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Spearman's Rank Correlation Coefficient</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2259/411-chi-squared-independence-tests.html">4.11 Chi Squared Independence tests</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2413/411-chi-squared-goodness-of-fit-and-the-t-test.html">4.11 Chi Squared Goodness of Fit and the T-Test</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"> 4.3 Central tendency and Dispersion <a href="#" class="mark-page-favorite pull-right" data-pid="2189" 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">4.3 Central tendency and Dispersion</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"> <div class="bg-cover" style="background-image: url('/media/mathsapplications/averages.png');"> </div> <img src="../../averages-1.png" style="display: none"> <div class="content"> <p class="theading">Mean, Mode, Median, Range, Variance, Standard Deviation</p> <p class="text">Measures of central tendency, such as the mean, median and mode, are very useful ways of representing large amounts of data with just one value. They can be very useful in forming conclusions about data, but they can also misrepresent data and hide information. Measures of dispersion, such as range, interquartile range, variance and standard deviation are ways of representing how spread out the data is.</p> </div> </div> <div class="panel panel-has-colored-body panel-turquoise"> <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"> <div> <p><meta charset="utf-8"></p> <p dir="ltr"><b id="docs-internal-guid-bd928f18-7fff-8591-d823-8728102aa679">In this unit you should learn to…</b></p> <ul> <li aria-level="1" dir="ltr">Find the mode, median and mean of a set of data</li> <li aria-level="1" dir="ltr">Find the range and interquartile range of a set of data</li> <li aria-level="1" dir="ltr">Find the variance and standard deviation of a set of data</li> <li aria-level="1" dir="ltr">Know what happens to the summary statistics when data is transformed. </li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" 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-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Mode</p> </div> </div> <div class="panel-body"> <div> <p>Finding the mode from a list and from a frequency table</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/701584701?h=1b99aa0463&badge=0&autopause=0&player_id=0&app_id=58479"></iframe></div> <script src="../../../api/player.js"></script></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Median</p> </div> </div> <div class="panel-body"> <div> <p>Finding the median from a list and from a frequency table</p> <p style="text-align: center"> </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/701595647?h=e654d200f3&badge=0&autopause=0&player_id=0&app_id=58479"></iframe></div> <script src="../../../api/player.js"></script> <p> </p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Mean</p> </div> </div> <div class="panel-body"> <div> <p>Finding the mean from a list and a frequency table</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/701591308?h=39ebe51d6d&badge=0&autopause=0&player_id=0&app_id=58479"></iframe></div> <script src="../../../api/player.js"></script></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>4. Range and Interquartile Range</p> </div> </div> <div class="panel-body"> <div> <p>Finding the range and interquartile range of data</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/701597211?h=aea980ffd8&badge=0&autopause=0&player_id=0&app_id=58479"></iframe></div> <script src="../../../api/player.js"></script></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Variance and Standard Deviation</p> </div> </div> <div class="panel-body"> <p>Finding the variance and the standard deviation from data</p> <p style="text-align: center"> </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/701600996?h=2221a53857&badge=0&autopause=0&player_id=0&app_id=58479"></iframe></div> <script src="../../../api/player.js"></script> <p> </p> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <ul class="gallery layout-row thumb48"><li class=""><a href="../../../std-galleries/15-211/mode-1.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mode-1-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/mode-2.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mode-2-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/mode-3.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mode-3-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/mode-4.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mode-4-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/median-1.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/median-1-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/median-2.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/median-2-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/median-3.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/median-3-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/median-4.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/median-4-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/median-5.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/median-5-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/median-6.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/median-6-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/median-7.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/median-7-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/mean-1.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mean-1-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/mean-2.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mean-2-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-211/mean-3.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mean-3-thumb128.jpg" class="thumb"></a></li><li><a href="../../../std-galleries/15-211/mean-4.png" data-fancybox="gallery-211" class="fancy"><img src="../../../std-galleries/15-211/mean-4-thumb128.jpg" class="thumb"></a></li></ul> <ul class="gallery layout-row thumb48"><li class=""><a href="../../../std-galleries/15-212/range-1.png" data-fancybox="gallery-212" class="fancy"><img src="../../../std-galleries/15-212/range-1-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-212/range-2.png" data-fancybox="gallery-212" class="fancy"><img src="../../../std-galleries/15-212/range-2-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-212/v1.png" data-fancybox="gallery-212" class="fancy"><img src="../../../std-galleries/15-212/v1-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-212/v2.png" data-fancybox="gallery-212" class="fancy"><img src="../../../std-galleries/15-212/v2-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-212/t1.png" data-fancybox="gallery-212" class="fancy"><img src="../../../std-galleries/15-212/t1-thumb128.jpg" class="thumb"></a></li><li><a href="../../../std-galleries/15-212/t2.png" data-fancybox="gallery-212" class="fancy"><img src="../../../std-galleries/15-212/t2-thumb128.jpg" class="thumb"></a></li></ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself </p> </div> </div> <div class="panel-body"> <div> <p>Self Checking quiz</p> <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="15-522-2189"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the mean and median of this set of numbers?</p><p>8, 5, 6, 9, 2, 5, 3, 6</p></div><div class="q-answer"><p>Mean = <input type="text" style="height: auto;" data-c="5.5"> <span class="review"></span></p><p>Median = <input type="text" style="height: auto;" data-c="5.5"> <span class="review"></span></p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC</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>What is the the standard deviation of this set of numbers? (<em>Exact or 3.s.f</em>)</p><p>42, 63, 62, 59, 48, 33, 41</p></div><div class="q-answer"><p>Standard deviation = <input type="text" style="height: auto;" data-c="10.9"> <span class="review"></span> </p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC</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>Work out the mean and the standard deviation of the data from the following frequency table (<em>exact or 3.s.f</em>):</p><p><img alt="" src="../../images/rn/frequency-1.png" style="width: 300px; height: 199px;"></p></div><div class="q-answer"><p>Mean = <input type="text" style="height: auto;" data-c="3.23"> <span class="review"></span> </p><p>Standard deviation = <input type="text" style="height: auto;" data-c="1.32"> <span class="review"></span> </p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC - make sure list 2 is set as frequency</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>A set of newborn babies in a hospital were weighed in kilograms. Calculate an estimation of the mean and standard deviation from the data collected in the frequency table below (<em>Exact or 3.s.f</em>)</p><p><img alt="" src="../../images/rn/frequency-20.png" style="width: 200px; height: 163px;"></p></div><div class="q-answer"><p>Mean: <input type="text" style="height: auto;" data-c="3.42"> <span class="review"></span> </p><p>Standard deviation: <input type="text" style="height: auto;" data-c="0.841"> <span class="review"></span> </p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC - make sure list 2 is set as frequency</p><p>To estimate <span class="math-tex">\(x\)</span>, calculate the midpoint between the limits</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> <div class="panel-footer"> <div> </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"> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with 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"> <p dir="ltr" id="docs-internal-guid-de94c8f7-7fff-b265-1837-bd0c6b290d0c">The table below shows the number of languages a class of 25 children speak</p> <table align="left" dir="ltr" style="width:718px;"> <colgroup> <col> <col> <col> <col> <col> <col> </colgroup> <tbody> <tr> <td> <p dir="ltr">No of languages</p> </td> <td> <p dir="ltr">1</p> </td> <td> <p dir="ltr">2</p> </td> <td> <p dir="ltr">3</p> </td> <td> <p dir="ltr">4</p> </td> <td> <p dir="ltr">5</p> </td> </tr> <tr> <td> <p dir="ltr">Frequency</p> </td> <td> <p dir="ltr">7</p> </td> <td> <p dir="ltr">p</p> </td> <td> <p dir="ltr">6</p> </td> <td> <p dir="ltr">3</p> </td> <td> <p dir="ltr">1</p> </td> </tr> </tbody> </table> <div style="clear:both;"> </div> <ol style="list-style-type:lower-alpha;"> <li dir="ltr"> <p dir="ltr" role="presentation">Find the value of p </p> </li> </ol> <p dir="ltr" style="margin-left: 36pt; text-align: right;">(1)</p> <ol style="list-style-type:lower-alpha;"> <li dir="ltr" value="2"> <p dir="ltr" role="presentation">Use your graphical display calculator to find</p> <ol style="list-style-type:lower-roman;"> <li dir="ltr"> <p dir="ltr" role="presentation">The median number of languages spoken</p> </li> <li dir="ltr"> <p dir="ltr" role="presentation">The mean number of languages spoken</p> </li> <li dir="ltr"> <p dir="ltr" role="presentation">The standard deviation of the number of languages spoken</p> </li> </ol> </li> </ol> <p dir="ltr" style="margin-left: 108pt; text-align: right;">(3)</p> <ol style="list-style-type:lower-alpha;"> <li dir="ltr" value="3"> <p dir="ltr" role="presentation">Showing your working, identify any outliers in this data. </p> </li> </ol> <p dir="ltr" style="margin-left: 36pt; text-align: right;">(2)</p> <p style="text-align: right;"><br> (6 marks)</p> <p>Video solution </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/598259802?h=df42f2f040"></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"> <div> <p dir="ltr" id="docs-internal-guid-4d123a0b-7fff-79d4-f9ef-963020ddea50">The mean of the ten numbers listed below is 11.</p> <p dir="ltr"> 12, 9, 14, 10, 19, 8, 13, 11, a, b</p> <p dir="ltr">(a) Write down an equation in terms of a and b.</p> <p dir="ltr" style="text-align: right;">(2)</p> <p dir="ltr"> The mode of these ten numbers is nine and a is less than b. </p> <p dir="ltr">(b) Write down the value of</p> <p dir="ltr" style="margin-left:36pt;">(i) a;</p> <p dir="ltr" style="margin-left:36pt;">(ii) b.</p> <p dir="ltr" style="text-align: right;">(2)</p> <p dir="ltr">(c) Find the median of the ten numbers.</p> <p dir="ltr" style="text-align: right;">(2)</p> <p dir="ltr" style="text-align: right;">(Total 6 marks)</p> <p dir="ltr">Video solution</p> </div> <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/598267666?badge=0&autopause=0&player_id=0&app_id=58479&h=6553ec539e"></iframe></div> <script src="../../../api/player.js"></script></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"> <div> <p dir="ltr" id="docs-internal-guid-575ef551-7fff-4ac0-1636-66477f1832f0">10 dogs were weighed and their results written down in kilograms. The following box and whisker plot was used to represent their weights. </p> <p dir="ltr"><img height="316" src="../../../r5dtYfu8-AWhbCBtBdiRd-XFm-nD3maQf7F3_eNiCUMjAbhzyhlLdfyCtWrjPdjLV3xf4TTs_tU9RDk7FAwgbkXwHyejdr31OTespLqsZvpOS0s9Ym3DaVgcURF93Q" style="margin-top:0px;" width="678"></p> <p dir="ltr">a) Write down</p> <p dir="ltr" style="margin-left:36pt;">(i) the median weight;</p> <p dir="ltr" style="margin-left:36pt;">(ii) the 75th percentile weight;</p> <p dir="ltr" style="margin-left:36pt;">(iii) the range of marks.</p> <p dir="ltr" style="margin-left: 36pt; text-align: right;">(4)</p> <p dir="ltr">b) How many dogs weighed more than 39 kg?</p> <p dir="ltr" style="text-align: right;">(2)</p> <p dir="ltr">The mean weight of the dogs was found to be 40.5kg, and the standard deviation 13.8. After being put on a strict exercise routine, all the dogs had lost 2kg the next time they were weighed. </p> <p dir="ltr">c) What is the new mean and standard deviation of the dogs? </p> <p dir="ltr" style="text-align: right;">(2)</p> <p dir="ltr" style="text-align: right;">(Total 8 marks)</p> </div> <p>Video solution</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/598276058?badge=0&autopause=0&player_id=0&app_id=58479&h=a644fc5f54"></iframe></div> <script src="../../../api/player.js"></script></section> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2189"> <div class="panel-heading">MY PROGRESS</div> <div 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