{"id":343,"date":"2022-02-21T17:54:02","date_gmt":"2022-02-21T17:54:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/?post_type=chapter&#038;p=343"},"modified":"2022-05-20T16:46:48","modified_gmt":"2022-05-20T16:46:48","slug":"interpreting-the-mean-and-median-of-a-dataset-forming-connections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/chapter\/interpreting-the-mean-and-median-of-a-dataset-forming-connections\/","title":{"raw":"Interpreting the Mean and Median of a Dataset: Forming Connections","rendered":"Interpreting the Mean and Median of a Dataset: Forming Connections"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>objectives for this activity<\/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\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this activity, you'll use a distribution of professional basketball salaries to see that medians are resistant to influence from skew and outliers, while means are not. Importantly, means, in certain circumstances, 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\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>\r\n<h3 id=\"IdentMislead\">Misleading Claims<\/h3>\r\nIn fact, the median salary among Texas NBA players was $[latex]1,577,320[\/latex]. The mean salary was $[latex]5,262,279[\/latex]. Use this information to complete Questions 4-6.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\n[ohm_question hide_question_numbers=1]240626[\/ohm_question]\r\n\r\n[reveal-answer q=\"594557\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"594557\"]Use the salary information given above and compare it to the answer choices.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\n[ohm_question hide_question_numbers=1]241109[\/ohm_question]\r\n\r\n[reveal-answer q=\"319249\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"319249\"]Use your own judgement to estimate the values.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241110[\/ohm_question]\r\n\r\n[reveal-answer q=\"17406\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"17406\"]Use what you know about sensitivity to skew\/outliers.[\/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;\">[Guidance: \"Consider your answers to Questions 3 - 6. [voice over images of the dotplot with the vertical lines drawn] What did you consider to be a \"typical\" salary? What characteristic of this variable's distribution caused the mean to be different from the median?\"]<\/span>\r\n\r\n<\/div>\r\nNow consider the following scenario.\u00a0An NBA recruiter for the Houston Rockets approaches a promising college basketball player and says, \u201cthe typical salary among Texas NBA players is $[latex]5,262,279[\/latex].\u201d\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241111[\/ohm_question]\r\n\r\n[reveal-answer q=\"523741\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"523741\"]Use what you know about sensitivity to skew\/outliers.[\/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;\">[insert a sub-summary here. \"How did you your answer the question, \"is the recruiter's statement misleading?\" Did you consider the mean to be a \"typical\" salary among these NBA players? What could the recruiter have said instead? That it is\u00a0<em>likely\u00a0<\/em>a player would make $5.3 million by joining the team? That is is\u00a0<em>possible\u00a0<\/em>for some highly skilled and talented players? Or would it have been less misleading for the recruiter to have emphasized the median salary of $1.58 million? If you were in the prospective player's position, would you have asked to see the distribution to make your own assessment? Which value would you have used, mean or median, if you were in the recruiter's position?\"]<\/span>\r\n\r\n<\/div>\r\nYou've seen that the mean, under certain conditions, can be a misleading indicator of a \"typical\" observation value, such as the salary of a professional basketball player. Now try to apply this understanding to some other types of data collections.\r\n<h3 id=\"MeanOrMedian\">Appropriate Measures of Center<\/h3>\r\nThree situations are given below in which data is collected on a quantitative variable. For each, visualize what the distribution might look like and make predictions about the shape of the distribution (skewed or symmetric?), the relationship between the mean and median (will they be similar or will the mean be smaller or greater than the median?), and whether or not it would be appropriate to use the mean to represent a \"typical\" observation. Use what you learned about resistance in the previous section,\u00a0<em><a href=\"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/what-to-know-about-4c\/\">What to Know About Interpreting the Mean and Median of a Data Set: 4C<\/a>,<\/em> to guide you.\r\n\r\n<strong>Situation 1: Data are collected on incomes in New York City.<\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\n[ohm_question hide_question_numbers=1]241112[\/ohm_question]\r\n\r\n[reveal-answer q=\"157117\"]Hint[\/reveal-answer][hidden-answer a=\"157117\"]<span style=\"color: #000000;\">To help visualize what the data set might look like, imagine the possible range of salaries in a large, densely populated city. How many incomes will be at the lower end of the range? How many incomes will be at the higher end of the range?<\/span>\u00a0[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\n[ohm_question hide_question_numbers=1]241113[\/ohm_question]\r\n\r\n[reveal-answer q=\"422721\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"422721\"]Use what you know about sensitivity to skew\/outliers.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\n[ohm_question hide_question_numbers=1]241114[\/ohm_question]\r\n\r\n[reveal-answer q=\"362479\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"362479\"]Use what you know about sensitivity to skew\/outliers.[\/hidden-answer]\r\n\r\n<\/div>\r\n<span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\"><strong>Situation 2: Data are collected on GPAs at a local college.<\/strong>\r\n<\/span>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\n[ohm_question hide_question_numbers=1]241115[\/ohm_question]\r\n\r\n<span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">[reveal-answer q=\"668026\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"668026\"]<span style=\"color: #000000;\">Consider the range of possible GPAs and their frequencies. Do students typically perform average to above average? Do most pass their classes? Where would likely outliers appear in the distribution?<\/span>[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 12<\/h3>\r\n[ohm_question hide_question_numbers=1]241116[\/ohm_question]\r\n\r\n[reveal-answer q=\"385366\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"385366\"]Use what you know about sensitivity to skew\/outliers.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 13<\/h3>\r\n[ohm_question hide_question_numbers=1]241117[\/ohm_question]\r\n\r\n[reveal-answer q=\"197297\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"197297\"]Use what you know about sensitivity to skew\/outliers.[\/hidden-answer]\r\n\r\n<\/div>\r\n<span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\"><strong>Situation 3: Data are collected on peoples\u2019 body temperatures.<\/strong>\r\n<\/span>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 14<\/h3>\r\n[ohm_question hide_question_numbers=1]241118[\/ohm_question]\r\n\r\n<span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">[reveal-answer q=\"731082\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"731082\"]<span style=\"color: #000000;\">Would you expect to see extreme values in a distribution of people's body temperatures?<\/span>[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 15<\/h3>\r\n[ohm_question hide_question_numbers=1]241120[\/ohm_question]\r\n\r\n[reveal-answer q=\"156796\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"156796\"]Use what you know about sensitivity to skew\/outliers.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 16<\/h3>\r\n[ohm_question hide_question_numbers=1]241121[\/ohm_question]\r\n\r\n[reveal-answer q=\"672649\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"672649\"]Use what you know about sensitivity to skew\/outliers.[\/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;\">[Wrap-up: Provide a transition from these particular examples to larger situations in which a quantitative variable would tend to be skewed or symmetric: if the data would tend toward a bunched-up group of values but contain some extreme values, what would the shape of the distribution look like? If data were distributed on the graph \"as though it had fallen through a funnel onto a plane\" what would it look like? Then show and discuss the simulation at <a style=\"background-color: #e6daf7;\" href=\"https:\/\/dcmathpathways.shinyapps.io\/MeanvsMedian\/\">https:\/\/dcmathpathways.shinyapps.io\/MeanvsMedian\/<\/a> .Finally, show some distributions and ask viewers to predict the relationship between mean and median. ]<\/span>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>objectives for this activity<\/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\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<p>In this activity, you&#8217;ll use a distribution of professional basketball salaries to see that medians are resistant to influence from skew and outliers, while means are not. Importantly, means, in certain circumstances, 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.<\/p>\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-343-1\" href=\"#footnote-343-1\" 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<h3 id=\"IdentMislead\">Misleading Claims<\/h3>\n<p>In fact, the median salary among Texas NBA players was $[latex]1,577,320[\/latex]. The mean salary was $[latex]5,262,279[\/latex]. Use this information to complete Questions 4-6.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm240626\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=240626&theme=oea&iframe_resize_id=ohm240626\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q594557\">Hint<\/span><\/p>\n<div id=\"q594557\" class=\"hidden-answer\" style=\"display: none\">Use the salary information given above and compare it to the answer choices.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241109\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241109&theme=oea&iframe_resize_id=ohm241109\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q319249\">Hint<\/span><\/p>\n<div id=\"q319249\" class=\"hidden-answer\" style=\"display: none\">Use your own judgement to estimate the values.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241110\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241110&theme=oea&iframe_resize_id=ohm241110\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q17406\">Hint<\/span><\/p>\n<div id=\"q17406\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Guidance: &#8220;Consider your answers to Questions 3 &#8211; 6. [voice over images of the dotplot with the vertical lines drawn] What did you consider to be a &#8220;typical&#8221; salary? What characteristic of this variable&#8217;s distribution caused the mean to be different from the median?&#8221;]<\/span><\/p>\n<\/div>\n<p>Now consider the following scenario.\u00a0An NBA recruiter for the Houston Rockets approaches a promising college basketball player and says, \u201cthe typical salary among Texas NBA players is $[latex]5,262,279[\/latex].\u201d<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241111\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241111&theme=oea&iframe_resize_id=ohm241111\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q523741\">Hint<\/span><\/p>\n<div id=\"q523741\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[insert a sub-summary here. &#8220;How did you your answer the question, &#8220;is the recruiter&#8217;s statement misleading?&#8221; Did you consider the mean to be a &#8220;typical&#8221; salary among these NBA players? What could the recruiter have said instead? That it is\u00a0<em>likely\u00a0<\/em>a player would make $5.3 million by joining the team? That is is\u00a0<em>possible\u00a0<\/em>for some highly skilled and talented players? Or would it have been less misleading for the recruiter to have emphasized the median salary of $1.58 million? If you were in the prospective player&#8217;s position, would you have asked to see the distribution to make your own assessment? Which value would you have used, mean or median, if you were in the recruiter&#8217;s position?&#8221;]<\/span><\/p>\n<\/div>\n<p>You&#8217;ve seen that the mean, under certain conditions, can be a misleading indicator of a &#8220;typical&#8221; observation value, such as the salary of a professional basketball player. Now try to apply this understanding to some other types of data collections.<\/p>\n<h3 id=\"MeanOrMedian\">Appropriate Measures of Center<\/h3>\n<p>Three situations are given below in which data is collected on a quantitative variable. For each, visualize what the distribution might look like and make predictions about the shape of the distribution (skewed or symmetric?), the relationship between the mean and median (will they be similar or will the mean be smaller or greater than the median?), and whether or not it would be appropriate to use the mean to represent a &#8220;typical&#8221; observation. Use what you learned about resistance in the previous section,\u00a0<em><a href=\"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/what-to-know-about-4c\/\">What to Know About Interpreting the Mean and Median of a Data Set: 4C<\/a>,<\/em> to guide you.<\/p>\n<p><strong>Situation 1: Data are collected on incomes in New York City.<\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241112\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241112&theme=oea&iframe_resize_id=ohm241112\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157117\">Hint<\/span><\/p>\n<div id=\"q157117\" class=\"hidden-answer\" style=\"display: none\"><span style=\"color: #000000;\">To help visualize what the data set might look like, imagine the possible range of salaries in a large, densely populated city. How many incomes will be at the lower end of the range? How many incomes will be at the higher end of the range?<\/span>\u00a0<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241113\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241113&theme=oea&iframe_resize_id=ohm241113\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q422721\">Hint<\/span><\/p>\n<div id=\"q422721\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241114\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241114&theme=oea&iframe_resize_id=ohm241114\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q362479\">Hint<\/span><\/p>\n<div id=\"q362479\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\"><strong>Situation 2: Data are collected on GPAs at a local college.<\/strong><br \/>\n<\/span><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241115\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241115&theme=oea&iframe_resize_id=ohm241115\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\"><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q668026\">Hint<\/span><\/p>\n<div id=\"q668026\" class=\"hidden-answer\" style=\"display: none\"><span style=\"color: #000000;\">Consider the range of possible GPAs and their frequencies. Do students typically perform average to above average? Do most pass their classes? Where would likely outliers appear in the distribution?<\/span><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 12<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241116\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241116&theme=oea&iframe_resize_id=ohm241116\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q385366\">Hint<\/span><\/p>\n<div id=\"q385366\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 13<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241117\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241117&theme=oea&iframe_resize_id=ohm241117\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q197297\">Hint<\/span><\/p>\n<div id=\"q197297\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\"><strong>Situation 3: Data are collected on peoples\u2019 body temperatures.<\/strong><br \/>\n<\/span><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 14<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241118\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241118&theme=oea&iframe_resize_id=ohm241118\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\"><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731082\">Hint<\/span><\/p>\n<div id=\"q731082\" class=\"hidden-answer\" style=\"display: none\"><span style=\"color: #000000;\">Would you expect to see extreme values in a distribution of people&#8217;s body temperatures?<\/span><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 15<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241120\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241120&theme=oea&iframe_resize_id=ohm241120\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q156796\">Hint<\/span><\/p>\n<div id=\"q156796\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 16<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm241121\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=241121&theme=oea&iframe_resize_id=ohm241121\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q672649\">Hint<\/span><\/p>\n<div id=\"q672649\" class=\"hidden-answer\" style=\"display: none\">Use what you know about sensitivity to skew\/outliers.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Wrap-up: Provide a transition from these particular examples to larger situations in which a quantitative variable would tend to be skewed or symmetric: if the data would tend toward a bunched-up group of values but contain some extreme values, what would the shape of the distribution look like? If data were distributed on the graph &#8220;as though it had fallen through a funnel onto a plane&#8221; what would it look like? Then show and discuss the simulation at <a style=\"background-color: #e6daf7;\" href=\"https:\/\/dcmathpathways.shinyapps.io\/MeanvsMedian\/\">https:\/\/dcmathpathways.shinyapps.io\/MeanvsMedian\/<\/a> .Finally, show some distributions and ask viewers to predict the relationship between mean and median. ]<\/span><\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-343-1\">\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-343-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":175116,"menu_order":48,"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-343","chapter","type-chapter","status-publish","hentry"],"part":1252,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/343","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":12,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/343\/revisions"}],"predecessor-version":[{"id":1261,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapters\/343\/revisions\/1261"}],"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\/343\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/media?parent=343"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/pressbooks\/v2\/chapter-type?post=343"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/contributor?post=343"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/exemplarstatistics\/wp-json\/wp\/v2\/license?post=343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}