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<h1 class="center">Weighted Mean</h1>
<p class="center">Also called Weighted Average</p>
<p class="center larger">A mean where some values contribute more than others.</p>
<h2>Mean</h2>
<p>When we do a simple <a href="../mean.html">mean</a> (or average), we give equal weight to each number. </p>
<p>Here is the mean of 1, 2, 3 and 4:</p>
<p class="center"><img src="images/weighted-average-equal.svg" alt="weighted average equal"></p>
<p>Add up the numbers, divide by how many numbers:</p>
<p class="center larger">Mean = &nbsp;<span class="intbl"><em>1 + 2 + 3 + 4</em><strong>4</strong></span>&nbsp; = &nbsp;<span class="intbl"><em>10</em><strong>4</strong></span>&nbsp; = 2.5</p>
<h2>Weights</h2>
<p>We could think that each of those numbers has a "weight" of <span class="hilite">¼</span> (because there are 4 numbers):</p>
<p class="center larger">Mean = <span class="hilite">¼</span> × 1 + <span class="hilite">¼</span> × 2 + <span class="hilite">¼</span> × 3 + <span class="hilite">¼</span> × 4<br>
= 0.25 + 0.5 + 0.75 + 1 = <b>2.5</b></p>
<p>Same answer.</p>
<p>&nbsp;</p>
<p>Now let's change the weight of <b>3</b> to <span class="hilite">0.7</span>, and the weights of the other numbers to <span class="hilite">0.1</span> so <b>the total of the weights is still 1</b>:</p>
<p class="center"><img src="images/weighted-average-a.svg" alt="weighted average more weight"></p>
<p class="center larger">Mean = <span class="hilite">0.1</span> × 1 + <span class="hilite">0.1</span> × 2 + <span class="hilite">0.7</span> × 3 + <span class="hilite">0.1</span> × 4<br>
= 0.1 + 0.2 + 2.1 + 0.4 = <b>2.8</b></p>
<p class="center">This <b>weighted mean</b> is now a little higher ("pulled" there by the weight of 3).</p>
<p>&nbsp;</p>
<div class="beach">
<table align="center" border="0">
<tbody>
<tr>
<td align="center">
<p>When some values get more weight than others,<br>
the central point (the mean) can change:</p></td>
<td><img src="images/weighted-average-seesaw.svg" alt="weighted average seesaw"></td>
</tr>
</tbody></table>
</div>
<h2>Decisions</h2>
<p>Weighted means can help with decisions where some things are more important than others:</p>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/camera.jpg" alt="camera" height="91" width="120"></p>
<h3>Example: Sam wants to buy a new camera, and decides on the following rating system:</h3>
<ul>
<li>Image Quality <b>50%</b></li>
<li>Battery Life <b>30%</b></li>
<li>Zoom Range <b>20%</b></li>
</ul>
<p>The Sonu camera gets 8 (out of 10) for Image Quality, 6 for Battery Life and 7 for Zoom Range</p>
<p>The Conan camera gets 9 for Image Quality, 4 for Battery Life and 6 for Zoom Range</p>
<p>Which camera is best?</p>
<p class="center larger">Sonu: 0.5 × 8 + 0.3 × 6 + 0.2 × 7 = 4 + 1.8 + 1.4 = <b>7.2</b></p>
<p class="center larger">Conan: 0.5 × 9 + 0.3 × 4 + 0.2 × 6 = 4.5 + 1.2 + 1.2 = <b>6.9</b></p>
<p>Sam decides to buy the Sonu.</p>
</div>
<p>&nbsp;</p>
<h2>What if the Weights Don't Add to 1?</h2>
<p>When the weights don't add to 1, divide by the sum of weights.</p>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/lunch.jpg" alt="lunch" height="103" width="200"></p>
<h3>Example: Alex usually eats lunch 7 times a week, but some weeks only gets 1, 2, or 5 lunches. </h3>
<p>Alex had lunch:</p>
<ul>
<li>on 2 weeks: only one lunch for the whole week</li>
<li>on 14 weeks: 2 lunches each week</li>
<li>on 8 weeks: 5 lunches each week</li>
<li>on 32 weeks: 7 lunches each week</li>
</ul>
<p>What is the <b>mean number of lunches</b> Alex has each week? </p>
<p>&nbsp;</p>
<p>Use "Weeks" as the weighting:</p>
<p class="center larger"> Weeks × Lunches = 2 × 1 + 14 × 2 + 8 × 5 + 32 × 7<br>
= 2 + 28 + 40 + 224 = <b>294</b></p>
<p>Also add up the weeks:</p>
<p class="center larger"> Weeks = 2 + 14 + 8 + 32 = <b>56</b></p>
<p>Divide total lunches by total weeks:</p>
<p class="center larger"><span class="larger">Mean = &nbsp;</span><span class="intbl"><em><span class="larger">294</span></em><strong><span class="larger">56</span></strong></span><span class="larger">&nbsp; = 5.25</span></p>
<p>It looks like this:</p>
<p class="center"><img src="images/weighted-average-b.svg" alt="weighted average 5.25"></p>
</div>
<p>But it is often better to use a table to make sure you have all the numbers correct:</p>
<div class="example">
<h3>Example (continued):</h3>
<p>Let's use:</p>
<ul>
<li><b>w</b> for the number of weeks (the weight)</li>
<li><b>x</b> for lunches (the value we want the mean of)</li>
</ul>
<p>Multiply <b>w</b> by <b>x</b>, sum up <b>w</b> and sum up <b>wx</b>:</p>
<div class="beach">
<table align="center" border="0">
<tbody>
<tr>
<th align="center">Weight<br>
w</th>
<th align="center" width="100">Lunches<br>
x</th>
<th align="right" width="100"><br>
wx</th>
</tr>
<tr>
<td align="center">2</td>
<td align="center">1</td>
<td align="right">2</td>
</tr>
<tr>
<td align="center">14</td>
<td align="center">2</td>
<td align="right">28</td>
</tr>
<tr>
<td align="center">8</td>
<td align="center">5</td>
<td align="right">40</td>
</tr>
<tr>
<td align="center">32</td>
<td align="center">7</td>
<td align="right">224</td>
</tr>
<tr>
<td align="center" height="23"><span class="center larger">Σw = 56</span></td>
<td align="center">&nbsp;</td>
<td align="right"><span class="center larger">Σwx = 294</span></td>
</tr>
</tbody></table>
</div>
<p class="center">The symbol <b><span class="large">Σ</span></b> (<a href="../algebra/sigma-notation.html">Sigma</a>) means "Sum Up"</p>
<p>Divide <b>Σwx</b> by <b>Σw</b>:</p>
<p class="center larger"><span class="larger">Mean = &nbsp;</span><span class="intbl"><em><span class="larger">294</span></em><strong><span class="larger">56</span></strong></span><span class="larger">&nbsp; = 5.25</span></p>
<p>(Same answer as before.)</p>
</div>
<p>And that leads us to our formula:</p>
<div class="def">
<p class="center larger"><span class="larger">Weighted Mean = &nbsp;</span><span class="intbl"><em><span class="center larger">Σwx</span></em><strong><span class="center larger">Σw</span></strong></span></p>
</div>
<p>In other words: multiply each weight <b>w</b> by its matching value <b>x</b>, sum that all up, and divide by the sum of weights. </p>
<h2>Summary</h2>
<ul class="larger">
<li><b>Weighted Mean</b>: A mean where some values contribute more than others.</li>
<li>When the weights add to 1: just multiply each weight by the matching value and sum it all up</li>
<li>Otherwise, multiply each weight <b>w</b> by its matching value <b>x</b>, sum that all up, and divide by the sum of weights:
<div class="def">
<p class="center larger">Weighted Mean = &nbsp;<span class="intbl"><em><span class="center">Σwx</span></em><strong><span class="center">Σw</span></strong></span></p>
</div>
</li>
</ul>
<p>&nbsp;</p>
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