{"id":3927,"date":"2022-03-16T18:01:05","date_gmt":"2022-03-16T18:01:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3927"},"modified":"2022-03-17T03:18:50","modified_gmt":"2022-03-17T03:18:50","slug":"instructor-guide-forming-connections-interpreting-the-mean-and-median-of-a-dataset","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/instructor-guide-forming-connections-interpreting-the-mean-and-median-of-a-dataset\/","title":{"raw":"Instructor Guide Interpreting the Mean and Median of a Dataset: Forming Connections","rendered":"Instructor Guide Interpreting the Mean and Median of a Dataset: Forming Connections"},"content":{"raw":"<h2>Overview<\/h2>\r\n<ul>\r\n \t<li>Students will explore how skew and outliers influence different\u00a0measures of center (mean vs. median).<\/li>\r\n \t<li>Students will also interrogate misleading\u00a0statements about means.<\/li>\r\n \t<li>This activity connects back to calculations of measures of center, and connects forward to boxplots and analyses of univariate data.<\/li>\r\n \t<li><span style=\"background-color: #ffff99;\">C2, S2, C4, O2, C4, C3, C5, S4, V1 \u2190 Link to EBTP descriptions\u00a0<\/span><\/li>\r\n<\/ul>\r\n<h3>Prerequisite assumptions<\/h3>\r\nStudents should be able to do each of the following after completing the <em>What to Know<\/em> assignment.\r\n<ul>\r\n \t<li>Calculate and interpret the mean and median of a data set.<\/li>\r\n \t<li>Identify whether a data set is left-skewed, symmetric, or right-skewed.<\/li>\r\n \t<li>Make connections between the distribution of a dataset and how the mean and median relate.<\/li>\r\n<\/ul>\r\n<h3>Intended goals for this activity<\/h3>\r\nAfter completing this activity, students should understand that medians are resistant to influence from skew and outliers, that means are not resistant to influence from skew and outliers, and that means,\u00a0in certain circumstances, can be misleading. They should be able to identify misleading claims made using means and suggest the most appropriate\r\nmeasure of center to use in different situations.\r\n<h2>Synchronous Delivery and Activity Flow<\/h2>\r\nThe sample activity delivery below assumes a face-to-face class meeting but can be adapted to a fully online or hybrid delivery by using break-out rooms for pairs and small groups.\r\n<h3>Frame the activity (3 minutes)<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Have students read and answer Question 1 independently. Then, ask\u00a0students to discuss their answers in pairs before sharing with the full\u00a0group. V1<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Transition to the in-class activity by briefly discussing the Objectives\u00a0for the activity.<\/li>\r\n<\/ul>\r\n<h3>Activity Flow (18 minutes)<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Have students work on the dotplot and analysis in Questions 2 - 6 together.\r\n<ul>\r\n \t<li aria-level=\"1\">For Question 5, emphasize that students\u00a0do not need to\u00a0get the exact percentages; it\u2019s far more important to get the\u00a0general percentages from reasoning than to get their counts and\u00a0calculations exactly correct.<\/li>\r\n \t<li aria-level=\"1\">Questions 3 and 6 are good places for a whole class discussion after the groups have a chance to work on them together. C5<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Give students time to think about Question 7 independently and to write out their responses. Then,\r\n<ul>\r\n \t<li>First, ask for a student who thinks the mean is \u201ctypical\u201d to share\u00a0their reasoning.<\/li>\r\n \t<li>Second, ask for students who think the mean is not \u201ctypical\u201d to\u00a0share their reasonings.<\/li>\r\n \t<li>Third, return to the first sharer and ask if they have been\u00a0convinced that the mean is not actually \u201ctypical.\u201d\u00a0S4<\/li>\r\n \t<li>\u00a0Students should leave this discussion feeling the median is the\u00a0best representation of the \u201ctypical\u201d salary in this dataset. The next\u00a0question will broaden this idea. C5<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">For Questions 8 - 16,\u00a0Have students answer this question in small groups, and then ask\u00a0several groups to share their answers with the whole class. Ensure\u00a0that students include not just specific examples in their explanations\r\nbut also general patterns that can apply to multiple datasets (e.g.,\u00a0contains right skew, contains outliers, etc.)<\/li>\r\n \t<li aria-level=\"1\">After students have discussed, show the simulation linked below on\u00a0a projector. Illustrate the varying relationships between the mean and\u00a0median after low and high outliers are added. Also, illustrate how\u00a0similar the mean and median are when the distribution is symmetric. C3<\/li>\r\n \t<li aria-level=\"1\">Simulation: https:\/\/dcmathpathways.shinyapps.io\/MeanvsMedian\/.<\/li>\r\n<\/ul>\r\n<h3>Wrap-up\/transition (5 minutes)<\/h3>\r\n<ul>\r\n \t<li>As a final formative assessment, consider drawing various\u00a0distribution shapes (left skew, right skew, symmetric) on the board,\u00a0and have students describe the predicted relationship between the\u00a0median and mean. C4<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Have students refer back to the Objectives for the activity and identify ones they recognized completing. S2, C4, O2<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Assign the homework or\u00a0<em>Practice<\/em>\u00a0and any <em>What to Know<\/em> pages for the <em>Forming Connections<\/em> activities you plan to complete in the next class meeting. C2<\/li>\r\n<\/ul>","rendered":"<h2>Overview<\/h2>\n<ul>\n<li>Students will explore how skew and outliers influence different\u00a0measures of center (mean vs. median).<\/li>\n<li>Students will also interrogate misleading\u00a0statements about means.<\/li>\n<li>This activity connects back to calculations of measures of center, and connects forward to boxplots and analyses of univariate data.<\/li>\n<li><span style=\"background-color: #ffff99;\">C2, S2, C4, O2, C4, C3, C5, S4, V1 \u2190 Link to EBTP descriptions\u00a0<\/span><\/li>\n<\/ul>\n<h3>Prerequisite assumptions<\/h3>\n<p>Students should be able to do each of the following after completing the <em>What to Know<\/em> assignment.<\/p>\n<ul>\n<li>Calculate and interpret the mean and median of a data set.<\/li>\n<li>Identify whether a data set is left-skewed, symmetric, or right-skewed.<\/li>\n<li>Make connections between the distribution of a dataset and how the mean and median relate.<\/li>\n<\/ul>\n<h3>Intended goals for this activity<\/h3>\n<p>After completing this activity, students should understand that medians are resistant to influence from skew and outliers, that means are not resistant to influence from skew and outliers, and that means,\u00a0in certain circumstances, can be misleading. They should be able to identify misleading claims made using means and suggest the most appropriate<br \/>\nmeasure of center to use in different situations.<\/p>\n<h2>Synchronous Delivery and Activity Flow<\/h2>\n<p>The sample activity delivery below assumes a face-to-face class meeting but can be adapted to a fully online or hybrid delivery by using break-out rooms for pairs and small groups.<\/p>\n<h3>Frame the activity (3 minutes)<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Have students read and answer Question 1 independently. Then, ask\u00a0students to discuss their answers in pairs before sharing with the full\u00a0group. V1<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Transition to the in-class activity by briefly discussing the Objectives\u00a0for the activity.<\/li>\n<\/ul>\n<h3>Activity Flow (18 minutes)<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Have students work on the dotplot and analysis in Questions 2 &#8211; 6 together.\n<ul>\n<li aria-level=\"1\">For Question 5, emphasize that students\u00a0do not need to\u00a0get the exact percentages; it\u2019s far more important to get the\u00a0general percentages from reasoning than to get their counts and\u00a0calculations exactly correct.<\/li>\n<li aria-level=\"1\">Questions 3 and 6 are good places for a whole class discussion after the groups have a chance to work on them together. C5<\/li>\n<\/ul>\n<\/li>\n<li>Give students time to think about Question 7 independently and to write out their responses. Then,\n<ul>\n<li>First, ask for a student who thinks the mean is \u201ctypical\u201d to share\u00a0their reasoning.<\/li>\n<li>Second, ask for students who think the mean is not \u201ctypical\u201d to\u00a0share their reasonings.<\/li>\n<li>Third, return to the first sharer and ask if they have been\u00a0convinced that the mean is not actually \u201ctypical.\u201d\u00a0S4<\/li>\n<li>\u00a0Students should leave this discussion feeling the median is the\u00a0best representation of the \u201ctypical\u201d salary in this dataset. The next\u00a0question will broaden this idea. C5<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">For Questions 8 &#8211; 16,\u00a0Have students answer this question in small groups, and then ask\u00a0several groups to share their answers with the whole class. Ensure\u00a0that students include not just specific examples in their explanations<br \/>\nbut also general patterns that can apply to multiple datasets (e.g.,\u00a0contains right skew, contains outliers, etc.)<\/li>\n<li aria-level=\"1\">After students have discussed, show the simulation linked below on\u00a0a projector. Illustrate the varying relationships between the mean and\u00a0median after low and high outliers are added. Also, illustrate how\u00a0similar the mean and median are when the distribution is symmetric. C3<\/li>\n<li aria-level=\"1\">Simulation: https:\/\/dcmathpathways.shinyapps.io\/MeanvsMedian\/.<\/li>\n<\/ul>\n<h3>Wrap-up\/transition (5 minutes)<\/h3>\n<ul>\n<li>As a final formative assessment, consider drawing various\u00a0distribution shapes (left skew, right skew, symmetric) on the board,\u00a0and have students describe the predicted relationship between the\u00a0median and mean. C4<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Have students refer back to the Objectives for the activity and identify ones they recognized completing. S2, C4, O2<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Assign the homework or\u00a0<em>Practice<\/em>\u00a0and any <em>What to Know<\/em> pages for the <em>Forming Connections<\/em> activities you plan to complete in the next class meeting. C2<\/li>\n<\/ul>\n","protected":false},"author":25777,"menu_order":16,"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-3927","chapter","type-chapter","status-publish","hentry"],"part":3890,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3927","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":7,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3927\/revisions"}],"predecessor-version":[{"id":4062,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3927\/revisions\/4062"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/3890"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3927\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3927"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3927"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3927"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3927"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}