{"id":452,"date":"2021-12-20T14:34:35","date_gmt":"2021-12-20T14:34:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=452"},"modified":"2022-02-18T15:17:28","modified_gmt":"2022-02-18T15:17:28","slug":"forming-connections-in-4c","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/forming-connections-in-4c\/","title":{"raw":"Forming Connections in Interpreting the Mean and Median of a Dataset: 4C - 24","rendered":"Forming Connections in Interpreting the Mean and Median of a Dataset: 4C &#8211; 24"},"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<span style=\"background-color: #ffff00;\">There are two LOs identified in DC and present in the content:\r\n1) Identify misleading claims made using means\u00a0 &lt;------------------Add this one and an h3 tag that jumps to it in content.<\/span>\r\n<span style=\"background-color: #ffff00;\">2) Suggest the most appropriate measure of center to use in different situations<\/span>\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\nFrom a financial perspective, would you encourage this player to drop out of college? Explain.\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 dataset.\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 dataset.\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 dataset. 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 dataset (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\nDescribe the shape of the distribution in the dotplot. Comment on any visible skew or outliers.\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\nUsing the dotplot, make an estimate of the \u201ctypical\u201d salary. Explain the reasoning for your estimate.\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\">Identify misleading claims made using means<\/h3>\r\nIn fact, the median salary among Texas NBA players was $1,577,320. The mean salary was $5,262,279. 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]240626[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\nThere are 61 players in this dataset. Fill in the table below with your best estimates of the percentage of players who have salaries above the mean and the percentage of salaries above the median? Your answers do not have to be exact. Just use your own reasoning to estimate these.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Percentage of Salaries Above the Mean<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Percentage of Salaries Above the Median<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\"><\/td>\r\n<td style=\"width: 50%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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\nExplain briefly, in your own words, why the answers for the mean and median are different in Question 5.\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 $5,262,279.\u201d\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\nIs the statement made by the recruiter misleading? Why or why not?\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\">Identify under which conditions it is appropriate to use the mean as a measure 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 Dataset: 4C<\/a>,<\/em> to guide you.\r\n\r\n<strong>Situation 1: Data are collected on incomes in New York City.<\/strong>\u00a0[reveal-answer q=\"157117\"]Hint[\/reveal-answer][hidden-answer a=\"157117\"]<span style=\"color: #008080;\">To help visualize what the dataset 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<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\nWhat do you think the shape of the dataset's distribution will be?\r\n<p style=\"padding-left: 30px;\">a) The distribution will be skewed right.\r\nb) The distribution will be roughly symmetric\r\nc) The distribution will be skewed left.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\nWill the mean be higher or lower than the median, or will they be similar?\r\n<p style=\"padding-left: 30px;\">a) The mean will be higher than median.\r\n<span style=\"font-size: 1rem; background-color: initial;\">b) The mean will be lower than median.\r\nc) The mean and median will be similar.<\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\nConsidering the shape of the data, will it be\u00a0appropriate to use the mean as a measure of center (representing a \"typical\" data value)?\r\n<p style=\"padding-left: 30px;\">a) Yes, the mean will be appropriate as measure of center.\r\nb) No, the mean will be misleading as a measure of center; use the median instead.<\/p>\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[reveal-answer q=\"668026\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"668026\"]<span style=\"color: #008080;\">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<div class=\"textbox key-takeaways\">\r\n<h3>question 11<\/h3>\r\nWhat do you think the shape of the dataset's distribution will be?\r\n<p style=\"padding-left: 30px;\">a) The distribution will be skewed right.\r\nb) The distribution will be roughly symmetric\r\nc) The distribution will be skewed left.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 12<\/h3>\r\nWill the mean be higher or lower than the median, or will they be similar?\r\n<p style=\"padding-left: 30px;\">a) The mean will be higher than median.\r\n<span style=\"font-size: 1rem; background-color: initial;\">b) The mean will be lower than median.\r\nc) The mean and median will be similar.<\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 13<\/h3>\r\nConsidering the shape of the data, will it be\u00a0appropriate to use the mean as a measure of center (representing a \"typical\" data value)?\r\n<p style=\"padding-left: 30px;\">a) Yes, the mean will be appropriate as measure of center.\r\nb) No, the mean will be misleading as a measure of center; use the median instead.<\/p>\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[reveal-answer q=\"731082\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"731082\"]<span style=\"color: #008080;\">Would you expect to see extreme values in a distribution of people's body temperatures?<\/span>[\/hidden-answer]<\/span>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 14<\/h3>\r\nWhat do you think the shape of the dataset's distribution will be?\r\n<p style=\"padding-left: 30px;\">a) The distribution will be skewed right.\r\nb) The distribution will be roughly symmetric\r\nc) The distribution will be skewed left.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 15<\/h3>\r\nWill the mean be higher or lower than the median, or will they be similar?\r\n<p style=\"padding-left: 30px;\">a) The mean will be higher than median.\r\n<span style=\"font-size: 1rem; background-color: initial;\">b) The mean will be lower than median.\r\nc) The mean and median will be similar.<\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 16<\/h3>\r\nConsidering the shape of the data, will it be\u00a0appropriate to use the mean as a measure of center (representing a \"typical\" data value)?\r\n<p style=\"padding-left: 30px;\">a) Yes, the mean will be appropriate as measure of center.\r\nb) No, the mean will be misleading as a measure of center; use the median instead.<\/p>\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<p><span style=\"background-color: #ffff00;\">There are two LOs identified in DC and present in the content:<br \/>\n1) Identify misleading claims made using means\u00a0 &lt;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;Add this one and an h3 tag that jumps to it in content.<\/span><br \/>\n<span style=\"background-color: #ffff00;\">2) Suggest the most appropriate measure of center to use in different situations<\/span><\/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>From a financial perspective, would you encourage this player to drop out of college? Explain.<\/p>\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 dataset.<\/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 dataset.<\/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 dataset. 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 dataset (2017-2018). (2018) Kaggle. Retrieved from https:\/\/www.kaggle.com\/koki25ando\/salary\u00a0\" id=\"return-footnote-452-1\" href=\"#footnote-452-1\" 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>Describe the shape of the distribution in the dotplot. Comment on any visible skew or outliers.<\/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>Using the dotplot, make an estimate of the \u201ctypical\u201d salary. Explain the reasoning for your estimate.<\/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\">Identify misleading claims made using means<\/h3>\n<p>In fact, the median salary among Texas NBA players was $1,577,320. The mean salary was $5,262,279. 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&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>There are 61 players in this dataset. Fill in the table below with your best estimates of the percentage of players who have salaries above the mean and the percentage of salaries above the median? Your answers do not have to be exact. Just use your own reasoning to estimate these.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Percentage of Salaries Above the Mean<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\"><strong>Percentage of Salaries Above the Median<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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>Explain briefly, in your own words, why the answers for the mean and median are different in Question 5.<\/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 $5,262,279.\u201d<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p>Is the statement made by the recruiter misleading? Why or why not?<\/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\">Identify under which conditions it is appropriate to use the mean as a measure 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 Dataset: 4C<\/a>,<\/em> to guide you.<\/p>\n<p><strong>Situation 1: Data are collected on incomes in New York City.<\/strong>\u00a0<\/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: #008080;\">To help visualize what the dataset 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 class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p>What do you think the shape of the dataset&#8217;s distribution will be?<\/p>\n<p style=\"padding-left: 30px;\">a) The distribution will be skewed right.<br \/>\nb) The distribution will be roughly symmetric<br \/>\nc) The distribution will be skewed left.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p>Will the mean be higher or lower than the median, or will they be similar?<\/p>\n<p style=\"padding-left: 30px;\">a) The mean will be higher than median.<br \/>\n<span style=\"font-size: 1rem; background-color: initial;\">b) The mean will be lower than median.<br \/>\nc) The mean and median will be similar.<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p>Considering the shape of the data, will it be\u00a0appropriate to use the mean as a measure of center (representing a &#8220;typical&#8221; data value)?<\/p>\n<p style=\"padding-left: 30px;\">a) Yes, the mean will be appropriate as measure of center.<br \/>\nb) No, the mean will be misleading as a measure of center; use the median instead.<\/p>\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><\/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: #008080;\">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 class=\"textbox key-takeaways\">\n<h3>question 11<\/h3>\n<p>What do you think the shape of the dataset&#8217;s distribution will be?<\/p>\n<p style=\"padding-left: 30px;\">a) The distribution will be skewed right.<br \/>\nb) The distribution will be roughly symmetric<br \/>\nc) The distribution will be skewed left.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 12<\/h3>\n<p>Will the mean be higher or lower than the median, or will they be similar?<\/p>\n<p style=\"padding-left: 30px;\">a) The mean will be higher than median.<br \/>\n<span style=\"font-size: 1rem; background-color: initial;\">b) The mean will be lower than median.<br \/>\nc) The mean and median will be similar.<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 13<\/h3>\n<p>Considering the shape of the data, will it be\u00a0appropriate to use the mean as a measure of center (representing a &#8220;typical&#8221; data value)?<\/p>\n<p style=\"padding-left: 30px;\">a) Yes, the mean will be appropriate as measure of center.<br \/>\nb) No, the mean will be misleading as a measure of center; use the median instead.<\/p>\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><\/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: #008080;\">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 class=\"textbox key-takeaways\">\n<h3>question 14<\/h3>\n<p>What do you think the shape of the dataset&#8217;s distribution will be?<\/p>\n<p style=\"padding-left: 30px;\">a) The distribution will be skewed right.<br \/>\nb) The distribution will be roughly symmetric<br \/>\nc) The distribution will be skewed left.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 15<\/h3>\n<p>Will the mean be higher or lower than the median, or will they be similar?<\/p>\n<p style=\"padding-left: 30px;\">a) The mean will be higher than median.<br \/>\n<span style=\"font-size: 1rem; background-color: initial;\">b) The mean will be lower than median.<br \/>\nc) The mean and median will be similar.<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 16<\/h3>\n<p>Considering the shape of the data, will it be\u00a0appropriate to use the mean as a measure of center (representing a &#8220;typical&#8221; data value)?<\/p>\n<p style=\"padding-left: 30px;\">a) Yes, the mean will be appropriate as measure of center.<br \/>\nb) No, the mean will be misleading as a measure of center; use the median instead.<\/p>\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-452-1\">\u00a0<em>NBA player salary dataset (2017-2018).<\/em> (2018) Kaggle. Retrieved from https:\/\/www.kaggle.com\/koki25ando\/salary\u00a0 <a href=\"#return-footnote-452-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":25777,"menu_order":18,"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-452","chapter","type-chapter","status-publish","hentry"],"part":621,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/452","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":44,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/452\/revisions"}],"predecessor-version":[{"id":3345,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/452\/revisions\/3345"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/621"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/452\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=452"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=452"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=452"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=452"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}