<div dir="ltr"><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:small;color:#000000">This one is new to me.  </div><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:small;color:#000000"><br></div><div class="gmail_default" style="font-family:comic sans ms,sans-serif;font-size:small;color:#000000"><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">During a faculty meeting, a group of 9th grade teachers decided they needed to further understand what the optimal duration of study is for students to achieve satisfactory results. So, they decided to gather the approximate number of hours students were studying, and then compare to the student’s test scores.</p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">Mr. Simpson convinced the faculty that more data means better results, and so all of the teachers integrated their cross-course data for the analysis.</p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">The results were astounding. To everyone’s confusion, the less a student studied, the higher they tend to score on tests.</p><div class="gmail-ui_qtext_image_outer" style="display:inline-block;margin-bottom:1em;overflow:hidden;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px"><div class="gmail-ui_qtext_image_wrapper" style="line-height:0"><img class="gmail-landscape gmail-ui_qtext_image gmail-zoomable_in gmail-zoomable_in_feed" src="https://qph.fs.quoracdn.net/main-qimg-2eacb9e9772a722b8162e83ae1502c6d" style="border: 0px; color: transparent; animation-duration: 0.001s; animation-name: insQ_100; cursor: -webkit-zoom-in; border-radius: 3px; max-width: 100%;"></div></div><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">In fact, the coefficient associated with this correlation was -0.7981, a strongly negative relationship.</p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">Should they be encouraging their students to study less? How in the world could data be backing up such a claim? Surely something was missing.</p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">After discussing the results, the teachers agreed they should consult the school’s statistician, Mrs. Paradox. After Mr. Simpson explained to Mrs. Paradox what they had found in their results, Mrs. Paradox suggested they analyze each course’s data individually.</p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">So, they went ahead and analyzed Phys. Ed. and proceeded to have their minds blown.</p><div class="gmail-ui_qtext_image_outer" style="display:inline-block;margin-bottom:1em;overflow:hidden;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px"><div class="gmail-ui_qtext_image_wrapper" style="line-height:0"><img class="gmail-landscape gmail-ui_qtext_image gmail-zoomable_in gmail-zoomable_in_feed" src="https://qph.fs.quoracdn.net/main-qimg-37969d7ab08a1fb576e96e87f7421626" style="border: 0px; color: transparent; animation-duration: 0.001s; animation-name: insQ_100; cursor: -webkit-zoom-in; border-radius: 3px; max-width: 100%;"></div></div><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">A correlation of 0.6353! How in the statistical universe was this even possible?</p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">Mrs. Paradox then explained this as <b>Simpson’s Paradox, a statistical phenomenon where a seemingly strong relationship reverses or disappears when introduced to a third confounding variable.</b></p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">She convinced Mr. Simpson to plot all of the data once again, but then color-code each course separately to distinguish them from one another.</p><div class="gmail-ui_qtext_image_outer" style="display:inline-block;margin-bottom:1em;overflow:hidden;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px"><div class="gmail-ui_qtext_image_wrapper" style="line-height:0"><img class="gmail-landscape gmail-ui_qtext_image gmail-zoomable_in gmail-zoomable_in_feed" src="https://qph.fs.quoracdn.net/main-qimg-5fc5b7d12a883c70267d953cc98bb121" style="border: 0px; color: transparent; animation-duration: 0.001s; animation-name: insQ_100; cursor: -webkit-zoom-in; border-radius: 3px; max-width: 100%;"></div></div><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">After doing so, Mr. Simpson and the 9th grade faculty concluded that the relationship was indeed positive, and that the more hours a student studied, the higher the grade tends to be.</p><p class="gmail-ui_qtext_para gmail-u-ltr gmail-u-text-align--start" style="margin:0px 0px 1em;padding:0px;color:rgb(51,51,51);font-family:-apple-system,BlinkMacSystemFont,"Segoe UI",Roboto,Oxygen-Sans,Ubuntu,Cantarell,"Helvetica Neue",sans-serif;font-size:15px;direction:ltr">Including the course of study in the analysis completely reversed the relationship.</p><br class="gmail-Apple-interchange-newline"></div></div>