{"id":42,"date":"2022-05-20T16:59:04","date_gmt":"2022-05-20T16:59:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/interpreting-the-mean-and-median-of-a-dataset-forming-connections\/"},"modified":"2022-08-04T19:52:21","modified_gmt":"2022-08-04T19:52:21","slug":"interpreting-the-mean-and-median-of-a-dataset-forming-connections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/interpreting-the-mean-and-median-of-a-dataset-forming-connections\/","title":{"raw":"Interpreting the Mean and Median of a Dataset: Apply It 1","rendered":"Interpreting the Mean and Median of a Dataset: Apply It 1"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Goals<\/h3>\r\nDuring this activity, you will:\r\n<ul>\r\n \t<li><a href=\"#IdentMislead\">Identify misleading claims made using means<\/a><\/li>\r\n \t<li><a href=\"#MeanOrMedian\">Given characteristics of a distribution including skew and outliers, identify under which conditions it is appropriate to use the mean as a measure of center.<\/a><\/li>\r\n<\/ul>\r\nClick on a skill above to jump to its location in this activity.\r\n\r\n<\/div>\r\n<h2>Is It Worth It?<\/h2>\r\nConsider this scenario. A college basketball player is skilled enough to make an NBA roster and is thinking about dropping out of college this year.\r\n\r\n<strong><img class=\"wp-image-1008 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11194832\/Picture41-300x201.jpg\" alt=\"Lots of hundred dollars bills in a fan shape held in front of someone\" width=\"539\" height=\"361\" \/><\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\n[ohm_question hide_question_numbers=1]241106[\/ohm_question]\r\n\r\n[reveal-answer q=\"340259\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"340259\"]We'll be investigating salary potential in this activity, so for now document your initial thoughts.[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this activity, you'll use a distribution of professional basketball salaries to see how outliers influence different measures of center and how averages can be misleading.\r\n<div class=\"textbox examples\">\r\n<h3>recall<\/h3>\r\nBefore beginning this activity, take a moment to recall the meanings of the terms\u00a0<strong>left-skewed<\/strong>,<strong> right-skewed<\/strong>,<strong> symmetric<\/strong>, and <strong>outlier<\/strong>. You'll need to be able to use those terms to describe features of a data set.\r\n\r\nCore skill: [reveal-answer q=\"504559\"]Define\u00a0<em>skew<\/em> and\u00a0<em>outlier<\/em>[\/reveal-answer]\r\n\r\n[hidden-answer a=\"504559\"]\r\n\r\nWe say the quantitative variable is left-skewed, right-skewed, or symmetric if:\r\n<ul>\r\n \t<li><strong>left-skewed<\/strong>\u00a0(negative skew): most of the data is bunched up to the right of the graph with a tail of infrequent values to the left.<\/li>\r\n \t<li><strong>right-skewed<\/strong>\u00a0(positive skew): most of the data is bunched up to the left of the graph with a tail of infrequent values to the right.<\/li>\r\n \t<li><strong>symmetric:<\/strong>\u00a0values are similarly distributed\u00a0on either side of the mean\/median.<\/li>\r\n<\/ul>\r\nWe consider an\u00a0<strong>outlier<\/strong> to be an unusual or extreme value, given the other values in the data set.\r\n<br>You can find additional resources on skew and outlier in the Needs a Link lesson.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Intro: Starting from a sentence or two discussing Question 1, remind students that they have recently been working to calculate and interpret the mean and median of a data set. That is, the median is the value that splits the data in half, with half the observations above the mean and half below, regardless of the presence of skew or outliers. The median is fixed. But the mean is not; it gets pulled to the left or right of the mean under the presence of skew or outliers. The mean is sensitive to extreme values. So when we see that the mean is higher than the median, we say that it has been \"pulled to the right,\" and we understand the quantitative variable is skewed right. Likewise, if the mean is smaller, we'll say it's been \"pulled to the left,\" and we understand the quantitative variable is skewed left. If the mean and median are similar, though, we understand that the distribution is symmetric. In this activity, we'll use a distribution of professional basketball salaries to explore how skew arises in a quantitative variable and why we must be careful to consider all the characteristics of a quantitative variable's distribution before deciding if the mean or median would be more responsible to use as a measure of a \"typical\" value. ]<\/span>\r\n\r\n<\/div>\r\nBelow is a dotplot of NBA salaries[footnote]\u00a0<em>NBA player salary data set (2017-2018).<\/em> (2018) Kaggle. Retrieved from https:\/\/www.kaggle.com\/koki25ando\/salary\u00a0[\/footnote] for Texas players in the 2017\u20132018 season:\r\n\r\n<strong><img class=\"alignnone wp-image-1009\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11194840\/Picture42-300x62.png\" alt=\"A dotplot labeled &quot;Texas Player Salaries ($),&quot; numbered in increments of 5 million from 0 to 25 million. There are several high stacks of dots between 0 and 5 million. Above 5 million, there is only one stack and it has two dots. There are also several individual dots. One dot near 0 is labeled &quot;Chris Johnson: $25,000.&quot; Another dot is labeled &quot;Chris Paul: $24,599,495.&quot; One more dot is labeled &quot;James Harden: $28,299,399.&quot;\" width=\"948\" height=\"196\" \/><\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\n[ohm_question hide_question_numbers=1]241107[\/ohm_question]\r\n\r\n[reveal-answer q=\"182366\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"182366\"]Visually assess the distribution to describe it.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\n[ohm_question hide_question_numbers=1]241108[\/ohm_question]\r\n\r\n[reveal-answer q=\"613861\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"613861\"]What do <em>you<\/em> think? Support your estimate using what you know about distributions.[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Goals<\/h3>\n<p>During this activity, you will:<\/p>\n<ul>\n<li><a href=\"#IdentMislead\">Identify misleading claims made using means<\/a><\/li>\n<li><a href=\"#MeanOrMedian\">Given characteristics of a distribution including skew and outliers, identify under which conditions it is appropriate to use the mean as a measure of center.<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this activity.<\/p>\n<\/div>\n<h2>Is It Worth It?<\/h2>\n<p>Consider this scenario. A college basketball player is skilled enough to make an NBA roster and is thinking about dropping out of college this year.<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1008 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11194832\/Picture41-300x201.jpg\" alt=\"Lots of hundred dollars bills in a fan shape held in front of someone\" width=\"539\" height=\"361\" \/><\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241106\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241106&theme=oea&iframe_resize_id=ohm241106\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q340259\">Hint<\/span><\/p>\n<div id=\"q340259\" class=\"hidden-answer\" style=\"display: none\">We&#8217;ll be investigating salary potential in this activity, so for now document your initial thoughts.<\/div>\n<\/div>\n<\/div>\n<p>In this activity, you&#8217;ll use a distribution of professional basketball salaries to see how outliers influence different measures of center and how averages can be misleading.<\/p>\n<div class=\"textbox examples\">\n<h3>recall<\/h3>\n<p>Before beginning this activity, take a moment to recall the meanings of the terms\u00a0<strong>left-skewed<\/strong>,<strong> right-skewed<\/strong>,<strong> symmetric<\/strong>, and <strong>outlier<\/strong>. You&#8217;ll need to be able to use those terms to describe features of a data set.<\/p>\n<p>Core skill: <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q504559\">Define\u00a0<em>skew<\/em> and\u00a0<em>outlier<\/em><\/span><\/p>\n<div id=\"q504559\" class=\"hidden-answer\" style=\"display: none\">\n<p>We say the quantitative variable is left-skewed, right-skewed, or symmetric if:<\/p>\n<ul>\n<li><strong>left-skewed<\/strong>\u00a0(negative skew): most of the data is bunched up to the right of the graph with a tail of infrequent values to the left.<\/li>\n<li><strong>right-skewed<\/strong>\u00a0(positive skew): most of the data is bunched up to the left of the graph with a tail of infrequent values to the right.<\/li>\n<li><strong>symmetric:<\/strong>\u00a0values are similarly distributed\u00a0on either side of the mean\/median.<\/li>\n<\/ul>\n<p>We consider an\u00a0<strong>outlier<\/strong> to be an unusual or extreme value, given the other values in the data set.<br \/>\n<br \/>You can find additional resources on skew and outlier in the Needs a Link lesson.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Intro: Starting from a sentence or two discussing Question 1, remind students that they have recently been working to calculate and interpret the mean and median of a data set. That is, the median is the value that splits the data in half, with half the observations above the mean and half below, regardless of the presence of skew or outliers. The median is fixed. But the mean is not; it gets pulled to the left or right of the mean under the presence of skew or outliers. The mean is sensitive to extreme values. So when we see that the mean is higher than the median, we say that it has been &#8220;pulled to the right,&#8221; and we understand the quantitative variable is skewed right. Likewise, if the mean is smaller, we&#8217;ll say it&#8217;s been &#8220;pulled to the left,&#8221; and we understand the quantitative variable is skewed left. If the mean and median are similar, though, we understand that the distribution is symmetric. In this activity, we&#8217;ll use a distribution of professional basketball salaries to explore how skew arises in a quantitative variable and why we must be careful to consider all the characteristics of a quantitative variable&#8217;s distribution before deciding if the mean or median would be more responsible to use as a measure of a &#8220;typical&#8221; value. ]<\/span><\/p>\n<\/div>\n<p>Below is a dotplot of NBA salaries<a class=\"footnote\" title=\"\u00a0NBA player salary data set (2017-2018). (2018) Kaggle. Retrieved from https:\/\/www.kaggle.com\/koki25ando\/salary\u00a0\" id=\"return-footnote-42-1\" href=\"#footnote-42-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> for Texas players in the 2017\u20132018 season:<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1009\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11194840\/Picture42-300x62.png\" alt=\"A dotplot labeled &quot;Texas Player Salaries ($),&quot; numbered in increments of 5 million from 0 to 25 million. There are several high stacks of dots between 0 and 5 million. Above 5 million, there is only one stack and it has two dots. There are also several individual dots. One dot near 0 is labeled &quot;Chris Johnson: $25,000.&quot; Another dot is labeled &quot;Chris Paul: $24,599,495.&quot; One more dot is labeled &quot;James Harden: $28,299,399.&quot;\" width=\"948\" height=\"196\" \/><\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241107\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241107&theme=oea&iframe_resize_id=ohm241107\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q182366\">Hint<\/span><\/p>\n<div id=\"q182366\" class=\"hidden-answer\" style=\"display: none\">Visually assess the distribution to describe it.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241108\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241108&theme=oea&iframe_resize_id=ohm241108\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q613861\">Hint<\/span><\/p>\n<div id=\"q613861\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think? Support your estimate using what you know about distributions.<\/div>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-42-1\">\u00a0<em>NBA player salary data set (2017-2018).<\/em> (2018) Kaggle. Retrieved from https:\/\/www.kaggle.com\/koki25ando\/salary\u00a0 <a href=\"#return-footnote-42-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":17533,"menu_order":35,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-42","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/42","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions"}],"predecessor-version":[{"id":576,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions\/576"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapters\/42\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/media?parent=42"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/pressbooks\/v2\/chapter-type?post=42"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/contributor?post=42"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/alphamodule\/wp-json\/wp\/v2\/license?post=42"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}