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Statistical Edutainment: Reversing Comparisons

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Title: Statistical Edutainment: Reversing Comparisons
Language: English
Authors: Lesser, Lawrence M. (ORCID 0000-0001-5762-3987), Pearl, Dennis K. (ORCID 0000-0003-1981-1826)
Source: Teaching Statistics: An International Journal for Teachers. Aut 2019 41(3):118-122.
Availability: Wiley-Blackwell. 350 Main Street, Malden, MA 02148. Tel: 800-835-6770; Tel: 781-388-8598; Fax: 781-388-8232; e-mail: cs-journals@wiley.com; Web site: http://www.wiley.com/WileyCDA
Peer Reviewed: Y
Page Count: 5
Publication Date: 2019
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Statistics, Mathematics Instruction, Instructional Materials, Student Motivation, Educational Strategies
DOI: 10.1111/test.12203
ISSN: 0141-982X
Abstract: Educational fun items can be readily found and used to increase student motivation and learning. We illustrate resources and strategies for a specific piece of content: Simpson's paradox.
Abstractor: As Provided
Entry Date: 2019
Accession Number: EJ1225702
Database: ERIC
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  Value: <anid>AN0138203798;d8y01sep.19;2019Aug24.05:30;v2.2.500</anid> <title id="AN0138203798-1">Statistical edutainment: Reversing comparisons </title> <p>Educational fun items can be readily found and used to increase student motivation and learning. We illustrate resources and strategies for a specific piece of content: Simpson's paradox.</p> <p>Keywords: cartoons; edutainment; poems; Simpson's paradox; songs; teaching statistics</p> <hd id="AN0138203798-2">INTRODUCTION</hd> <p>By "edutainment", we mean a portmanteau of "education" and "entertainment." These are fun items, such as poems, songs, cartoons, puzzles, and jokes, that are related to learning objectives in introductory statistics, have been classroom tested (with reflection upon best practices for classroom use), and do not require a great deal of time or talent to use in a classroom. Our comments here are informed by years of classroom testing, empirical research, and our curating an international collection of fun items for introductory statistics at https://<ulink href="http://www.CAUSEweb.org/fun">www.CAUSEweb.org/fun</ulink>. This widely visited collection currently has over 800 items spanning 11 modalities of fun: art gallery, cartoon, game, joke, magic, poem, puzzle, quote, song, story, and video. The collection is searchable by modality of fun, statistics content topic, and by keywords. Such items can also help reverse common instructor hesitations such as those in Lesser et al.[<reflink idref="bib1" id="ref1">1</reflink>]</p> <hd id="AN0138203798-3">SIMPSON'S PARADOX</hd> <p>Speaking of reversal, let us apply these ideas to a particular statistics phenomenon known as Simpson's paradox. Typically, this can be described as the direction of a comparison between two variables being reversed upon aggregation of subgroups. Alternatively, the direction of comparison between two variables can be partially or wholly reversed by splitting into subgroups, that is, by the effect of a third variable. As a hypothetical example, two anti‐inflammatory drugs for fast pain relief might be tested, with 200 (consenting) patients using drug A and 400 using drug B, with overall success (pain reduction) after just 30 minutes of 8% for drug A and 8.5% for drug B. But when the patients are split by age group (less than 50, greater than or equal to 50), it might be found that this comparison is reversed in one or both groups, for example, as in Table.</p> <p>Hypothetical data for success for two drugs, split by age group</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">Age < 50</th><th align="left" /><th align="left">Age ≥ 50</th><th align="left" /><th align="left">Overall % Success</th></tr><tr><th align="left" /><th>Sample Size</th><th>Number and % Success</th><th>Sample Size</th><th>Number and % Success</th><th /></tr></thead><tbody valign="top"><tr><td align="left">Drug A</td><td>150</td><td>9 (6%)</td><td>50</td><td>7 (14%)</td><td>16 of 200 = 8%</td></tr><tr><td align="left">Drug B</td><td>100</td><td>4 (4%)</td><td>300</td><td>30 (10%)</td><td>34 of 400 = 8.5%</td></tr></tbody></table> </ephtml> </p> <p>Simpson's paradox (which is mentioned in the <emph>2016 GAISE College Report</emph>, pp. 10, 38) is accessible enough to be verified by a middle school student's knowledge of fractions and yet nuanced and complex enough to be interesting to college and university students. See Lesser[<reflink idref="bib2" id="ref2">2</reflink>] and Schield[<reflink idref="bib3" id="ref3">3</reflink>] for background and many representations of Simpson's paradox, including table, circle graph (bubble chart), platform scale, trapezoid (BK plot), unit square, determinants, and vector geometry. This topic is also a good vehicle to show that educational fun items can be found for almost <emph>any</emph> topic in statistics.</p> <p>Seeing not just that it <emph>can</emph> happen but also getting intuition into <emph>when</emph> it happens requires more active exploration, such as by doing an active lesson such as the crumpled paper ball toss activity of Gou and Zhang[<reflink idref="bib4" id="ref4">4</reflink>] or by using web apps such as http://www.math.usu.edu/schneit/Statlets/Simpson%26apos%3Bs%20Paradox/[<reflink idref="bib5" id="ref5">5</reflink>] or https://rstudio.aws.science.psu.edu:3838/Boast/Multivariable_Topics/Paradox/.</p> <hd id="AN0138203798-4">SIMPSON'S PARADOX: SONG</hd> <p>As demonstrated in Table , the most concrete way to represent Simpson's paradox is with a table of numbers, but a fun song can both introduce the paradox and lead to examining or creating a table of data.</p> <p>A Simpson's paradox song that award‐winning performer/writer of educational science songs Monty Harper was commissioned to write for the authors' NSF‐funded Project SMILES has a specific example actually built into the lyric at https://<ulink href="http://www.CAUSEweb.org/cause/resources/fun/songs/no-one-counted-simpsons-paradox">www.CAUSEweb.org/cause/resources/fun/songs/no-one-counted-simpsons-paradox</ulink>.</p> <p>The first two verses describe a situation for a student choir in which the mean male height increased and the mean female height increased compared to last year's choir, and yet the overall mean decreased. The final verse explains qualitatively how this is possible: Well we had more girls this yearAnd fewer boys this yearAnd girls are mostly shorterThan boys as a ruleSo our girls got tallerAnd our boys got tallerWhile our choir got shorterAnd we all got schooled</p> <p>To demonstrate their conceptual understanding, students can construct their own dataset that would be consistent with the song. Table has an example of such a data set for a middle school choir with 10 students each year.</p> <p>For additional fun and engagement, Monty's song can also be explored by students in an interactive version for which students first provide inputs (in response to prompts) that get inserted and highlighted in the song upon playback not unlike the phrasal word template game Mad Libs that was also repurposed for statistics education by Trumpower.[<reflink idref="bib6" id="ref6">6</reflink>] The interactive version may be found on the website for the NSF‐funded Project SMILES at https://<ulink href="http://www.CAUSEweb.org/smiles/songs/simpsons%5fparadox">www.CAUSEweb.org/smiles/songs/simpsons%5fparadox</ulink>. This allows students to demonstrate further understanding by constructing a new context (ie, to apply or transfer their understanding to a scenario other than heights and genders of choir members).</p> <p>Another example in the CAUSEweb.org collection at https://<ulink href="http://www.CAUSEweb.org/cause/resources/fun/videos/simpsons-paradox">www.CAUSEweb.org/cause/resources/fun/videos/simpsons-paradox</ulink> is a 1‐minute song video for a parody (of the Queen hit "Another One Bites the Dust") by US teacher Mary McLellan from Aledo High School in Texas. The song declares the punchline of Simpson's paradox and the video includes a table with specific numbers.</p> <p>While not all students love statistics (yet), all students love music, so the use of song to engage students is powerful. Teachers hesitating out of uncertainty about effectiveness can find reassurance from literature (eg, <ulink href="http://singaboutscience.org/wp/educating/research/">http://singaboutscience.org/wp/educating/research/</ulink>). Teachers who have hesitated out of perceived lack of talent can simply pull up a CAUSEweb song or video and press "play."</p> <hd id="AN0138203798-5">SIMPSON'S PARADOX: CARTOON</hd> <p>Another popular modality of educational fun in statistics is cartoons and there have been studies (eg,[<reflink idref="bib7" id="ref7">7</reflink>]), cartoon books (eg,[<reflink idref="bib8" id="ref8">8</reflink>]), and cartoon lessons (eg,[<reflink idref="bib9" id="ref9">9</reflink>]). Figure is a cartoon from London‐based cartoonist John Landers, based on an idea by Dennis Pearl. After a moment to enjoy the cartoon's humor, students can be asked "If most men in the crowd liked the joke and most women in the crowd liked the joke, is it possible that most people in the total audience disliked the joke?" To reach the correct answer of "no" to this question, students have to go through an active discussion to recognize explicitly that (a) the value of a weighted mean must be in between the means of the two subgroups and (b) Simpson's paradox must involve a third variable, but the cartoon has only two ("gender" and "joke enjoyment"). Speaking of impossibilities, an enrichment challenge is to explore why what Friedlander and Wagon[<reflink idref="bib10" id="ref10">10</reflink>] call a "double Simpson's paradox" cannot happen.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/D8Y/01sep19/test12203-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="test12203-fig-0001.jpg" title="1 Simpson's paradox cartoon from CAUSEweb.org collection, https://www.CAUSEweb.org/cause/resources/fun/cartoons/Simpsons‐Comic [Colour figure can be viewed at wileyonlinelibrary.com]" /> </p> <p></p> <p>Another Simpson's paradox cartoon drawn by Landers (Figure) was added to the CAUSEweb collection after the first author's doubly corny caption was judged the winner in a CAUSE cartoon caption contest. (This monthly contest https://<ulink href="http://www.CAUSEweb.org/cause/caption-contest/">www.CAUSEweb.org/cause/caption-contest/</ulink> has been open to all students and instructors for 3 years.) The cartoon's visual image of a cornfield may give students a vivid way to remember "Cornfield's Conditions," which are lesser‐known conditions for Simpson's paradox to be possible but not guaranteed. They can be viewed as a "minimum effect size necessary for a potential confounder to explain an observed association assuming the association is totally spurious" (,[<reflink idref="bib11" id="ref11">11</reflink>] p. 2). For example, in the case of comparing drugs A and B in Table , the overall difference between drugs A and B is 0.5% (A = 8.5%, B = 8%), but Simpson's paradox is indeed possible because there is a confounder ("age") with a larger difference of approximately 5.4% (age < 50 = 5.2%, age ≥ 50 = 10.57%). In the case of Table , where we are interested in the difference in heights between choir members in consecutive years, Simpson's paradox is indeed possible because the 4.8‐cm difference between years (162.4 and 157.6 cm) is less than the 11.2‐cm difference (boys 165.6 cm; girls 154.4 cm) for the confounding variable of gender.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/D8Y/01sep19/test12203-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="test12203-fig-0002.jpg" title="2 Winner of June 2018 CAUSEweb cartoon caption contest https://www.CAUSEweb.org/cause/resources/fun/cartoons/corn‐fields‐condition [Colour figure can be viewed at wileyonlinelibrary.com]" /> </p> <p></p> <p>Heights of boys and girls in a choir</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">Number of Girls (Mean Height)</th><th align="left">Number of Boys (Mean Height)</th><th align="left">Overall</th></tr></thead><tbody valign="top"><tr><td align="left">Last year</td><td>8 (155 cm)</td><td>2 (168 cm)</td><td>10 (157.6 cm)</td></tr><tr><td align="left">This year</td><td>2 (152 cm)</td><td>8 (165 cm)</td><td>10 (162.4 cm)</td></tr></tbody></table> </ephtml> </p> <hd id="AN0138203798-8">SIMPSON'S PARADOX: JOKE</hd> <p>Because cartoons implicitly contain a verbal and/or visual joke, it is natural to also see jokes as a potential modality of educational fun. Indeed, Simpson's paradox can itself be viewed as having the structure of a joke. As <ulink href="http://michaelnielsen.org/reinventing%5fexplanation/">http://michaelnielsen.org/reinventing%5fexplanation/</ulink> relates, "Simpson's paradox ... has a premise which leads us to expect a particular conclusion, and then BAM, there's a switch and we see that the situation can be understood in a totally different way. It's challenging to get right, though, because the audience may not immediately get the punchline, unless they've been prepared in advance. That requires careful design." More generally, the use of jokes/humor in statistics has been researched (eg,[<reflink idref="bib12" id="ref12">12</reflink>]) and an excellent website of statistics humor (beyond the CAUSEweb collection) is at <ulink href="http://my.ilstu.edu/~gcramsey/Gallery.html">http://my.ilstu.edu/~gcramsey/Gallery.html</ulink>.</p> <p>There is also humor (and truth) in a special case of Simpson's paradox called the Will Rogers Phenomenon, in which moving an element from one set to another raises the mean of each set while, of course, the overall mean would stay the same at the two time points. This has had popular usage, with Rogers quipping (in the 1930s) that "When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states" and with New Zealand's then‐prime minister Robert Muldoon quipping similarly a half‐century later at Australia's expense.[<reflink idref="bib13" id="ref13">13</reflink>] Students can actively explore with numbers how this can happen by being asked to make up two sets of numbers {a,b} and {c,d,e} where c is less than d and e but greater than a and b. Students readily see that moving c to the other set raises the mean of each set, and with further playing with numbers can see that c does not even have to be the lowest value of the second set. Real‐life context can be added from the field of medicine. Improved cancer detection tools allow earlier diagnosis and therefore increased classification into the more severe stages. This stage migration improved patient survival statistics in each stage (eg,[<reflink idref="bib14" id="ref14">14</reflink>]).</p> <hd id="AN0138203798-9">SIMPSON'S PARADOX: POEM</hd> <p>It may be no surprise that another modality of educational fun in statistics is poetry, given that statistics poems have appeared on the cover of statistics journals (eg, <ulink href="http://www.radstats.org.uk/journal/issue117/),">http://www.radstats.org.uk/journal/issue117/),</ulink> featured in mathematical poetry blogs (eg, https://poetrywithmathematics.blogspot.com/2014/10/abc-of-statistics.html) and statistics magazines.[<reflink idref="bib15" id="ref15">15</reflink>] Here is a poem[<reflink idref="bib16" id="ref16">16</reflink>] about Simpson's paradox (ie, verse for a reverse!) that is reprinted with permission from <emph>The Mathematical Intelligencer</emph>: "Confounded"3 of 8 poemsI submittedto the classic journalwere accepted,while 1 of 3 my rival did were, so I won.2 of 3 poems I sent to the modern journal were accepted,while my rival had 3 of 5, so I won.But overall, my rival had half of hers accepted and I did not,so she wonafter all. I wasconfounded! I foundthat numbers don't lie,but don't explain why. Whytry comparing if comparison can bereversed with a Peterson rollby underdog wrestlingdata, rival, or self?When my parts aresummed,am I less than someof my parts?</p> <p>After reading the poem, classes could try questions such as</p> <p></p> <ulist> <item> Verify that 3/8 > 1/3, 2/3 > 3/5 but 5/11 < 4/8, thus showing that there is a reversal of comparison between the author and his rival upon aggregation across the two types of journals.</item> <p></p> <item> Create a physical or visual representation of this reversal (see,[<reflink idref="bib2" id="ref17">2</reflink>] for ideas).</item> <p></p> <item> Does the word "confounded" in the poem align with its everyday meaning, its statistical meaning, or both? Explain. (Also, students may need to look up what the "Peterson roll" wrestling move does.)</item> <p></p> <item> For an enrichment challenge: The total number of poems submitted (ie, the sum of the four denominators) is 19. Can you change the numbers in the poem so that a reversal still happens with a total less than 19? (spoiler: see problem 5321 in http://ssma.play-cello.com/wp-content/uploads/2016/03/Feb-2015.pdf)</item> </ulist> <hd id="AN0138203798-10">DISCUSSION</hd> <p>Making statistical ideas engaging and accessible to broad audiences is an important skill for all educators and communicators. Simpson's paradox provides a mechanism to stress the importance of understanding multivariable thinking. With a variety of modalities of fun items, we have used Simpson's paradox to demonstrate how educational fun items can be efficiently and memorably used to introduce, illustrate, or illuminate a statistical topic. We look forward to sharing more ideas and examples in the future. If you have any comments on this column or suggestions for the CAUSEweb.org fun collection, please email us.</p> <hd id="AN0138203798-11">ACKNOWLEDGMENT</hd> <p>This work was supported by Project SMILES, NSF/EHR/DUE 1544426 (PSU), 1544237 (UTEP). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.</p> <ref id="AN0138203798-12"> <title> REFERENCES </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> L. M. Lesser et al., Using fun in the statistics classroom: An exploratory study of college instructors' hesitations and motivations, J Stat Educ, 21 (1) (2013), 1 – 33, https://doi.org/10.1080/10691898.2013.11889659.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> L. Lesser, Representations of reversal: An exploration of Simpson's paradox, In The Roles of Representation in School Mathematics, A. A. Cuoco, F. R. Curcio (eds.), National Council of Teachers of Mathematics, Reston, VA, 2001, 129 – 145 <ulink href="http://www.statlit.org/pdf/2001LesserNCTM.pdf">http://www.statlit.org/pdf/2001LesserNCTM.pdf</ulink>.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref3" type="bt">3</bibl> <bibtext> M. Schield, Understanding confounding from lurking variables using graphs, Stats: The Magazine for Students of Statistics, 46 (2006), 14 – 18 https://<ulink href="http://www.CAUSEweb.org/cause/filebrowser/download/193">www.CAUSEweb.org/cause/filebrowser/download/193</ulink>.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref4" type="bt">4</bibl> <bibtext> J. Gou and F. Zhang, Experience Simpson's paradox in the classroom, Am Stat, 71 (1) (2017), 61 – 66.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref5" type="bt">5</bibl> <bibtext> K. Schneiter, An applet for the investigation of Simpson's paradox, J Stat Educ, 21 (1) (2013), 1 – 20.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref6" type="bt">6</bibl> <bibtext> D. Trumpower, Mad Libs statistics: A 'happy' activity, Teach Stat, 32 (1) (2010), 17 – 20.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref7" type="bt">7</bibl> <bibtext> A. A. Özdoğru and R. F. McMorris, Humorous cartoons in college textbooks: Student perceptions and learning, Humor, 26 (1) (2013), 135 – 154.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref8" type="bt">8</bibl> <bibtext> G. Klein and A. Dabney, The cartoon introduction to statistics, Hill and Wang, New York, 2013.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref9" type="bt">9</bibl> <bibtext> L. Lesser, Classroom notes: One in ten, Teach Stat, 40 (1) (2018), 33 – 34.</bibtext> </blist> <blist> <bibtext> R. Friedlander and S. Wagon, Double Simpson's paradox, Math Mag, 66 (4) (1993), 268.</bibtext> </blist> <blist> <bibtext> M. Schield, Simpson's paradox and Cornfield's conditions, In JSM Proceedings, Section on Statistical Education, 1999, 106 – 111.</bibtext> </blist> <blist> <bibtext> D. L. Neumann, M. Hood, M. M. Neumann, Statistics? You must be joking: The application and evaluation of humor when teaching statistics, J Stat Educ, 17 (2) (2009), https://doi.org/10.1080/10691898.2009.11889525.</bibtext> </blist> <blist> <bibtext> H. Orsman and J. Moore, Heinemann dictionary of New Zealand quotations, Heinemann, Auckland, 1988.</bibtext> </blist> <blist> <bibtext> M. P. Sormani, The Will Rogers phenomenon: The effect of different diagnostic criteria, J Neurol Sci, 287 S1 (2009), S46 – S49.</bibtext> </blist> <blist> <bibtext> J. Champkin, Eveline Pye: Poetry in numbers, Significance, 8 (3) (2011), 127 – 130.</bibtext> </blist> <blist> <bibtext> L. Lesser, Confounded, Math Intell, 32 (4) (2010), 53, https://doi.org/10.1007/2Fs00283-009-9127-x.</bibtext> </blist> </ref> <aug> <p>By Lawrence M. Lesser and Dennis K. Pearl</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref10"></nolink> <nolink nlid="nl2" bibid="bib11" firstref="ref11"></nolink> <nolink nlid="nl3" bibid="bib12" firstref="ref12"></nolink> <nolink nlid="nl4" bibid="bib13" firstref="ref13"></nolink> <nolink nlid="nl5" bibid="bib14" firstref="ref14"></nolink> <nolink nlid="nl6" bibid="bib15" firstref="ref15"></nolink> <nolink nlid="nl7" bibid="bib16" firstref="ref16"></nolink>
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      – PersonEntity:
          Name:
            NameFull: Lesser, Lawrence M.
      – PersonEntity:
          Name:
            NameFull: Pearl, Dennis K.
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          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2019
          Identifiers:
            – Type: issn-print
              Value: 0141-982X
          Numbering:
            – Type: volume
              Value: 41
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Teaching Statistics: An International Journal for Teachers
              Type: main
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