#8220;tail&#8221; of infrequent values on the right (upper) end of the distribution.<\/p>\n<dl id=\"fs-id1170572229190\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>symmetric<\/dt>\n<dd id=\"fs-id1170572482689\">the left and right sides of the distribution (closely) mirror each other. If you drew a vertical line down the center of the distribution and folded the distribution in half, the left and right sides would closely match one another.<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572482683\" class=\"definition\">\n<dt>outlier<\/dt>\n<dd id=\"fs-id1170572482689\">an unusual or extreme value, given the other values in the data set.<\/dd>\n<\/dl>\n<p>&nbsp;<\/p>\n<h2>My Skills Checklist:<\/h2>\n<ul>\n<li>I can identify medians as being resistant to influence from skew and outliers.<\/li>\n<li>I can identify means as not being resistant to influence from skew and outliers.<\/li>\n<li>I can identify, in certain circumstances, when the mean is misleading.<\/li>\n<li>I can identify misleading claims made using means and suggest the most appropriate measure of center to use in different situations.<\/li>\n<\/ul>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5772\/2022\/02\/08025259\/Screen-Shot-2022-03-07-at-6.52.16-PM-225x193.png\" alt=\"Check mark list on clipboard\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Topic Complete \u2013 now test your understanding in the Self-Check.<\/strong><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-345\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">Public domain content<\/div><ul class=\"citation-list\"><li>Roller hockey ball overlaid with a green check. <strong>Authored by<\/strong>: Parutakupiu. <strong>Provided by<\/strong>: Wikimedia Commons. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Rollerhockeyball_check.svg\">https:\/\/commons.wikimedia.org\/wiki\/File:Rollerhockeyball_check.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":175116,"menu_order":49,"template":"","meta":{"_candela_citation":"[{\"type\":\"pd\",\"description\":\"Roller hockey ball overlaid with a green check\",\"author\":\"Parutakupiu\",\"organization\":\"Wikimedia Commons\",\"url\":\"https:\/\/commons.wikimedia.org\/wiki\/File:Rollerhockeyball_check.svg\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-345","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/345","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/users\/175116"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/345\/revisions"}],"predecessor-version":[{"id":1160,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/345\/revisions\/1160"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/parts\/1252"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/345\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=345"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=345"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=345"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}