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294 lines
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<title>Calculating the mean from a frequency table</title>
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<h1 align="center">The Mean from a Frequency Table</h1>
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<p>It is easy to calculate the <a href="../mean.html">Mean</a>:</p>
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<p align="center"><b>Add up</b> all the numbers, <br>
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then <b>divide by how many</b> numbers there are.</p>
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<div class="example">
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<h3>Example: What is the Mean of these numbers?</h3>
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<center>
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<p class="largemono">6, 11, 7 </p>
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</center>
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<ul>
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<li>Add the numbers: <b>6 + 11 + 7 = 24</b></li>
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<li>Divide by <i>how many</i> numbers (there are 3 numbers): <b>24 ÷
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3 = 8</b></li>
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</ul>
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<center>
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<h3><b>The Mean is 8</b></h3>
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</center>
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</div>
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<p>But sometimes we don't have a simple list of numbers, it might be a frequency table like this (the "frequency" says how often they occur):</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<th align="center" width="100">Score</th>
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<th align="center" width="100">Frequency</th>
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</tr>
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<tr>
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<td align="center">1</td>
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<td align="center">2</td>
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</tr>
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<tr>
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<td align="center">2</td>
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<td align="center">5</td>
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</tr>
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<tr>
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<td align="center">3</td>
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<td align="center">4</td>
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</tr>
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<tr>
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<td align="center">4</td>
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<td align="center">2</td>
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</tr>
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<tr>
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<td align="center">5</td>
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<td align="center">1</td>
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</tr>
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</tbody></table>
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<p align="center"><i>(it says that score 1 occurred 2 times, score 2 occurred 5 times, etc)</i></p>
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<p>We could list all the numbers like this:</p>
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<p class="center larger">Mean = <span class="intbl"><em>1+1 + 2+2+2+2+2 + 3+3+3+3 + 4+4 + 5</em><strong>(how many numbers)</strong></span></p>
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<p>But rather than do lots of adds (like 3+3+3+3) it is easier to use multiplication:</p>
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<p class="center larger">Mean = <span class="intbl">
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<em><span class="hilite">2×</span>1 + <span class="hilite">5×</span>2 + <span class="hilite">4×</span>3 + <span class="hilite">2×</span>4 + <span class="hilite">1×</span>5</em>
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<strong>(how many numbers)</strong>
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</span></p>
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<p>And rather than count how many numbers there are, we can add up the frequencies:</p>
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<p class="center larger">Mean = <span class="intbl">
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<em><span class="hilite">2×</span>1 + <span class="hilite">5×</span>2 + <span class="hilite">4×</span>3 + <span class="hilite">2×</span>4 + <span class="hilite">1×</span>5</em>
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<strong><span class="hilite">2 + 5 + 4 + 2 + 1</span></strong>
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</span></p>
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<p>And now we calculate:</p>
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<p class="center larger">Mean = <span class="intbl">
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<em>2 + 10 + 12 + 8 + 5</em>
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<strong>14</strong>
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</span> <br>
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= <span class="intbl"><em>37</em><strong>14</strong></span>
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= <b>2.64...</b></p>
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<p>And that is how to calculate the mean from a frequency table!</p>
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<p>Here is another example:</p>
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<div class="example">
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<h3>Example: Parking Spaces per House in Hampton Street</h3>
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<p>Isabella went up and down the street to find out how many parking spaces each house has. Here are her results:</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<th align="center" width="100">Parking<br>
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Spaces</th>
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<th align="center" width="100">Frequency</th>
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</tr>
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<tr>
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<td align="center">1</td>
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<td align="center">15</td>
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</tr>
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<tr>
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<td align="center">2</td>
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<td align="center">27</td>
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</tr>
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<tr>
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<td align="center">3</td>
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<td align="center">8</td>
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</tr>
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<tr>
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<td align="center">4</td>
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<td align="center">5</td>
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</tr>
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</tbody></table>
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<p>What is the mean number of Parking Spaces?</p>
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<p>Answer:</p>
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<div class="tbl">
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<div class="row"><span class="left">Mean = </span><span class="right"><span class="intbl"><em><span class="hilite">15×</span>1 + <span class="hilite">27×</span>2 + <span class="hilite">8×</span>3 + <span class="hilite">5×</span>4</em><strong>15 + 27 + 8 + 5</strong></span></span></div>
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<div class="row"><span class="left">= </span><span class="right"><span class="intbl"><em>15 + 54 + 24 + 20</em><strong>55</strong></span></span></div>
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<div class="row"><span class="left">= </span><span class="right">2.05...</span></div>
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</div>
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<p class="large" align="center">The Mean is <b>2.05</b> (to 2 decimal places)</p>
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<p><i>(much easier than adding all numbers separately!)</i></p>
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</div>
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<h2>Notation</h2>
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<p>Now you know how to do it, let's do that last example again, but using formulas.</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><img src="../algebra/images/sigma.gif" alt="Sigma" height="34" width="32"></td>
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<td align="center">This symbol (called Sigma) means
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"sum up"<br>
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(read more at <a href="../algebra/sigma-notation.html">Sigma Notation</a>)</td>
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</tr>
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</tbody></table>
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<p>So we can say "add up all frequencies" this way: </p>
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<p align="center"><img src="images/sum-f.gif" alt="sum f" height="29" width="43"><i><br>
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</i>(where <span class="times"><i>f</i></span> is frequency)</p>
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<p>And we can use it like this:</p>
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<p align="center"><img src="images/sum-f-example.gif" alt="sum f" height="28" width="243"></p>
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<p>Likewise we can add up "frequency times score" this way:</p>
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<p align="center"><img src="images/sum-fx-example.gif" alt="sum fx" height="30" width="403"><br>
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(where <span class="times"><i>f</i></span> is frequency and <span class="times"><i>x</i></span> is the matching score)</p>
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<p>And the formula for calculating the mean from a frequency table is:</p>
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<div class="def">
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<p align="center"><img src="images/mean-frequency-table.gif" alt="mean of a frequency table" height="52" width="87"></p>
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<p align="center"><i>The <span class="times">x</span> with the bar on top says "the mean of <span class="times">x</span>"</i></p>
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</div>
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<p>So now we are ready to do our example above, but with correct notation.</p>
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<div class="example">
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<h3>Example: Calculate the Mean of this Frequency Table</h3>
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<table align="center" border="1">
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<tbody>
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<tr>
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<th align="center" width="100"><span class="times"><i>x</i></span></th>
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<th align="center" width="100"><span class="times"><i>f</i></span></th>
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</tr>
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<tr>
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<td align="center">1</td>
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<td align="center">15</td>
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</tr>
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<tr>
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<td align="center">2</td>
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<td align="center">27</td>
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</tr>
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<tr>
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<td align="center">3</td>
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<td align="center">8</td>
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</tr>
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<tr>
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<td align="center">4</td>
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<td align="center">5</td>
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</tr>
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</tbody></table>
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<p>And here it is:</p>
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<p class="center large"><span class="times" style="text-decoration:overline;">x</span> = <span class="intbl"><em><span style="font: 140% Georgia, Arial; ">Σ</span>fx</em><strong><span style="font: 150% Georgia, Arial;">Σ</span>f</strong></span> = <span class="intbl"><em>15×1 + 27×2 + 8×3 + 5×4</em><strong>15+27+8+5</strong></span><br>
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= <b>2.05...</b></p>
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<!-- x = SIGfx/SIGf = 15*1+27*2+8*3+5*4/15+27+8+5 = 2.05... -->
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</div>
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<p align="center">There you go! You can use sigma notation.</p>
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<h2>Calculate in the Table</h2>
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<p> It is often better to do
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the
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calculations <b>in</b> the table.</p>
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<div class="example">
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<h3>Example: (continued)</h3>
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<p>From the previous example, calculate <span class="hilite"><span class="times"><i>f × x</i></span></span> in the right-hand column and then do totals:</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<th align="center" width="100"><span class="times"><i>x</i></span></th>
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<th align="center" width="100"><span class="times"><i>f</i></span></th>
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<th align="center" width="100"><span class="times"><i>fx</i></span></th>
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</tr>
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<tr>
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<td align="center">1</td>
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<td align="center">15</td>
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<td class="hilite" align="center">15</td>
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</tr>
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<tr>
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<td align="center">2</td>
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<td align="center">27</td>
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<td class="hilite" align="center">54</td>
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</tr>
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<tr>
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<td align="center">3</td>
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<td align="center">8</td>
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<td class="hilite" align="center">24</td>
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</tr>
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<tr>
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<td align="center">4</td>
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<td align="center">5</td>
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<td class="hilite" align="center">20</td>
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</tr>
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<tr class="hilite">
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<td align="center"><b>TOTALS:</b></td>
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<td align="center"><b>55</b></td>
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<td align="center"><b>113</b></td>
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</tr>
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</tbody></table>
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<p>And the Mean is then easy:</p>
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<p class="center large">Mean = <span class="intbl"><em>113</em><strong>55</strong></span> = <b>2.05...</b></p>
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<!-- Mean = 113/55 = 2.05... -->
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</div><p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(694, 695, 1464, 1465, 1466, 1467, 3060, 3061, 3062, 3063);</script>
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</div>
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<div class="related"><a href="../mean.html">Mean</a>
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<a href="frequency-distribution.html">Frequency Distribution</a>
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<a href="index.html">Data Index</a></div>
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Copyright © 2019 MathsIsFun.com
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