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from grouped frequencies.">
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<h1 style="text-align:center">Mean, Median and Mode<br>
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from Grouped Frequencies</h1>
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<p class="center"><img src="images/arrow-mean-median-mode.svg" alt="mean median mode"></p>
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<p class="center"><i>Explained with Three Examples</i></p>
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<h2>The Race and the Naughty Puppy</h2>
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<p>This starts with some raw data (<b>not a grouped frequency yet</b>) ...</p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/runners.jpg" alt="runners" height="177" width="200"></p>
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<h3> Alex timed 21 people in the sprint race, to the nearest second: </h3>
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<p class="center">59, 65, 61, 62, 53, 55, 60, 70, 64, 56, 58, 58, 62, 62, 68, 65, 56, 59, 68, 61, 67</p>
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</div>
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<p>To find the <a href="../mean.html"><span><span style="text-decoration:underline">Mean</span></span></a> Alex adds up all the numbers, then divides by how many numbers:</p>
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<p><b>Mean = <span class="intbl"><em>59 + 65 + 61 + 62 + 53 + 55 + 60 + 70 + 64 + 56 + 58 + 58 + 62 + 62 + 68 + 65 + 56 + 59 + 68 + 61 + 67</em>21</span> <br>
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<span style="visibility:hidden">Mean</span> = 61.38095...</b></p>
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<p> </p>
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<p>To find the <a href="../median.html"><span><span style="text-decoration:underline">Median</span></span></a> Alex places the numbers in value order and finds the middle number.</p>
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<p class="center"><img src="images/grouped-freq-1.gif" alt="frequency" height="56" width="360"></p>
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<p>In this case the median is the 11<sup>th</sup> number:</p>
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<p class="center">53, 55, 56, 56, 58, 58, 59, 59, 60, 61, <span class="hilite">61</span>, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70</p>
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<p class="center"><b>Median = 61 </b> </p>
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<p>To find the <a href="../mode.html">Mode</a>,
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or modal value, Alex places the numbers in value order then counts how many
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of each number. The Mode is the number which appears most often
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(there can be more than one mode):</p>
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<p class="center">53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, <span class="hilite">62, 62, 62</span>, 64, 65, 65, 67, 68, 68, 70</p>
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<p class="center">62 appears three times, more often than the other values, so <b>Mode = 62</b></p>
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<h2>Grouped Frequency Table</h2>
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<p>Alex then makes a <a href="frequency-distribution-grouped.html">Grouped Frequency Table</a>:</p>
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<div class="simple">
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<table style="border-collapse:collapse" align="center" width="217">
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<tbody>
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<tr valign="top">
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<th align="center" width="85">Seconds</th>
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<th align="center" width="77">Frequency</th>
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</tr>
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<tr valign="top">
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<td align="center">51 - 55</td>
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<td align="center">2</td>
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</tr>
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<tr valign="top">
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<td align="center">56 - 60</td>
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<td align="center">7</td>
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</tr>
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<tr valign="top">
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<td align="center">61 - 65</td>
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<td align="center">8</td>
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</tr>
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<tr valign="top">
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<td align="center">66 - 70</td>
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<td align="center">4</td>
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</tr>
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</tbody></table>
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</div>
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<p class="center"><img src="images/grouped-freq-2.gif" alt="frequency with groups" height="99" width="360"></p>
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<p>So 2 runners took between 51 and 55 seconds, 7 took between 56 and 60 seconds, etc</p>
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<h2>Oh No!</h2>
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<div class="center80">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/arrow-pup-rip.jpg" alt="puppy rips"></p>
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<p class="center larger">Suddenly all the original data gets lost (naughty pup!)</p>
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<p class="center larger"> <br>
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<b>Only the Grouped Frequency Table survived ...</b></p>
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</div>
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<p class="center">... can we help Alex calculate the Mean, Median and Mode from just that table? </p>
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<p>
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</p><center>
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</center>
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<p></p>
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<p>The answer is ... no we can't. Not accurately anyway. But, we can make <span style="font-weight:bold">estimates</span>.</p>
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<h2>Estimating the Mean from Grouped Data</h2>
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<p>So all we have left is:</p>
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<div class="simple">
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<table style="border-collapse:collapse" align="center" width="217">
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<tbody>
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<tr valign="top">
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<th align="center" width="85">Seconds</th>
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<th align="center" width="77">Frequency</th>
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</tr>
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<tr valign="top">
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<td align="center">51 - 55</td>
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<td align="center">2</td>
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</tr>
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<tr valign="top">
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<td align="center">56 - 60</td>
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<td align="center">7</td>
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</tr>
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<tr valign="top">
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<td align="center">61 - 65</td>
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<td align="center">8</td>
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</tr>
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<tr valign="top">
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<td align="center">66 - 70</td>
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<td align="center">4</td>
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</tr>
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</tbody></table>
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</div>
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<br>
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<div class="def">
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<p>The groups (51-55, 56-60, etc), also called <span style="font-weight:bold">class intervals</span>, are of <span style="font-weight:bold">width</span> 5 </p>
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<p>The <span style="font-weight:bold">midpoints </span>are in the middle of each class: 53, 58, 63 and 68</p>
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</div>
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<p class="center"><img src="images/grouped-freq-3.gif" alt="grouped frequency" height="99" width="360"></p>
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<p>We can estimate the <span style="font-weight:bold">Mean</span> by using the <b>midpoints</b>.</p>
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<div class="center80">
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<p>So, how does this work?</p>
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<p>Think about the 7 runners in the group <b>56 - 60</b>: all we know is that they ran somewhere between 56 and 60 seconds:</p>
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<ul>
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<li>Maybe all seven of them did 56 seconds,</li>
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<li>Maybe all seven of them did 60 seconds,</li>
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<li>But it is more likely that there is a spread of numbers: some at 56,
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some at 57, etc</li>
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</ul>
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<p>So we take an average and <b>assume</b> that all seven of them took 58 seconds.</p>
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</div>
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<p>Let's now make the table using midpoints:</p>
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<div class="simple">
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<table style="border-collapse:collapse" align="center" width="217">
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<tbody>
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<tr valign="top">
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<th align="center" width="85">Midpoint</th>
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<th align="center" width="77">Frequency</th>
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</tr>
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<tr valign="top">
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<td align="center">53</td>
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<td align="center">2</td>
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</tr>
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<tr valign="top">
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<td align="center">58</td>
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<td align="center">7</td>
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</tr>
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<tr valign="top">
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<td align="center">63</td>
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<td align="center">8</td>
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</tr>
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<tr valign="top">
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<td align="center">68</td>
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<td align="center">4</td>
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</tr>
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</tbody></table>
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</div>
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<p class="center"><img src="images/grouped-freq-3.gif" alt="grouped frequency" height="99" width="360"></p>
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<p>Our thinking is: "2 people took 53 sec, 7 people took 58 sec, 8 people took 63 sec and 4 took 68 sec". In other words we <b>imagine</b> the data looks like this:</p>
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<p class="center">53, 53, 58, 58, 58, 58, 58, 58, 58, 63, 63, 63, 63, 63, 63, 63, 63, 68, 68, 68, 68</p>
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<p>Then we add them all up and divide by 21. The quick way to do it is to multiply each midpoint by each frequency:</p>
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<div class="simple">
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<table style="border-collapse:collapse" align="center">
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<tbody>
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<tr valign="top">
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<th align="center" width="90">Midpoint<br>
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x</th>
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<th align="center" width="90">Frequency<br>
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f</th>
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<th align="center" width="100">Midpoint × Frequency<br>
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fx</th>
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</tr>
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<tr valign="top">
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<td align="center">53</td>
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<td align="center">2</td>
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<td align="center">106</td>
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</tr>
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<tr valign="top">
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<td align="center">58</td>
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<td align="center">7</td>
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<td align="center">406</td>
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</tr>
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<tr valign="top">
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<td align="center">63</td>
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<td align="center">8</td>
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<td align="center">504</td>
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</tr>
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<tr valign="top">
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<td align="center">68</td>
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<td align="center">4</td>
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<td align="center">272</td>
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</tr>
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<tr valign="top">
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<td align="center">Totals:</td>
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<td style="border-top: 1px solid black;" align="center"><b>21</b></td>
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<td style="border-top: 1px solid black;" align="center"><b>1288</b></td>
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</tr>
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</tbody></table>
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</div>
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<p>And then our <b>estimate</b> of the mean time to complete the race is:</p>
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<div style="margin-left:109pt;text-align:left"></div>
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<p class="center larger"><b>Estimated Mean</b> = <span class="intbl"><em><span class="center">1288</span></em><strong>21</strong></span> = <b>61.333...</b></p>
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<p>Very close to the exact answer we got earlier.</p>
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<h2>Estimating the Median from Grouped Data</h2>
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<p>Let's look at our data again:</p>
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<div class="simple">
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<table style="border-collapse:collapse" align="center" width="217">
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<tbody>
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<tr valign="top">
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<th align="center" width="85">Seconds</th>
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<th align="center" width="77">Frequency</th>
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</tr>
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<tr valign="top">
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<td align="center">51 - 55</td>
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<td align="center">2</td>
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</tr>
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<tr valign="top">
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<td align="center">56 - 60</td>
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<td align="center">7</td>
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</tr>
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<tr valign="top">
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<td align="center">61 - 65</td>
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<td align="center">8</td>
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</tr>
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<tr valign="top">
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<td align="center">66 - 70</td>
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<td align="center">4</td>
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</tr>
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</tbody></table>
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</div>
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<p class="center"><img src="images/grouped-freq-3.gif" alt="grouped frequency" height="99" width="360"></p>
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<p>The median is the middle value, which in our case is the 11<sup>th</sup> one, which is in the 61 - 65 group:</p>
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<p class="center larger">We can say "the <b>median group</b> is 61 - 65"</p>
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<p>But if we want an estimated <b>Median value</b> we need to look more closely at the 61 - 65 group.</p>
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<div class="center80">
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<p>We call it "61 - 65", but it really includes values from 60.5 up to (but not including) 65.5. </p>
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<p>Why? Well, the values are in whole seconds, so a real time of 60.5 is measured as 61. Likewise 65.4 is measured as 65.</p>
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</div>
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<p>At 60.5 we already have <b>9</b> runners, and by the next boundary at 65.5 we have <b>17</b> runners. By drawing a straight line in between we can pick out where the median frequency of <b>n/2</b> runners is:</p>
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<p class="center"><img src="images/grouped-freq-4.gif" alt="grouped frequency " height="230" width="360"></p>
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<p>And this handy formula does the calculation:</p>
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<p class="center larger"><b>Estimated Median</b> = L + <span class="intbl"><em>(n/2) − B</em><strong>G</strong></span> × w</p>
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<p>where:</p>
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<ul>
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<li><b>L</b> is the lower class boundary of the group containing the median </li>
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<li><b>n</b> is the total number of values </li>
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<li><b>B</b> is the cumulative frequency of the groups before the median group </li>
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<li><b>G</b> is the frequency of the median group </li>
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<li><b>w</b> is the group width </li>
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</ul>
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<p>For our example:</p>
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<ul>
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<li><b>L</b> = 60.5</li>
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<li><b>n</b> = 21</li>
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<li><b>B</b> = 2 + 7 = 9</li>
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<li><b>G</b> = 8</li>
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<li><b>w</b> = 5</li>
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</ul>
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<div class="tbl">
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<div class="row"><span class="left"><b>Estimated Median</b></span><span class="right">= 60.5 + <span class="intbl"> <em>(21/2) − 9</em> <strong>8</strong></span> × 5 </span></div>
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<div class="row"><span class="left"> </span><span class="right">= 60.5 + 0.9375</span></div>
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<div class="row"><span class="left"> </span><span class="right">= <b>61.4375</b></span></div>
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</div>
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<h2>Estimating the Mode from Grouped Data</h2>
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<p>Again, looking at our data:</p>
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<div class="simple">
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<table style="border-collapse:collapse" align="center" width="217">
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||
<tbody>
|
||
<tr valign="top">
|
||
<th align="center" width="85">Seconds</th>
|
||
<th align="center" width="77">Frequency</th>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center">51 - 55</td>
|
||
<td align="center">2</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center">56 - 60</td>
|
||
<td align="center">7</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center">61 - 65</td>
|
||
<td align="center">8</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center">66 - 70</td>
|
||
<td align="center">4</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>We can easily find the modal group (the group with the highest
|
||
frequency), which is <b>61 - 65</b></p>
|
||
<p class="center larger">We can say "the <b>modal group</b> is 61 - 65"</p>
|
||
<p>But the actual <span style="font-weight:bold">Mode </span>may not even be in that group! Or there may be more than one mode. Without the raw data we don't really know.</p>
|
||
<p>But, we can <span style="font-weight:bold">estimate</span> the Mode using the following formula:</p>
|
||
<p class="center larger"><b>Estimated Mode</b> = L + <span class="intbl"><em>f<sub>m</sub> − f<sub>m-1</sub></em><strong>(f<sub>m</sub> − f<sub>m-1</sub>) + (f<sub>m</sub> − f<sub>m+1</sub>)</strong></span> × w<b></b></p>
|
||
<p>where:</p>
|
||
<ul>
|
||
<li>L is the lower class boundary of the modal group </li>
|
||
<li>f<sub>m-1</sub> is the frequency of the group before the modal group</li>
|
||
<li>f<sub>m</sub> is the frequency of the modal group</li>
|
||
<li>f<sub>m+1</sub> is the frequency of the group after the modal group</li>
|
||
<li>w is the group width</li>
|
||
</ul>
|
||
<p>In this example:</p>
|
||
<ul>
|
||
<li>L = 60.5</li>
|
||
<li>f<sub>m-1</sub> = 7</li>
|
||
<li>f<sub>m</sub> = 8</li>
|
||
<li>f<sub>m+1</sub> = 4</li>
|
||
<li>w = 5</li>
|
||
</ul>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><b>Estimated Mode</b></span><span class="right">= 60.5 + <span class="intbl"><em>8 − 7</em><strong>(8 − 7) + (8 − 4)</strong></span> × 5 </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= 60.5 + (1/5) × 5 </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= <b>61.5</b></span></div>
|
||
</div>
|
||
<p> </p>
|
||
<div class="center80">
|
||
<p><b>Our final result is:</b></p>
|
||
<ul>
|
||
<li>Estimated Mean: <b>61.333...</b></li>
|
||
<li>Estimated Median: <b>61.4375</b></li>
|
||
<li>Estimated Mode: <b>61.5</b></li>
|
||
</ul>
|
||
<p>(Compare that with the true Mean, Median and Mode of <b>61.38..., 61 and 62</b> that we got at the very start.)</p>
|
||
</div>
|
||
<p> </p>
|
||
<p>And that is how it is done.</p>
|
||
<p>Now let us look at two more examples, and get some more practice along the way!</p>
|
||
<h2>Baby Carrots Example</h2>
|
||
<p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/carrots.jpg" alt="carrots" height="75" width="200"></p>
|
||
<p><span class="larger">Example: You grew fifty baby carrots using special soil. You dig them up and measure their lengths (to the nearest mm) and group the results</span>:</p>
|
||
<div class="simple">
|
||
<table align="center" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<th align="center" width="100">Length (mm)</th>
|
||
<th align="center" width="100">Frequency</th>
|
||
</tr>
|
||
<tr>
|
||
<td align="center"><span style="text-align:center">150 - 154</span></td>
|
||
<td align="center">5</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center"><span style="text-align:center">155 - 159</span></td>
|
||
<td align="center">2</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">160 - 164</td>
|
||
<td align="center">6</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">165 - 169</td>
|
||
<td align="center">8</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">170 - 174</td>
|
||
<td align="center">9</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center"><span style="text-align:center">175 - 179</span></td>
|
||
<td align="center">11</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">180 - 184</td>
|
||
<td align="center">6</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">185 - 189</td>
|
||
<td align="center">3</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<div style="margin-left:147pt;text-align:left"></div>
|
||
<p> </p>
|
||
<h3>Mean</h3>
|
||
<div class="simple">
|
||
<table align="center" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<th align="center" width="100">Length (mm)</th>
|
||
<th align="center" width="100">Midpoint<br>
|
||
x</th>
|
||
<th align="center" width="100">Frequency<br>
|
||
f</th>
|
||
<th align="center" width="100"><br>
|
||
fx</th>
|
||
</tr>
|
||
<tr>
|
||
<td align="center"><span style="text-align:center">150 - 154</span></td>
|
||
<td align="center">152</td>
|
||
<td align="center">5</td>
|
||
<td align="center">760</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center"><span style="text-align:center">155 - 159</span></td>
|
||
<td align="center">157</td>
|
||
<td align="center">2</td>
|
||
<td align="center">314</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">160 - 164</td>
|
||
<td align="center">162</td>
|
||
<td align="center">6</td>
|
||
<td align="center">972</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">165 - 169</td>
|
||
<td align="center">167</td>
|
||
<td align="center">8</td>
|
||
<td align="center">1336</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">170 - 174</td>
|
||
<td align="center">172</td>
|
||
<td align="center">9</td>
|
||
<td align="center">1548</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center"><span style="text-align:center">175 - 179</span></td>
|
||
<td align="center">177</td>
|
||
<td align="center">11</td>
|
||
<td align="center">1947</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">180 - 184</td>
|
||
<td align="center">182</td>
|
||
<td align="center">6</td>
|
||
<td align="center">1092</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center">185 - 189</td>
|
||
<td align="center">187</td>
|
||
<td align="center">3</td>
|
||
<td align="center">561</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center"> </td>
|
||
<td align="center">Totals:</td>
|
||
<td style="border-top: 1px solid black;" align="center"><b>50</b></td>
|
||
<td style="border-top: 1px solid black;" align="center"><b>8530</b></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<br>
|
||
<div style="margin-left:86pt;text-align:left"></div>
|
||
<p class="center larger"><b>Estimated Mean</b> = <span class="intbl"><em><span class="center">8530</span></em><strong>50</strong></span> = <b>170.6 mm</b></p>
|
||
<p> </p>
|
||
<h3>Median</h3>
|
||
<p>The Median is the mean of the 25<sup>th</sup> and the 26<sup>th</sup> length, so is in the <b>170 - 174</b> group:</p>
|
||
<ul>
|
||
<li><b>L</b> = 169.5 (the lower class boundary of the 170 - 174 group)</li>
|
||
<li><b>n</b> = 50</li>
|
||
<li><b>B</b> = 5 + 2 + 6 + 8 = 21</li>
|
||
<li><b>G</b> = 9</li>
|
||
<li><b>w</b> = 5</li>
|
||
</ul>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><b>Estimated Median</b></span><span class="right">= 169.5 + <span class="intbl"><em>(50/2) − 21</em><strong>9</strong></span> × 5 </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= 169.5 + 2.22... </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= <span style="text-align:center"><b>171.7 mm</b> (to 1 decimal)</span></span></div>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Mode</h3>
|
||
<p>The Modal group is the one with the highest frequency,
|
||
which is <b>175 - 179</b>:</p>
|
||
<ul>
|
||
<li>L = 174.5 (the lower class boundary of the 175 - 179 group)</li>
|
||
<li>f<sub>m-1</sub> = 9</li>
|
||
<li>f<sub>m</sub> = 11</li>
|
||
<li>f<sub>m+1</sub> = 6</li>
|
||
<li>w = 5</li>
|
||
</ul>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><b>Estimated Mode</b></span><span class="right">= 174.5 + <span class="intbl"><em>11 − 9</em><strong>(11 − 9) + (11 − 6)</strong></span> × 5</span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= 174.5 + 1.42... </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= <b>175.9 mm</b> (to
|
||
1 decimal)</span></div>
|
||
</div>
|
||
<h2>Age Example</h2>
|
||
<p>Age is a special case. </p>
|
||
<p>When we say "Sarah
|
||
is 17" she stays
|
||
"17" up until her eighteenth birthday. <br>
|
||
She might be 17 years and 364 days old and still be called "17".</p>
|
||
<p>This changes the midpoints and class boundaries.</p>
|
||
<p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/tropical-island.jpg" alt="tropical island" height="100" width="150"></p>
|
||
<p class="larger">Example: The ages of the 112 people who live on a tropical island are
|
||
grouped as follows:</p>
|
||
<div class="beach">
|
||
<table style="border-collapse:collapse" align="center">
|
||
<tbody>
|
||
<tr valign="top">
|
||
<th align="center" width="80" valign="top">Age</th>
|
||
<th align="center" width="80">Number</th>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">0 - 9</td>
|
||
<td align="center">20</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">10 - 19</td>
|
||
<td align="center">21</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">20 - 29</td>
|
||
<td align="center">23</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">30 - 39</td>
|
||
<td align="center">16</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">40 - 49</td>
|
||
<td align="center">11</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">50 - 59</td>
|
||
<td align="center">10</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">60 - 69</td>
|
||
<td align="center">7</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">70 - 79</td>
|
||
<td align="center">3</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">80 - 89</td>
|
||
<td align="center">1</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>A child in the first group <b>0 - 9 </b>could be
|
||
almost 10 years old. So the midpoint for this group is <b>5</b> <span>not
|
||
4.5</span></p>
|
||
<p> The midpoints are 5, 15, 25, 35, 45, 55, 65, 75 and 85</p>
|
||
<p>Similarly, in the calculations of Median and Mode, we will use the
|
||
class boundaries 0, 10, 20 etc</p>
|
||
<h3>Mean</h3>
|
||
<div class="beach">
|
||
<table style="border-collapse:collapse" align="center">
|
||
<tbody>
|
||
<tr valign="top">
|
||
<th align="center" width="80" valign="top">Age</th>
|
||
<th align="center" width="80">Midpoint<br>
|
||
x</th>
|
||
<th align="center" width="80">Number<br>
|
||
f</th>
|
||
<th align="center" width="80"><br>
|
||
fx</th>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">0 - 9</td>
|
||
<td align="center">5</td>
|
||
<td align="center">20</td>
|
||
<td align="center">100</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">10 - 19</td>
|
||
<td align="center">15</td>
|
||
<td align="center">21</td>
|
||
<td align="center">315</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">20 - 29</td>
|
||
<td align="center">25</td>
|
||
<td align="center">23</td>
|
||
<td align="center">575</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">30 - 39</td>
|
||
<td align="center">35</td>
|
||
<td align="center">16</td>
|
||
<td align="center">560</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">40 - 49</td>
|
||
<td align="center">45</td>
|
||
<td align="center">11</td>
|
||
<td align="center">495</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">50 - 59</td>
|
||
<td align="center">55</td>
|
||
<td align="center">10</td>
|
||
<td align="center">550</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">60 - 69</td>
|
||
<td align="center">65</td>
|
||
<td align="center">7</td>
|
||
<td align="center">455</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">70 - 79</td>
|
||
<td align="center">75</td>
|
||
<td align="center">3</td>
|
||
<td align="center">225</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top">80 - 89</td>
|
||
<td align="center">85</td>
|
||
<td align="center">1</td>
|
||
<td align="center">85</td>
|
||
</tr>
|
||
<tr valign="top">
|
||
<td align="center" valign="top"> </td>
|
||
<td align="center">Totals:</td>
|
||
<td style="border-top: 1px solid black;" align="center"><b>112</b></td>
|
||
<td style="border-top: 1px solid black;" align="center"><b>3360</b></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<div style="margin-left:72pt;text-align:left"></div>
|
||
<br>
|
||
<div style="margin-left:109pt;text-align:left"></div>
|
||
<p class="center larger"><b>Estimated Mean</b> = <span class="intbl"><em><span class="center">3360</span></em><strong>112</strong></span> = <b>30</b></p>
|
||
<p> </p>
|
||
<h3>Median</h3>
|
||
<p>The Median is the mean of the ages of the 56<sup>th</sup> and the 57<sup>th</sup> people, so is in the 20 - 29 group:</p>
|
||
<ul>
|
||
<li><b>L</b> = 20 (the lower class boundary of the class interval
|
||
containing the median)</li>
|
||
<li><b>n</b> = 112</li>
|
||
<li><b>B</b> = 20 + 21 = 41</li>
|
||
<li><b>G</b> = 23</li>
|
||
<li><b>w</b> = 10</li>
|
||
</ul>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><b>Estimated Median</b></span><span class="right">= 20 + <span class="intbl"><em>(112/2) − 41</em><strong>23</strong></span> × 10 </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= 20 + 6.52... </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= <span style="text-align:center"><b>26.5</b> (to 1 decimal)</span></span></div>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Mode</h3>
|
||
<p>The Modal group is the one with the highest frequency,
|
||
which is 20 - 29:</p>
|
||
<ul>
|
||
<li>L = 20 (the lower class boundary of the modal class)</li>
|
||
<li>f<sub>m-1</sub> = 21</li>
|
||
<li>f<sub>m</sub> = 23</li>
|
||
<li>f<sub>m+1</sub> = 16</li>
|
||
<li>w = 10</li>
|
||
</ul>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><b>Estimated Mode</b></span><span class="right">= 20 + <span class="intbl"><em>23 − 21</em><strong>(23 − 21) + (23 − 16)</strong></span> × 10 </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= 20 + 2.22... </span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">= <b>22.2</b> (to
|
||
1 decimal)</span></div>
|
||
</div>
|
||
<h2>Summary</h2>
|
||
<ul class="larger">
|
||
<li> For grouped data, we cannot find the exact Mean, Median and Mode,
|
||
we can only give <span style="font-weight:bold">estimates.</span> </li>
|
||
<li> To estimate the <b>Mean</b> use the <b>midpoints</b> of the class intervals:
|
||
|
||
|
||
|
||
|
||
<p class="center larger"><b>Estimated Mean</b> = <span class="intbl"><em>Sum of (Midpoint × Frequency)</em><strong>Sum of Frequency</strong></span></p>
|
||
</li>
|
||
<li> To estimate the <b>Median</b> use:
|
||
|
||
|
||
|
||
|
||
<p class="center larger"><b>Estimated Median</b> = L + <span class="intbl"><em>(n/2) − B</em><strong>G</strong></span> × w</p>
|
||
<p>where:</p>
|
||
<ul>
|
||
<li><b>L</b> is the lower class boundary of the group containing the median </li>
|
||
<li><b>n</b> is the total number of data </li>
|
||
<li><b>B</b> is the cumulative frequency of the groups before the median group </li>
|
||
<li><b>G</b> is the frequency of the median group </li>
|
||
<li><b>w</b> is the group width </li>
|
||
</ul>
|
||
|
||
</li>
|
||
<li>To estimate the <b>Mode</b> use:
|
||
|
||
|
||
|
||
|
||
<p class="center larger"><b>Estimated Mode</b> = L + <span class="intbl"><em>f<sub>m</sub> − f<sub>m-1</sub></em><strong>(f<sub>m</sub> − f<sub>m-1</sub>) + (f<sub>m</sub> − f<sub>m+1</sub>)</strong></span> × w<b></b></p>
|
||
<p>where:</p>
|
||
<ul>
|
||
<li>L is the lower class boundary of the modal group </li>
|
||
<li>f<sub>m-1</sub> is the frequency of the group before the modal group</li>
|
||
<li>f<sub>m</sub> is the frequency of the modal group</li>
|
||
<li>f<sub>m+1</sub> is the frequency of the group after the modal group</li>
|
||
<li>w is the group width</li>
|
||
</ul>
|
||
</li></ul>
|
||
<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(8731, 8732, 8830, 8831, 8832, 8833, 8834, 8835, 8836, 8837);</script> </div>
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