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style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> 4.10 Spearman's Rank Correlation Coefficient <a href="#" class="mark-page-favorite pull-right" data-pid="2407" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1065/statistics-and-probability.html">Statistics and Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">4.10 Spearman's Rank Correlation Coefficient</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <div class="intro-card" readonly="false"><img class="intro-image" readonly="true" src="../../page-image-1.jpg" style="width: 1920px; height: 1280px;"> <div class="content" readonly="true"> <p class="text">Analysing how closely two things are correlated is incredibly useful to ascertain exactly how strongly one variable affects another. Whilst helpful if the data is approximately linear, Pearson's Product Moment Correlation Coefficient can be insufficient when analysing connections in the real world. This is especially the case when variables are difficult to measure or quantify, such as judging the quality of different dancers, or the taste of different foods, for example. Hence, we may need a different correlation coefficient in specific cases.</p> </div> </div> <div class="panel panel-has-colored-body panel-turquoise"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>In this unit you should learn to…</p> <ul> <li>Understand the difference between Pearson's Product Moment Correlation Coefficient and Spearman's Rank Correlation Coefficient. </li> <li>Create a ranked table. </li> <li>Use your GDC to find Spearman's Rank Correlation Coefficient. </li> <li>Interpret what the Spearman's Rank Correlation Coefficient means in a specific context (see quiz questions and exam style questions for this). </li> </ul> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-orange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Pearson vs Spearman</p> </div> </div> <div class="panel-body"> <div> <p>What is the difference between Pearson's Product Moment Correlation Coefficient and Spearman's Rank Correlation Coefficient?</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/639682587"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Ranking Data</p> </div> </div> <div class="panel-body"> <div>How do you rank data before inserting it into the GDC?</div> <div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/640090409"></iframe></div> <p>Question: If you ranked the highest value first, rather than the lowest for both variables, do you think this will change <span class="math-tex">\(r_s\)</span>?</p> <p>Answer:</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Simply put, no, it will not change the answer. Ranking them lowest to highest, or highest to lowest is absolutely fine. However, if you rank one variable from lowest to highest, you must do the same for the other variable. Whichever way you choose to rank one variable, you must stick with that method for the other variable. </p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Using the GDC to find Spearman's Rank</p> </div> </div> <div class="panel-body">TI-84</div> <div class="panel-body"> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/548828143"></iframe></div> </section> </div> <div class="panel-body">TI-Nspire</div> <div class="panel-body"> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/546824739"></iframe></div> </section> <p>Casio</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/671692651"></iframe></div> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <p><img alt="" src="../../flashcard.png" style="width: 890px; height: 515px;"></p> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <h4>Self Checking Quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos above<em>.</em></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-635-2407"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate Spearman's rank correlation coefficient, <span class="math-tex">\(r_{s}\)</span>, for the following scatter graph. </p><p><img alt="" src="../../images/dans-images/picture1(1).png" style="width: 300px; height: 231px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0.725</label></p><p><label class="radio"><input type="radio"> 0.322</label></p><p><label class="radio"><input class="c" type="radio"> 0.964</label></p><p><label class="radio"><input type="radio"> 1</label></p></div><div class="q-explanation"><p>The data is strictly increasing, with the exeption of the final point, and given that the final point is still relatively close to the previous point, <span class="math-tex">\(r_{s}\)</span>, is high. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate Spearman's rank correlation coefficient, <span class="math-tex">\(r_{s}\)</span>, for the following scatter graph. </p><p><img alt="" src="../../images/dans-images/picture2.png" style="width: 300px; height: 232px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> -1</label></p><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input type="radio"> 0.999</label></p><p><label class="radio"><input class="c" type="radio"> 1</label></p></div><div class="q-explanation"><p>Given the data is strictly increasing (monotonic), <span class="math-tex">\(r_{s}\)</span> equals one. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate Spearman's rank correlation coefficient, <span class="math-tex">\(r_{s}\)</span>, for the following scatter graph. </p><p><img alt="" src="../../images/dans-images/picture3.png" style="width: 300px; height: 225px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> -1</label></p><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input type="radio"> -0.855</label></p></div><div class="q-explanation"><p>Given the data is strictly increasing (monotonic), <span class="math-tex">\(r_{s}\)</span> equals minus one. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Two judges score a diving competition with five competitors. After ranking the scores, the <span class="math-tex">\(r_{s}\)</span> value is found to be 0.952. Comment on <span class="math-tex">\(r_{s}\)</span>in this context.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> There is a strong, positive correlation between the ranks. The judges strongly agree with each other. </label></p><p><label class="radio"><input type="radio"> There is no significant correlation between the ranks. The judges do not agree with each other. </label></p><p><label class="radio"><input type="radio"> There is a moderate, positive correlation between the ranks. The judges partially agree with each other. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>The amount of time studying for a Biology test is compared with the assessment scores. The <span class="math-tex">\(r_{s}\)</span> value for the data was 0.650. Interpret this value.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> There is a strong, positive correlation between the ranks of study time and test scores. Study time is strongly associated with test scores. </label></p><p><label class="radio"><input type="radio"> There is a weak, positive correlation between the ranks of study time and test scores. Study time is weakly associated with test scores. </label></p><p><label class="radio"><input class="c" type="radio"> There is a moderate, positive correlation between the ranks of study time and test scores. Study time is moderately associated with test scores. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Before calulating Spearman's Rank Correlation Coeffiient, we create a ranked table. Create a ranked table for the data below. Rank from smallest to largest.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>x</td><td>2</td><td>2</td><td>5</td><td>3</td><td>8</td></tr><tr><td>y</td><td>3</td><td>2</td><td>3</td><td>4</td><td>7</td></tr></tbody></table></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" height="64" width="742"><tbody><tr><td>Rank, x</td><td> <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></td></tr><tr><td>Rank, y</td><td> <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>Given that the first position and second position are equal for the variable x, we average these positions to get 1.5 in each case. This is similarly the case for the second and third position for variable y, giving 2.5 for these positions. Note that it is equally valid to rank from largest to smallest, as long as you do this for bothh variables.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The temperature of coffee is taken over time. The results are given in the table below.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Time (t), mins</td><td>0</td><td>10</td><td>20</td><td>30</td><td>40</td></tr><tr><td>Temperature, T, celsius</td><td>68</td><td>45</td><td>30</td><td>25</td><td>25</td></tr></tbody></table><p>Find <span class="math-tex">\(r_{s}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> - 0.975</label></p><p><label class="radio"><input type="radio"> -1</label></p><p><label class="radio"><input type="radio"> 0.975</label></p></div><div class="q-explanation"><p>The ranked table should be:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rank Time</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>Rank Temperature</td><td>5</td><td>4</td><td>3</td><td>1.5</td><td>1.5</td></tr></tbody></table><p>After inserting this data into your GDC, you find <span class="math-tex">\(r_{s}\)</span> to be - 0.975.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>The table below shows a class sample of study time compared with scores on a test.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Study Time, t</td><td>1</td><td>1</td><td>5</td><td>3.5</td><td>1</td><td>4</td></tr><tr><td>Score, S</td><td>68</td><td>60</td><td>92</td><td>70</td><td>64</td><td>80</td></tr></tbody></table><p>Fill in the gaps in the ranked table below.</p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Study Time, t</td><td> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></td><td>6</td><td>4</td><td> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></td><td>5</td></tr><tr><td>Score, S</td><td>3</td><td>1</td><td>6</td><td>4</td><td>2</td><td>5</td></tr></tbody></table><p> </p></div><div class="q-explanation"><p>Given that the first, second and third position for time spent studying are the same, we will find the average of these positions by adding them and dividing by three. In this case, that gives a rank of 2 for all three empty boxes.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>From the list below, pick the advantages of using Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient.</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> Spearman's rank provides a more accurate correlation than Pearson's. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> Spearman's rank can be used on data which is not linear. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> Spearman's rank can be useful for data which is less easy to quanitfy. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> Spearman's rank is less effected by outliers. </label></p></div><div class="q-explanation"><p>See answers above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>A set of data is strictly increasing. If the final data point is brought below the previous data point, but so that it is still above all other data points. How will this effect <span class="math-tex">\(r_{s}\)</span>?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> It will decrease to zero. </label></p><p><label class="radio"><input class="c" type="radio"> It will decrease only slightly from 1. </label></p><p><label class="radio"><input type="radio"> It will decrease to 0.5.</label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam Style Questions</p> </div> </div> <div class="panel-body"> <p>The following are exam style questions on this topic that you should try on your own before checking against the video solution.</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"><img alt="" src="../../images/dans-images/picture1(2).png" style="width: 600px; height: 338px;"></div> <div class="panel-body">Video Solution</div> <div class="panel-body"> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/642074305"></iframe></div> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"><img alt="" src="../../images/dans-images/picture2(1).png" style="width: 600px; height: 373px;"></div> <div class="panel-body">Video Solution</div> <div class="panel-body"> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/642458553"></iframe></div> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"><img alt="" src="../../images/dans-images/picture3(1).png" style="width: 600px; height: 369px;"></div> <div class="panel-body">Video Solution</div> <div class="panel-body"> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/670864369"></iframe></div> </section> </div> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2407"> <div 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