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Distribution</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" 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"> Representing and Measuring Data <a href="#" class="mark-page-favorite pull-right" data-pid="687" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../549/stats-probability.html">Stats & Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Representing and Measuring Data</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"> <p><img alt="" src="../../files/data-handling/main-1.png" style="float: left; width: 100px; height: 100px;">This page deals with one-variable statistics. You need to know how to represent the data in tables and diagrams (histograms, cumulative frequency graphs and box and whisker diagrams), as well measure the data using measures of central tendency (mean, median and mode) and dispersion (interquartile range, standard deviation and variance). It is not a difficult topic, but attention to detail is required. It is recommended that you work through the revision notes carefully.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>On this page, you should learn about</p> <ul> <li>frequency distributions</li> <li>histograms</li> <li>cumulative frequency graphs</li> <li>mean, median,mode</li> <li>quartiles, percentiles</li> <li>range, interquartile range, standard deviation and variance</li> <li>box and whisker diagrams</li> <li>outliers</li> <li>the effect of constant changes to mean and standard deviation </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-footer"> <div> </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"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/data-handling/revision-notes_representing_and_representing_data.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/data-handling/revision-notes_representing_and_representing_data.pdf" target="_blank">here</a></p> </div> </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"> <p>Here is a quiz about some definitions in statistics</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#9c4d2bc0"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="9c4d2bc0"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Statistics Definitions <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-531-687" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Match up the term with the definition </p><p><span class="q-text-draggable draggable" draggable="true">data</span> <span class="q-text-draggable draggable" draggable="true">categorical variable</span> <span class="q-text-draggable draggable" draggable="true">quantitative variable</span> <span class="q-text-draggable draggable" draggable="true">population</span> <span class="q-text-draggable draggable" draggable="true">census</span> <span class="q-text-draggable draggable" draggable="true">parameter</span> <span class="q-text-draggable draggable" draggable="true">sample</span> <span class="q-text-draggable draggable" draggable="true">survey</span> <span class="q-text-draggable draggable" draggable="true">statistic</span></p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>information about the characteristics of a group of individuals</td><td> <input type="text" style="height: auto;" data-c="data"> <span class="review"></span> </td></tr><tr><td>the collection of information from a sample</td><td> <input type="text" style="height: auto;" data-c="survey"> <span class="review"></span> </td></tr><tr><td>a subset of the group of individuals that you want to study</td><td> <input type="text" style="height: auto;" data-c="sample"> <span class="review"></span> </td></tr><tr><td>a quantitative characteristic of the <strong>population</strong> that you’re interested in estimating or testing</td><td> <input type="text" style="height: auto;" data-c="parameter"> <span class="review"></span> </td></tr><tr><td>a quantitative characteristic of a <strong>sample </strong>that often helps estimate or test the population parameter</td><td> <input type="text" style="height: auto;" data-c="statistic"> <span class="review"></span> </td></tr><tr><td>the entire group of individuals you want to study</td><td> <input type="text" style="height: auto;" data-c="population"> <span class="review"></span> </td></tr><tr><td>describes a characteristic that has a numerical value</td><td> <input type="text" style="height: auto;" data-c="quantitative variable"> <span class="review"></span> </td></tr><tr><td>describes a characteristic which can be divided into categories</td><td> <input type="text" style="height: auto;" data-c="categorical variable"> <span class="review"></span> </td></tr><tr><td>a collection of information from the whole population</td><td> <input type="text" style="height: auto;" data-c="census"> <span class="review"></span> </td></tr></tbody></table></div><div class="q-explanation"></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz about boxplots, cumulative frequency and histograms</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#515bdd8e"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="515bdd8e"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Representing Data <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-347-687" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Here is a cumulative frequency graph showing the scores of students in a test.</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q1.png" style="width: 450px; height: 343px;"></p><p>How many students scored more than 80?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 27</label></p><p><label class="radio"><input type="radio"> 16</label></p><p><label class="radio"><input type="radio"> 73</label></p><p><label class="radio"><input type="radio"> 84</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q1-ans.png" style="width: 496px; height: 343px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Here is a cumulative frequency graph</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q2.png" style="width: 450px; height: 331px;"></p><p>Find the median</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 18</label></p><p><label class="radio"><input type="radio"> 45</label></p><p><label class="radio"><input class="c" type="radio"> 52</label></p><p><label class="radio"><input type="radio"> 30</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q2-ans.png" style="width: 450px; height: 329px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The following data set is displayed in a box-and-whisker plot</p><p>1, 3, 4, 4, 6, 6, 6, 7, 7, 8, 10, 15</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q3.png" style="width: 422px; height: 120px;"></p><p>Find the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p><em><strong>a </strong></em>= <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><strong><em>a </em></strong> is the lower quartile</p><p>There are different methods for finding the lower quartile, but with 12 items it will lie in between the 3rd and the 4th items. So the lower quartile is 4.</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Here is a cumulative frequency graph</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q4.png" style="width: 450px; height: 331px;"></p><p>Find the interquartile range</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 37</label></p><p><label class="radio"><input class="c" type="radio"> 19</label></p><p><label class="radio"><input type="radio"> 30</label></p><p><label class="radio"><input type="radio"> 24</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q4-ans.png" style="width: 450px; height: 330px;"></p><p>Q1 = 37</p><p>Q3 = 56</p><p>IQR = 56 - 37 = 19</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The following graph shows the scores of students in a test. The scores for grade 7 are missing from the graph. If the mean score for the whole class is 5.1, find the number of students who scored grade 7.</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q5.png" style="width: 450px; height: 359px;"></p></div><div class="q-answer"><p>Number of students scoring grade 7 = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>mean = <span class="math-tex">\(\frac {3+20+35+30+7x}{18+x}=5.1\)</span></p><p>88 + 7x = 7.1(18 + x)</p><p>88 + 7x = 91.8 + 5.1x</p><p>1.9x = 3.8</p><p>x = 2</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the IQs of grade 12 students</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q6.png" style="width: 450px; height: 325px;"></p><p>What score would a student in the 80th percentile score?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 88</label></p><p><label class="radio"><input type="radio"> 80</label></p><p><label class="radio"><input type="radio"> 9</label></p><p><label class="radio"><input class="c" type="radio"> 113</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q6-ans.png" style="width: 450px; height: 324px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Q7 The following histogram represents scores in a test</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q7.png" style="width: 450px; height: 299px;"></p><p>Which box-and-whisker plot best represents these data</p><p>A <img alt="" src="../../files/data-handling/quiz-displaying-data/q7.1.png" style="width: 424px; height: 71px;"></p><p>B <img alt="" src="../../files/data-handling/quiz-displaying-data/q7.2.png" style="width: 420px; height: 65px;"></p><p>C <img alt="" src="../../files/data-handling/quiz-displaying-data/q7.3.png" style="width: 424px; height: 77px;"></p><p>D <img alt="" src="../../files/data-handling/quiz-displaying-data/q7.4.png" style="width: 424px; height: 72px;"></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> D</label></p></div><div class="q-explanation"><p>The data are uniform and so evenly spread</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The following cumulative frequency graph shows the monthly income of 60 families</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q8.png" style="width: 450px; height: 323px;"></p><p>Which box-and-whisker plot best represents these data</p><p>A <img alt="" src="../../files/data-handling/quiz-displaying-data/q8a.png" style="width: 450px; height: 67px;"></p><p>B <img alt="" src="../../files/data-handling/quiz-displaying-data/q8b.png" style="width: 450px; height: 77px;"></p><p>C <img alt="" src="../../files/data-handling/quiz-displaying-data/q8c.png" style="width: 450px; height: 68px;"></p><p>D <img alt="" src="../../files/data-handling/quiz-displaying-data/q8d.png" style="width: 450px; height: 70px;"></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>From the cumulative frequency graph we can deduce that</p><p>Minimum = $10 000</p><p>Maximum = $60 000</p><p>Median is a little over $40 000 which is just over halfway in between these two values.</p><p>Hence A is the best fit.</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The box-and-whisker plot below shows the number of times students visit YouTube on a particular day</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q9.png" style="width: 450px; height: 115px;"></p><p>One student visits the site <strong><em>k</em></strong> times (<strong><em>k</em></strong> > 10). What is the least value of <strong><em>k</em></strong> if <strong><em>k </em></strong>is an outlier?</p></div><div class="q-answer"><p><em><strong>k</strong></em> = <input type="text" style="height: auto;" data-c="21"> <span class="review"></span></p></div><div class="q-explanation"><p>Outlier is defined as a data item which is more than 1.5 × interquartile range (IQR) from the nearest quartile.</p><p>IQR = 7</p><p>Upper boundary = 10 + 1.5 × 7 = 20.5</p><p>Upper boundary = 21</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The box-and-whisker plot below shows the grades in a test of 50 students</p><p><img alt="" src="../../files/data-handling/quiz-displaying-data/q10.png" style="width: 450px; height: 117px;"></p><p>How many students scored 5 or less?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>12</span></label> </p><p><label class="radio"> <input type="radio"> <span>26</span></label> </p><p><label class="radio"> <input type="radio"> <span>13</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>25</span></label> </p></div><div class="q-explanation"><p>This question is more challenging than it looks!</p><p>The median score is 5.5 which means that</p><ul><li>the 25th person score 5</li><li>the 26th person score is 6</li></ul></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz about mean and standard deviation</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#36f09b8f"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="36f09b8f"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Mean and Standard Deviation <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-355-687" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The number of goals scored per match in a season is given below</p><p><img alt="" src="../../files/data-handling/mean-and-stadard-deviation/quiz/q1.png" style="width: 191px; height: 242px;"></p><p>Given that the mean number of goals per match is 2.2 , find the value of <strong><em>k</em></strong>.</p></div><div class="q-answer"><p><em><strong> k </strong></em>= <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Mean = <span class="math-tex">\(\frac {0+6+10+3k+8+5+6}{2+6+k+2+1+1}=2.2\)</span></p><p>35+3k = 2.2(17 + k)</p><p>35 + 3k = 37.4 + 2.2k</p><p>0.8k = 2.4</p><p>k = 3</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> The histogram below shows the scores in an examination.</p><p>Calculate the mean score.</p><p><img alt="" src="../../files/data-handling/mean-and-stadard-deviation/quiz/q2.png" style="width: 450px; height: 303px;"></p></div><div class="q-answer"><p>Mean = <input type="text" style="height: auto;" data-c="41.8"> <span class="review"></span></p></div><div class="q-explanation"><p>The data from the histogram can be converted into a frequency table</p><p><img alt="" src="../../files/data-handling/mean-and-stadard-deviation/quiz/q2a.png" style="width: 233px; height: 211px;"></p><p>The midpoint of the scores needs to be used</p><p><img alt="" src="../../files/data-handling/mean-and-stadard-deviation/quiz/q2b.png" style="width: 232px; height: 212px;"></p><p>Enter these data into your graphical calculator. Don't forget to set the frequency as the second list.</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> The cumulative frequency graph shows the score in a test.</p><p>Calculate the standard deviation of the scores to 3 significant figures.</p><p><img alt="" src="../../files/data-handling/mean-and-stadard-deviation/quiz/q3.png" style="width: 450px; height: 350px;"></p></div><div class="q-answer"><p>Standard deviation = <input type="text" style="height: auto;" data-c="10.5"> <span class="review"></span></p></div><div class="q-explanation"><p>The data can be converted into a table</p><p><img alt="" src="../../files/data-handling/mean-and-stadard-deviation/quiz/q3a.png" style="width: 273px; height: 182px;"></p><p>We need a frequency table with midpoint scores</p><p><img alt="" src="../../files/data-handling/mean-and-stadard-deviation/quiz/q3b.png"></p><p>Enter these data into your graphical calculator. Don't forget to set the frequency as the second list.</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The mean of five numbers is 10.8. One of the numbers is 12 and another is 18.</p><p>The other three numbers are the same. Find the valule of one of the three numbers.</p></div><div class="q-answer"><p>number = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>Total of 5 numbers = 10.8 x 5 = 54</p><p>Let x be one of the three unknown numbers</p><p>12 + 18 + 3x = 54</p><p>3x = 24</p><p>x = 8</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>A student takes the bus to school. She records the time taken on 20 days.</p><p>She finds the</p><p><span class="math-tex">\(\sum_{i=1}^{20} x _i=580\)</span> and <span class="math-tex">\(\sum_{i=1}^{20} x _i^2=16 948\)</span></p><p>Find the standard deviation of the times taken to 3 significant figures</p></div><div class="q-answer"><p>standard deviation = <input type="text" style="height: auto;" data-c="2.53"> <span class="review"></span></p></div><div class="q-explanation"><p>standard deviation = <span class="math-tex">\(\sqrt {\frac{\sum_{i=1}^{20} x _i^2}{n}-\frac{\sum_{i=1}^{20} x _i}{n}}\)</span></p><p>= <span class="math-tex">\(\sqrt {\frac{16948}{20}-\frac{580}{20}}\)</span></p><p><span class="math-tex">\(\approx 2.52982\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The mean score of 3 students in a test is 69.</p><p>A 4th student takes the test. The mean of the 4 students is 71.</p><p>What is the score of the 4th student?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 75</label></p><p><label class="radio"><input class="c" type="radio"> 77</label></p><p><label class="radio"><input type="radio"> 73</label></p><p><label class="radio"><input type="radio"> 71</label></p></div><div class="q-explanation"><p>Sum of 3 students = 3x69 = 207</p><p>Sum of 4 students = 4x71 = 284</p><p>4th student scores 284 - 207 = 77</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>10 students in a class take a test.</p><p>One student gets grade 1, four get grade 6 and all the other students get grade 7.</p><p>How many students score above the mean grade of the class?</p></div><div class="q-answer"><p>Answer = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>Mean = <span class="math-tex">\(\frac{1+6\times 4+5\times 7}{10}=6\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>In a test, the mean score = 55 and the standard deviation = 5</p><p>The teacher increases the scores by 3 marks.</p><p>Which of the following is true for the new scores?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> mean = 55 , variance = 25</label></p><p><label class="radio"><input type="radio"> mean = 58 , variance = 5</label></p><p><label class="radio"><input class="c" type="radio"> mean = 58 , variance = 25</label></p><p><label class="radio"><input type="radio"> mean = 58 , variance = 75</label></p></div><div class="q-explanation"><p>In general, if each number in a data set is increased by <strong><em>b </em></strong></p><p>The mean is increased by <strong><em>b</em></strong></p><p>The standard deviation remains unchanged (and the variance remains unchanged)</p><p>Variance = (standard deviation)²</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>In a test, the mean score = 55 and the standard deviation = 5</p><p>The teacher doubles each of the scores</p><p>Which of the following is true for the new scores?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> mean = 110 , variance = 50</label></p><p><label class="radio"><input type="radio"> mean = 55 , variance = 100</label></p><p><label class="radio"><input type="radio"> mean = 55 , variance = 50</label></p><p><label class="radio"><input class="c" type="radio"> mean = 110 , variance = 100</label></p></div><div class="q-explanation"><p>In general, if each number in a data set is multiplied by <strong><em>a</em></strong></p><p>The mean is multipled by <strong><em>a</em></strong></p><p>The standard deviation is multiplied by |a|</p><p>The variance is multiplies by a²</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Consider the data set {k - 3 , k - 1 , k + 2 , k + 4}</p><p>Each number in the above data is now decreased by 4 and the new mean is 6.5.</p><p>Find the value of <strong><em>k</em></strong>.</p></div><div class="q-answer"><p><strong><em>k</em></strong> = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>old mean = 6.5</p><p>New mean = 10.5</p><p><span class="math-tex">\(\frac{k - 3+k-1+k+2+k+4}{4}=10.5\)</span></p><p><span class="math-tex">\(\frac{4k+2}{4}=10.5\)</span></p><p>k = 10</p></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </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" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <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"> <div> <div class="smart-object center" data-id="877"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">During week 1, a group of 60 athletes were asked to record the amount of water, X litres, that they consumed in that week. Here are the results</span></p> <p><span class="math-tex">\(\sum x=1470\)</span></p> <p><span class="math-tex">\(\sum x^2=36\ 132.6\)</span></p> <p>Calculate</p> <p style="margin-left: 40px;">a. the mean of X</p> <p style="margin-left: 40px;">b. the standard deviation of X</p> <p>During week 2, as part of a programme to improve their performance, the athletes were instructed to drink 10% more water. Assuming that they do this, find for week 2</p> <p style="margin-left: 40px;">c. the new mean</p> <p style="margin-left: 40px;">d. the new standard deviation</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>If every athlete drinks 10% more, then each piece of data is multiplied by 1.1</p> <p>Mean is multiplied by 1.1</p> Standard deviation is multiplied by 1.1</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/esq_representing_and_measuring_data1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/esq_representing_and_measuring_data1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border 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"> <div> <div class="smart-object center" data-id="878"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The graph below shows the scores in the SL Mathematics Analysis and Approaches for at Goodenough High School. There are 20 students, the minimum score was 2 and the mean score is 4.45</p> <p><img alt="" src="../../files/data-handling/esq2.bmp" style="width: 500px; height: 390px;"></p> <p>How many grade 4s and how many grade 7s were there?</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Let x be the number of 4s and y be the number of 7s.</p> <p><content>Write 2 equations for the information given.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Let x be the number of 4s and y be the number of 7s.</p> <p>There are 20 students altogether: 2 + 3 + x + 4 + 3 + y = 20</p> <p>x + y = 8</p> <p>The mean score is 4.45 which mean that the sum of all the grades is 4.45x20 = 89</p> <p>4 + 9 + 4x + 20 + 18 + 7y = 89</p> <p>4x + 7y = 38</p> <p>Solve the simultaneous equations:</p> <p>4x + 7(8 - x) = 38</p> <p>4x + 56 - 7x = 38</p> <p>-3x = -18</p> <p>x = 6</p> <p>y = 2</p> <p>6 students got a grade 4</p> <p>2 students go a grade 7</p> <p><img alt="" src="../../files/data-handling/esq2ans.bmp" style="width: 400px; height: 312px;"></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border 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"> <div> <div class="smart-object center" data-id="879"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span style="font-size:12.0pt;line-height:115%; font-family:" timesnewromanpsmt="">The cumulative frequency graph gives information about the lengths, in minutes, of 80 telephone calls.</span></p> <p><img alt="" src="../../files/data-handling/esq3a.png" style="width: 750px; height: 500px;"></p> <p>a. Find the median length of a phone call</p> <p>b. Find the interquartile range of the length of a phone call</p> <p>c. Find the number of phone calls that were more than 10 minutes in length</p> <p>d. The frequency table below shows the lengths of the 80 phone calls. Find values a, b and c.</p> <p><img alt="" src="../../files/data-handling/esq3b.png" style="width: 300px; height: 344px;"></p> <div> <p>e. These data contain some outliers. How many outliers are there?</p> </div> <p>f. Calculate an estimate of the mean length of a phone call</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Read values from the graph carefully<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/esq_representing_and_measuring_data3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/esq_representing_and_measuring_data3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div 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