{"id":428,"date":"2021-12-20T14:25:55","date_gmt":"2021-12-20T14:25:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=428"},"modified":"2022-02-17T20:09:32","modified_gmt":"2022-02-17T20:09:32","slug":"forming-connections-in-4a","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/forming-connections-in-4a\/","title":{"raw":"Forming Connections in Calculating Mean and Median of a Dataset: 4A - 18","rendered":"Forming Connections in Calculating Mean and Median of a Dataset: 4A &#8211; 18"},"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=\"#compareHist\">Compare the mean and median of groups by examining a histogram.<\/a><\/li>\r\n \t<li><a href=\"#calcGroups\">Use a data analysis tool to calculate the mean and median for multiple groups at once.<\/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>Feeling Sleepy!<\/h2>\r\nIn <a href=\"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/3a-forming-connections-with-categorical-variables\/\"><em>Forming Connections in Displaying Categorical Data: 3A<\/em><\/a>, we explored data from a study that asked college students whether they identified as owls, larks, or neither and collected other pieces of data related to academics, lifestyle, stress, and sleep . Recall that owls were defined as night people while larks represented morning people. In that activity, you used the dataset from the study to visualize categorical data via pie charts and bar graphs. Now, let's use the same dataset to explore distributions of quantitative variables. In this activity, you'll see the mean and median as numerical measures of the \"center\" of quantitative data.\r\n\r\n<strong><img class=\"aligncenter wp-image-988\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11190649\/Picture24-300x200.jpg\" alt=\"A young man resting his head on his arms on his desk, sleeping in a classroom. Other students in the background are awake and working at their desks.\" width=\"443\" height=\"295\" \/><\/strong>\r\n\r\nBefore we begin, consider the following question, which asks about differences between larks and owls with respect to factors contributing to quality of sleep.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\nHow might the quality of sleep differ among these groups? In particular, do you think one of these groups might consume more alcoholic drinks each week when compared to the others?\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[<span style=\"color: #ff0000;\">Intro<\/span>: Recall using histograms to estimate mean and median as well as using technology to calculate them. Recall lark = morning person, owl = night person, and neither responses. Let's discuss the variables of interest briefly: alcoholic drinks per week is self-explanatory. Poor sleep quality score: the higher the score, the worse the sleep quality. We want to use the tool to interpret graphs and compare centers. A highlight of the tool will be helpful]<\/span>\r\n\r\n<\/div>\r\nBefore we compare each of the groups (owl, lark, neither), let\u2019s consider two variables of interest for <em>all<\/em> college students in this study:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><em>Poor Sleep Quality Score<\/em> \u2013 a score indicating the average quality of sleep for the participants; the greater the score, the worse the quality of sleep.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><em>Alcoholic Drinks per Week<\/em> \u2013 the average number of alcoholic drinks consumed by the participants each week.<\/li>\r\n<\/ul>\r\nThe following histograms display the frequency of the participants\u2019 poor sleep quality scores and the number of alcoholic drinks consumed each week. Use the histograms to estimate the mean and median for each dataset.\r\n\r\n<strong><img class=\"alignnone wp-image-989\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11190654\/Picture25-300x114.png\" alt=\"Two histograms. Above them, there is a legend showing that green indicates Poor Sleep Quality Score and yellow indicates Number of Drinks Consumed Per Week. The first histogram is green. For 1, the count is approximately 2. For 2, the count is approximately 13. For 3, the count is approximately 25. For 4, the count is approximately 38. For 5, the count is approximately 40. For 6, the count is approximately 36. For 7, the count is approximately 20. For 8, the count is approximately 27. For 9, the count is approximately 19. For 10, the count is approximately 10. For 11, the count is approximately 11. For 12, the count is approximately 4. For 13, the count is approximately 2. For 14, the count is approximately 2. For 15, the count is approximately 3. For 18, the count is approximately 1. The second chart is yellow. For 0, the count is approximately 33. For 1, the count is 9. For 2, the count is approximately 15. For 3, the count is approximately 30. For 4, the count is approximately 17. For 5, the count is approximately 30. For 6, the count is approximately 22. For 7, the count is approximately 21. For 8, the count is approximately 13. For 9, the count is approximately 10. For 10, the count is approximately 26. For 12, the count is approximately 9. For 13, the count is approximately 3. For 14, the count is approximately 1. For 15, the count is approximately 3. For 18, the count is approximately 1. For 20, the count is approximately 2. For 24, the count is approximately 1.\" width=\"887\" height=\"337\" \/><\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\nFill in the table below with your estimations based only on the graphs of the distributions.\r\n<table style=\"border-collapse: collapse; width: 72.8346%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><\/td>\r\n<td style=\"width: 19.9475%; text-align: center;\"><strong>Mean<\/strong><\/td>\r\n<td style=\"width: 19.5537%; text-align: center;\"><strong>Median<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Poor Sleep Quality Score<\/strong><\/td>\r\n<td style=\"width: 19.9475%;\"><\/td>\r\n<td style=\"width: 19.5537%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Number of Drinks Consumed Per Week<\/strong><\/td>\r\n<td style=\"width: 19.9475%;\"><\/td>\r\n<td style=\"width: 19.5537%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow let's go to the technology to calculate the precise values for mean and median.\r\n<div class=\"textbox\">\r\n\r\nGo to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.\r\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Dataset<\/strong>\u00a0and select <strong>Sleep Study: Poor Sleep Quality Score<\/strong>. If desired, select <strong>Histogram<\/strong> under <strong>Choose Type of Plot<\/strong>. Under <strong>Select Binwidth for Histogram<\/strong>, use\u00a01 to emulate the histogram in the image above.<\/p>\r\n\r\n<\/div>\r\nThe tool will display descriptive statistics and a distribution for the quantitative variable\u00a0<em>PoorSleepQuality<\/em> from the study's dataset. Recall that this variable records individual responses to a measure of sleep quality, with higher numbers indicating poorer sleep.\r\n\r\nRecord the calculated values for mean and median of Poor Sleep Quality Score in the table in Question 3 below.\r\n<div class=\"textbox\">Step 4) Change the <strong>Dataset<\/strong> selection to <strong>Alcoholic Drinks Per Week<\/strong> and record the calculated values for mean and median of Alcoholic Drinks Per Week in the table below.<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\nFill in the table below with the calculations from the data analysis tool.\r\n<table style=\"border-collapse: collapse; width: 72.8346%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><\/td>\r\n<td style=\"width: 19.9475%; text-align: center;\"><strong>Mean<\/strong><\/td>\r\n<td style=\"width: 19.5537%; text-align: center;\"><strong>Median<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Poor Sleep Quality Score<\/strong><\/td>\r\n<td style=\"width: 19.9475%;\"><\/td>\r\n<td style=\"width: 19.5537%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Number of Drinks Consumed Per Week<\/strong><\/td>\r\n<td style=\"width: 19.9475%;\"><\/td>\r\n<td style=\"width: 19.5537%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"171533\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"171533\"]Make sure the Single Group tab is selected and read these values from \"Descriptive Statistics.\"[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nHow did the estimations you made in Question 2 compare with the actual calculations?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\nIn a few sentences, interpret your findings. What do the calculations for the mean and median suggest about the students\u2019 poor sleep quality scores and consumption of alcohol each week?\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>video placement<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[sub-summary: surfaces only after Q 2 - 5 have been answered. \"Did you find it difficult to estimate the centers of the data using the histograms? If your estimations compared well with the calculations from the tool, you may have instinctively used visual estimations of \"weight\" to estimate the mean or estimations of frequency counts to estimate the median. Show an analysis using these ideas on the two graphs to model the behavior of estimation for the students. Show the analysis of mean and median calculated by the tool.\u00a0 ]<\/span>\r\n\r\n<\/div>\r\nLet\u2019s now compare quality of sleep and alcohol consumption habits between those who identified as owls, larks, or neither.\r\n<h3 id=\"compareHist\">Compare the mean and median of groups by examining a histogram.<\/h3>\r\nThe following histograms illustrate the distributions of the Poor Sleep Quality Score variable based on whether the participants identified as owls, larks, or neither.\r\n\r\n<strong><img class=\"alignnone wp-image-1932 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/26143017\/Screen-Shot-2022-01-26-at-9.30.01-AM.png\" alt=\"Three histograms, labeled &quot;Poor Sleep Quality Score&quot; on the horizontal axis. At the top, there is a legend showing that green indicates lark, yellow indicates owl, and brown indicates neither. The first chart is green. For 0-1, the count is 1. For 2, the count is 10. For 3, the count is 10. For 4, the count is 12. For 5, the count is 12. For 6, the count is 5. For 7, the count is 5. For 8, the count is 8. For 9, the count is 8. For 10, the count is 4. For 11, the count is 4. For 12, the count is 2. For 13, the count is 2. For 14, the count is 2. For 15, the count is 2. The next plot is yellow. For 2, the count is 4. For 3, the count is 4, For 4, the count is 14. For 5, the count is 14. For 6, the count is 7. For 7, the count is 7. For 8, the count is 13. For 9, the count is 13. For 10, the count is 10. For 11, the count is 10. For 12, the count is 2. For 13, the count is 2. For 14, the count is 2. For 15, the count is 2. For 18, the count is 2. The next chart is brown. For 1, the count is 1. For 2, the count is 7. For 3, the count is 17. For 4, the count is 25. For 5, the count is 28. For 6, the count is 29. For 7, the count is 14. For 8, the count is 14. For 9, the count is 10. For 10, the count is 5. For 11, the count is 4. For 12, the count is 3. For 13, the count is 1. For 14, the count is 2. For 15, the count is 1.\" width=\"1396\" height=\"722\" \/><\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\nWhat do you notice in these histograms? In particular, based on their estimated means and medians, does it appear one group has a worse quality of sleep than the other groups? (Remember, the greater the score, the worse the quality of sleep.)\r\n\r\n[reveal-answer q=\"542244\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"542244\"]When centers appear similar, it can help to look for differences in the percent of responses that fall in the low and high ends of the distribution. Is there a group with more or fewer low scores or high scores? [\/hidden-answer]\r\n\r\n<span style=\"background-color: #e6daf7;\">[Feedback for Question 6 --\" that was a tricky analysis since the centers of all three groups were similar. We see that Larks and Neither reported a greater percentage of low scores and Owls appear to have reported a few extreme high scores\"]\u00a0<\/span>\r\n\r\n<\/div>\r\nThe following histograms illustrate the distributions of the Alcoholic Drinks per Week variable based on whether the participants identified as owls, larks, or neither.\r\n\r\n<strong><img class=\"alignnone wp-image-990\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11190658\/Picture26-300x167.png\" alt=\"Three histograms, labeled &quot;Alcoholic Drinks per Week&quot; on the horizontal axis. At the top, there is a legend showing that green indicates lark, yellow indicates owl, and brown indicates neither. The first chart is green. For 0-1, the count is approximately 8. For 2-3, the count is approximately 13. For 4-5, the count is approximately 7. For 6-7, the count is approximately 5. For 8-9, the count is approximately 1. For 10-11, the count is approximately 4. For 12-13, the count is approximately 2. For 20-21, the count is approximately 1. The next plot is yellow. For 0-1, the count is approximately 10. For 2-3, the count is approximately 4. For 4-5, the count is approximately 7. For 6-7, the count is approximately 8. For 8-9, the count is approximately 9. For 10-11, the count is approximately 4. For 12-13, the count is approximately 3. For 14-15, the count is approximately 4. The next chart is brown. For 0-1, the count is approximately 23. For 2-3, the count is approximately 29. For 4-5, the count is approximately 33. For 6-7, the count is approximately 30. For 8-9, the count is approximately 14. For 10-11, the count is approximately 19. For 12-13, the count is approximately 8.\" width=\"896\" height=\"499\" \/><\/strong>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 7<\/h3>\r\nWhat do you notice in these histograms? In particular, based on their estimated means and medians, does it appear one group consumes more alcoholic drinks each week?\r\n\r\n[reveal-answer q=\"247487\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"247487\"]Use a similar analysis to the one that applied to Question 6.[\/hidden-answer]\r\n\r\n<span style=\"background-color: #e6daf7;\">[Feedback for Question 7 --\" that was a tricky analysis since the centers of all three groups were similar. We see that all three histograms show students reported consuming 0 - 15 drinks per week with a few outliers in the larks and owls groups. The means appear quite similar but the median for the lark group may be less than the others.\"]\u00a0<\/span>\r\n\r\n<\/div>\r\n<h3 id=\"calcGroups\">Use a data analysis tool to calculate the mean and median for multiple groups at once<\/h3>\r\n<div class=\"textbox\">\r\n\r\nGo to the Describing and Exploring Quantitative Variables tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.\r\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Several Groups<\/strong> tab.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\r\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Dataset<\/strong> and select <strong>Sleep Study: Poor Sleep Quality Score<\/strong>. You may change the display to show histograms as desired.<\/p>\r\n\r\n<\/div>\r\n<p style=\"padding-left: 30px;\">The tool will display descriptive statistics and a distribution for the quantitative variable\u00a0<em>PoorSleepQuality<\/em>\u00a0categorized by whether a respondent identified as a lark, an owl, or as neither.\u00a0Using information from \"Descriptive Statistics,\" fill in the table in Question 8 below for Poor Sleep Quality Score.<\/p>\r\n\r\n<div class=\"textbox\">Step 4) Change the Dataset to <strong>Sleep Study: Alcoholic Drinks per Week<\/strong> and use the information from the tool to complete the table in Question 8 for Alcoholic Drinks Per Week.<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 8<\/h3>\r\nFollow the steps above to complete the table.\r\n<table style=\"border-collapse: collapse; width: 51.6094%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 11.433%;\" rowspan=\"2\"><\/td>\r\n<td style=\"width: 19.1159%; text-align: center;\" colspan=\"2\"><strong>Poor Sleep Quality Score<\/strong><\/td>\r\n<td style=\"width: 21.0587%; text-align: center;\" colspan=\"2\"><strong>Alcoholic Drinks Per Week<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 9.26873%; text-align: center;\"><strong>Mean<\/strong><\/td>\r\n<td style=\"width: 9.84721%; text-align: center;\"><strong>Median<\/strong><\/td>\r\n<td style=\"width: 11.1371%; text-align: center;\"><strong>Mean<\/strong><\/td>\r\n<td style=\"width: 9.92156%; text-align: center;\"><strong>Median<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 11.433%; text-align: center;\"><strong>Owl<\/strong><\/td>\r\n<td style=\"width: 9.26873%; text-align: center;\"><\/td>\r\n<td style=\"width: 9.84721%; text-align: center;\"><\/td>\r\n<td style=\"width: 11.1371%; text-align: center;\"><\/td>\r\n<td style=\"width: 9.92156%; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 11.433%; text-align: center;\"><strong>Lark<\/strong><\/td>\r\n<td style=\"width: 9.26873%; text-align: center;\"><\/td>\r\n<td style=\"width: 9.84721%; text-align: center;\"><\/td>\r\n<td style=\"width: 11.1371%; text-align: center;\"><\/td>\r\n<td style=\"width: 9.92156%; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 11.433%; text-align: center;\"><strong>Neither<\/strong><\/td>\r\n<td style=\"width: 9.26873%; text-align: center;\"><\/td>\r\n<td style=\"width: 9.84721%; text-align: center;\"><\/td>\r\n<td style=\"width: 11.1371%; text-align: center;\"><\/td>\r\n<td style=\"width: 9.92156%; text-align: center;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div align=\"left\"><\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 9<\/h3>\r\nHow do your calculations relate to the comments and estimations you provided in <a href=\"#Quest3\">Questions 3 - 5<\/a>? Is there anything that surprised you?\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 10<\/h3>\r\nReflect back on your interpretations, estimations, and calculations from Questions 2\u20139. What might you conclude about the quality of sleep and drinking habits of those who identified as owls, larks, or neither? Explain.\r\n\r\n[reveal-answer q=\"809806\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"809806\"]Try to make comparisons between means and medians using words like \"smaller\/larger\" and \"fewer\/more.\"[\/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: <\/span>\r\n<ul>\r\n \t<li><span style=\"background-color: #e6daf7;\">Provide a model answer for Question 10 to use statistical language for students to mimic: Let's consider the conclusions, if any, we can draw from this analysis. When you try to make statements drawing conclusions in a statistical analysis, take care not to make assumptions or statements of fact. Instead, use language that includes phrases such as, \"this suggests ...,\" or \"this group tends to... .\"<\/span><\/li>\r\n \t<li><span style=\"background-color: #e6daf7; font-size: 1rem; orphans: 1; text-align: initial;\">Reflect back to your response to Question 1 -- were you surprised by what you discovered from the data?\u00a0<\/span><\/li>\r\n \t<li><span style=\"background-color: #e6daf7;\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: #e6daf7;\">Reflect on the use of the histograms in the activity. Address the challenge of estimating the mean using the histograms. Encourage students to think about\u00a0<\/span><em style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: #e6daf7;\">why<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: #e6daf7;\"> they found it challenging. \"Do you think it is possible to look at a histogram and guess whether\u00a0the mean or median might be larger? If so, what characteristics or\u00a0features of a histogram might suggest that the mean is larger? What\u00a0characteristics or features of a histogram might suggest that the\u00a0median is larger?\u201d]<\/span><\/span><\/li>\r\n<\/ul>\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=\"#compareHist\">Compare the mean and median of groups by examining a histogram.<\/a><\/li>\n<li><a href=\"#calcGroups\">Use a data analysis tool to calculate the mean and median for multiple groups at once.<\/a><\/li>\n<\/ul>\n<p>Click on a skill above to jump to its location in this activity.<\/p>\n<\/div>\n<h2>Feeling Sleepy!<\/h2>\n<p>In <a href=\"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/3a-forming-connections-with-categorical-variables\/\"><em>Forming Connections in Displaying Categorical Data: 3A<\/em><\/a>, we explored data from a study that asked college students whether they identified as owls, larks, or neither and collected other pieces of data related to academics, lifestyle, stress, and sleep . Recall that owls were defined as night people while larks represented morning people. In that activity, you used the dataset from the study to visualize categorical data via pie charts and bar graphs. Now, let&#8217;s use the same dataset to explore distributions of quantitative variables. In this activity, you&#8217;ll see the mean and median as numerical measures of the &#8220;center&#8221; of quantitative data.<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-988\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11190649\/Picture24-300x200.jpg\" alt=\"A young man resting his head on his arms on his desk, sleeping in a classroom. Other students in the background are awake and working at their desks.\" width=\"443\" height=\"295\" \/><\/strong><\/p>\n<p>Before we begin, consider the following question, which asks about differences between larks and owls with respect to factors contributing to quality of sleep.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>How might the quality of sleep differ among these groups? In particular, do you think one of these groups might consume more alcoholic drinks each week when compared to the others?<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[<span style=\"color: #ff0000;\">Intro<\/span>: Recall using histograms to estimate mean and median as well as using technology to calculate them. Recall lark = morning person, owl = night person, and neither responses. Let&#8217;s discuss the variables of interest briefly: alcoholic drinks per week is self-explanatory. Poor sleep quality score: the higher the score, the worse the sleep quality. We want to use the tool to interpret graphs and compare centers. A highlight of the tool will be helpful]<\/span><\/p>\n<\/div>\n<p>Before we compare each of the groups (owl, lark, neither), let\u2019s consider two variables of interest for <em>all<\/em> college students in this study:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><em>Poor Sleep Quality Score<\/em> \u2013 a score indicating the average quality of sleep for the participants; the greater the score, the worse the quality of sleep.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><em>Alcoholic Drinks per Week<\/em> \u2013 the average number of alcoholic drinks consumed by the participants each week.<\/li>\n<\/ul>\n<p>The following histograms display the frequency of the participants\u2019 poor sleep quality scores and the number of alcoholic drinks consumed each week. Use the histograms to estimate the mean and median for each dataset.<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-989\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11190654\/Picture25-300x114.png\" alt=\"Two histograms. Above them, there is a legend showing that green indicates Poor Sleep Quality Score and yellow indicates Number of Drinks Consumed Per Week. The first histogram is green. For 1, the count is approximately 2. For 2, the count is approximately 13. For 3, the count is approximately 25. For 4, the count is approximately 38. For 5, the count is approximately 40. For 6, the count is approximately 36. For 7, the count is approximately 20. For 8, the count is approximately 27. For 9, the count is approximately 19. For 10, the count is approximately 10. For 11, the count is approximately 11. For 12, the count is approximately 4. For 13, the count is approximately 2. For 14, the count is approximately 2. For 15, the count is approximately 3. For 18, the count is approximately 1. The second chart is yellow. For 0, the count is approximately 33. For 1, the count is 9. For 2, the count is approximately 15. For 3, the count is approximately 30. For 4, the count is approximately 17. For 5, the count is approximately 30. For 6, the count is approximately 22. For 7, the count is approximately 21. For 8, the count is approximately 13. For 9, the count is approximately 10. For 10, the count is approximately 26. For 12, the count is approximately 9. For 13, the count is approximately 3. For 14, the count is approximately 1. For 15, the count is approximately 3. For 18, the count is approximately 1. For 20, the count is approximately 2. For 24, the count is approximately 1.\" width=\"887\" height=\"337\" \/><\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>Fill in the table below with your estimations based only on the graphs of the distributions.<\/p>\n<table style=\"border-collapse: collapse; width: 72.8346%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><\/td>\n<td style=\"width: 19.9475%; text-align: center;\"><strong>Mean<\/strong><\/td>\n<td style=\"width: 19.5537%; text-align: center;\"><strong>Median<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Poor Sleep Quality Score<\/strong><\/td>\n<td style=\"width: 19.9475%;\"><\/td>\n<td style=\"width: 19.5537%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Number of Drinks Consumed Per Week<\/strong><\/td>\n<td style=\"width: 19.9475%;\"><\/td>\n<td style=\"width: 19.5537%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Now let&#8217;s go to the technology to calculate the precise values for mean and median.<\/p>\n<div class=\"textbox\">\n<p>Go to the <em>Describing and Exploring Quantitative Variables<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.<\/p>\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Single Group<\/strong> tab.<\/p>\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Dataset<\/strong>\u00a0and select <strong>Sleep Study: Poor Sleep Quality Score<\/strong>. If desired, select <strong>Histogram<\/strong> under <strong>Choose Type of Plot<\/strong>. Under <strong>Select Binwidth for Histogram<\/strong>, use\u00a01 to emulate the histogram in the image above.<\/p>\n<\/div>\n<p>The tool will display descriptive statistics and a distribution for the quantitative variable\u00a0<em>PoorSleepQuality<\/em> from the study&#8217;s dataset. Recall that this variable records individual responses to a measure of sleep quality, with higher numbers indicating poorer sleep.<\/p>\n<p>Record the calculated values for mean and median of Poor Sleep Quality Score in the table in Question 3 below.<\/p>\n<div class=\"textbox\">Step 4) Change the <strong>Dataset<\/strong> selection to <strong>Alcoholic Drinks Per Week<\/strong> and record the calculated values for mean and median of Alcoholic Drinks Per Week in the table below.<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p>Fill in the table below with the calculations from the data analysis tool.<\/p>\n<table style=\"border-collapse: collapse; width: 72.8346%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><\/td>\n<td style=\"width: 19.9475%; text-align: center;\"><strong>Mean<\/strong><\/td>\n<td style=\"width: 19.5537%; text-align: center;\"><strong>Median<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Poor Sleep Quality Score<\/strong><\/td>\n<td style=\"width: 19.9475%;\"><\/td>\n<td style=\"width: 19.5537%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Number of Drinks Consumed Per Week<\/strong><\/td>\n<td style=\"width: 19.9475%;\"><\/td>\n<td style=\"width: 19.5537%;\"><\/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=\"q171533\">Hint<\/span><\/p>\n<div id=\"q171533\" class=\"hidden-answer\" style=\"display: none\">Make sure the Single Group tab is selected and read these values from &#8220;Descriptive Statistics.&#8221;<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>How did the estimations you made in Question 2 compare with the actual calculations?<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>In a few sentences, interpret your findings. What do the calculations for the mean and median suggest about the students\u2019 poor sleep quality scores and consumption of alcohol each week?<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[sub-summary: surfaces only after Q 2 &#8211; 5 have been answered. &#8220;Did you find it difficult to estimate the centers of the data using the histograms? If your estimations compared well with the calculations from the tool, you may have instinctively used visual estimations of &#8220;weight&#8221; to estimate the mean or estimations of frequency counts to estimate the median. Show an analysis using these ideas on the two graphs to model the behavior of estimation for the students. Show the analysis of mean and median calculated by the tool.\u00a0 ]<\/span><\/p>\n<\/div>\n<p>Let\u2019s now compare quality of sleep and alcohol consumption habits between those who identified as owls, larks, or neither.<\/p>\n<h3 id=\"compareHist\">Compare the mean and median of groups by examining a histogram.<\/h3>\n<p>The following histograms illustrate the distributions of the Poor Sleep Quality Score variable based on whether the participants identified as owls, larks, or neither.<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1932 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2021\/12\/26143017\/Screen-Shot-2022-01-26-at-9.30.01-AM.png\" alt=\"Three histograms, labeled &quot;Poor Sleep Quality Score&quot; on the horizontal axis. At the top, there is a legend showing that green indicates lark, yellow indicates owl, and brown indicates neither. The first chart is green. For 0-1, the count is 1. For 2, the count is 10. For 3, the count is 10. For 4, the count is 12. For 5, the count is 12. For 6, the count is 5. For 7, the count is 5. For 8, the count is 8. For 9, the count is 8. For 10, the count is 4. For 11, the count is 4. For 12, the count is 2. For 13, the count is 2. For 14, the count is 2. For 15, the count is 2. The next plot is yellow. For 2, the count is 4. For 3, the count is 4, For 4, the count is 14. For 5, the count is 14. For 6, the count is 7. For 7, the count is 7. For 8, the count is 13. For 9, the count is 13. For 10, the count is 10. For 11, the count is 10. For 12, the count is 2. For 13, the count is 2. For 14, the count is 2. For 15, the count is 2. For 18, the count is 2. The next chart is brown. For 1, the count is 1. For 2, the count is 7. For 3, the count is 17. For 4, the count is 25. For 5, the count is 28. For 6, the count is 29. For 7, the count is 14. For 8, the count is 14. For 9, the count is 10. For 10, the count is 5. For 11, the count is 4. For 12, the count is 3. For 13, the count is 1. For 14, the count is 2. For 15, the count is 1.\" width=\"1396\" height=\"722\" \/><\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p>What do you notice in these histograms? In particular, based on their estimated means and medians, does it appear one group has a worse quality of sleep than the other groups? (Remember, the greater the score, the worse the quality of sleep.)<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q542244\">Hint<\/span><\/p>\n<div id=\"q542244\" class=\"hidden-answer\" style=\"display: none\">When centers appear similar, it can help to look for differences in the percent of responses that fall in the low and high ends of the distribution. Is there a group with more or fewer low scores or high scores? <\/div>\n<\/div>\n<p><span style=\"background-color: #e6daf7;\">[Feedback for Question 6 &#8211;&#8221; that was a tricky analysis since the centers of all three groups were similar. We see that Larks and Neither reported a greater percentage of low scores and Owls appear to have reported a few extreme high scores&#8221;]\u00a0<\/span><\/p>\n<\/div>\n<p>The following histograms illustrate the distributions of the Alcoholic Drinks per Week variable based on whether the participants identified as owls, larks, or neither.<\/p>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-990\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/11190658\/Picture26-300x167.png\" alt=\"Three histograms, labeled &quot;Alcoholic Drinks per Week&quot; on the horizontal axis. At the top, there is a legend showing that green indicates lark, yellow indicates owl, and brown indicates neither. The first chart is green. For 0-1, the count is approximately 8. For 2-3, the count is approximately 13. For 4-5, the count is approximately 7. For 6-7, the count is approximately 5. For 8-9, the count is approximately 1. For 10-11, the count is approximately 4. For 12-13, the count is approximately 2. For 20-21, the count is approximately 1. The next plot is yellow. For 0-1, the count is approximately 10. For 2-3, the count is approximately 4. For 4-5, the count is approximately 7. For 6-7, the count is approximately 8. For 8-9, the count is approximately 9. For 10-11, the count is approximately 4. For 12-13, the count is approximately 3. For 14-15, the count is approximately 4. The next chart is brown. For 0-1, the count is approximately 23. For 2-3, the count is approximately 29. For 4-5, the count is approximately 33. For 6-7, the count is approximately 30. For 8-9, the count is approximately 14. For 10-11, the count is approximately 19. For 12-13, the count is approximately 8.\" width=\"896\" height=\"499\" \/><\/strong><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 7<\/h3>\n<p>What do you notice in these histograms? In particular, based on their estimated means and medians, does it appear one group consumes more alcoholic drinks each week?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q247487\">Hint<\/span><\/p>\n<div id=\"q247487\" class=\"hidden-answer\" style=\"display: none\">Use a similar analysis to the one that applied to Question 6.<\/div>\n<\/div>\n<p><span style=\"background-color: #e6daf7;\">[Feedback for Question 7 &#8211;&#8221; that was a tricky analysis since the centers of all three groups were similar. We see that all three histograms show students reported consuming 0 &#8211; 15 drinks per week with a few outliers in the larks and owls groups. The means appear quite similar but the median for the lark group may be less than the others.&#8221;]\u00a0<\/span><\/p>\n<\/div>\n<h3 id=\"calcGroups\">Use a data analysis tool to calculate the mean and median for multiple groups at once<\/h3>\n<div class=\"textbox\">\n<p>Go to the Describing and Exploring Quantitative Variables tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/\" target=\"_blank\" rel=\"noopener\">https:\/\/dcmathpathways.shinyapps.io\/EDA_quantitative\/<\/a>.<\/p>\n<p style=\"padding-left: 30px;\">Step 1) Select the <strong>Several Groups<\/strong> tab.<\/p>\n<p style=\"padding-left: 30px;\">Step 2) Locate the drop-down menu under <strong>Enter Data<\/strong> and select <strong>From Textbook<\/strong>.<\/p>\n<p style=\"padding-left: 30px;\">Step 3) Locate the drop-down menu under <strong>Dataset<\/strong> and select <strong>Sleep Study: Poor Sleep Quality Score<\/strong>. You may change the display to show histograms as desired.<\/p>\n<\/div>\n<p style=\"padding-left: 30px;\">The tool will display descriptive statistics and a distribution for the quantitative variable\u00a0<em>PoorSleepQuality<\/em>\u00a0categorized by whether a respondent identified as a lark, an owl, or as neither.\u00a0Using information from &#8220;Descriptive Statistics,&#8221; fill in the table in Question 8 below for Poor Sleep Quality Score.<\/p>\n<div class=\"textbox\">Step 4) Change the Dataset to <strong>Sleep Study: Alcoholic Drinks per Week<\/strong> and use the information from the tool to complete the table in Question 8 for Alcoholic Drinks Per Week.<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 8<\/h3>\n<p>Follow the steps above to complete the table.<\/p>\n<table style=\"border-collapse: collapse; width: 51.6094%;\">\n<tbody>\n<tr>\n<td style=\"width: 11.433%;\" rowspan=\"2\"><\/td>\n<td style=\"width: 19.1159%; text-align: center;\" colspan=\"2\"><strong>Poor Sleep Quality Score<\/strong><\/td>\n<td style=\"width: 21.0587%; text-align: center;\" colspan=\"2\"><strong>Alcoholic Drinks Per Week<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 9.26873%; text-align: center;\"><strong>Mean<\/strong><\/td>\n<td style=\"width: 9.84721%; text-align: center;\"><strong>Median<\/strong><\/td>\n<td style=\"width: 11.1371%; text-align: center;\"><strong>Mean<\/strong><\/td>\n<td style=\"width: 9.92156%; text-align: center;\"><strong>Median<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 11.433%; text-align: center;\"><strong>Owl<\/strong><\/td>\n<td style=\"width: 9.26873%; text-align: center;\"><\/td>\n<td style=\"width: 9.84721%; text-align: center;\"><\/td>\n<td style=\"width: 11.1371%; text-align: center;\"><\/td>\n<td style=\"width: 9.92156%; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 11.433%; text-align: center;\"><strong>Lark<\/strong><\/td>\n<td style=\"width: 9.26873%; text-align: center;\"><\/td>\n<td style=\"width: 9.84721%; text-align: center;\"><\/td>\n<td style=\"width: 11.1371%; text-align: center;\"><\/td>\n<td style=\"width: 9.92156%; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 11.433%; text-align: center;\"><strong>Neither<\/strong><\/td>\n<td style=\"width: 9.26873%; text-align: center;\"><\/td>\n<td style=\"width: 9.84721%; text-align: center;\"><\/td>\n<td style=\"width: 11.1371%; text-align: center;\"><\/td>\n<td style=\"width: 9.92156%; text-align: center;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div style=\"text-align: left;\"><\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 9<\/h3>\n<p>How do your calculations relate to the comments and estimations you provided in <a href=\"#Quest3\">Questions 3 &#8211; 5<\/a>? Is there anything that surprised you?<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 10<\/h3>\n<p>Reflect back on your interpretations, estimations, and calculations from Questions 2\u20139. What might you conclude about the quality of sleep and drinking habits of those who identified as owls, larks, or neither? Explain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q809806\">Hint<\/span><\/p>\n<div id=\"q809806\" class=\"hidden-answer\" style=\"display: none\">Try to make comparisons between means and medians using words like &#8220;smaller\/larger&#8221; and &#8220;fewer\/more.&#8221;<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>video placement<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[wrap-up: <\/span><\/p>\n<ul>\n<li><span style=\"background-color: #e6daf7;\">Provide a model answer for Question 10 to use statistical language for students to mimic: Let&#8217;s consider the conclusions, if any, we can draw from this analysis. When you try to make statements drawing conclusions in a statistical analysis, take care not to make assumptions or statements of fact. Instead, use language that includes phrases such as, &#8220;this suggests &#8230;,&#8221; or &#8220;this group tends to&#8230; .&#8221;<\/span><\/li>\n<li><span style=\"background-color: #e6daf7; font-size: 1rem; orphans: 1; text-align: initial;\">Reflect back to your response to Question 1 &#8212; were you surprised by what you discovered from the data?\u00a0<\/span><\/li>\n<li><span style=\"background-color: #e6daf7;\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: #e6daf7;\">Reflect on the use of the histograms in the activity. Address the challenge of estimating the mean using the histograms. Encourage students to think about\u00a0<\/span><em style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: #e6daf7;\">why<\/em><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: #e6daf7;\"> they found it challenging. &#8220;Do you think it is possible to look at a histogram and guess whether\u00a0the mean or median might be larger? If so, what characteristics or\u00a0features of a histogram might suggest that the mean is larger? What\u00a0characteristics or features of a histogram might suggest that the\u00a0median is larger?\u201d]<\/span><\/span><\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":25777,"menu_order":4,"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-428","chapter","type-chapter","status-publish","hentry"],"part":621,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/428","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\/428\/revisions"}],"predecessor-version":[{"id":3309,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/428\/revisions\/3309"}],"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\/428\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=428"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=428"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=428"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=428"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}