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<title>Mean Deviation</title>
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<h1 class="center">Mean Deviation</h1>
<p class="center">How far, on average, all values are from the middle.</p>
<h2>Calculating It</h2>
<p> Find the mean of all values ... use it to work out distances ... then find the mean of those distances! </p>
<p>In three steps:</p>
<ul>
<li>1. Find the <a href="../mean.html">mean</a> of all values</li>
<li>2. Find the <b>distance</b> of each value from that mean (subtract the mean from each value, ignore minus signs)</li>
<li>3. Then find the <b>mean of those distances</b> </li>
</ul>
<p>Like this:</p>
<div class="example">
<h3>Example: the <span class="center">Mean Deviation</span> of 3, 6, 6, 7, 8, 11, 15, 16</h3>
<p>Step 1: Find the <b>mean</b>:</p>
<p class="center larger">Mean = &nbsp;<span class="intbl"><em>3 + 6 + 6 + 7 + 8 + 11 + 15 + 16</em><strong>8</strong></span>&nbsp; = &nbsp;<span class="intbl"><em>72</em><strong>8</strong></span>&nbsp; = 9</p>
<p>Step 2: Find the <b>distance</b> of each value from that mean:</p>
<div class="beach">
<table border="0" align="center">
<tr>
<th width="120" align="center">Value</th>
<th width="120" align="center">Distance from 9</th>
</tr>
<tr>
<td align="center">3</td>
<td align="center">6</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">7</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">8</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">11</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">15</td>
<td align="center">6</td>
</tr>
<tr>
<td align="center">16</td>
<td align="center">7</td>
</tr>
</table>
</div>
<p>Which looks like this:</p>
<p class="center"><img src="images/mean-deviation.svg" alt="mean deviation" /></p>
<p><i>(No minus signs!)</i></p>
<p>Step 3. Find the <b>mean of those distances</b>:</p>
<p class="center larger"><b>Mean Deviation</b> = &nbsp;<span class="intbl"><em>6 + 3 + 3 + 2 + 1 + 2 + 6 + 7</em><strong>8</strong></span>&nbsp; = &nbsp;<span class="intbl"><em>30</em><strong>8</strong></span>&nbsp; = <b>3.75</b></p>
<p>&nbsp;</p>
<p>So, the <b>mean = 9</b>, and the <b>mean deviation = 3.75</b></p>
</div>
<p>It tells us how far, on average, all values are from the middle.</p>
<p>In that example the values are, on average, 3.75 away from the middle.</p>
<p class="words">For <b>deviation</b> just think <b>distance</b></p>
<h2>Formula</h2>
<p>The formula is:</p>
<div class="def"><p class="center large">Mean Deviation = <span class="intbl"><em>&Sigma;|x &minus; &mu;|</em><strong>N</strong></span></p>
</div>
<ul>
<li><b>&Sigma;</b> is <a href="../algebra/sigma-notation.html">Sigma</a>, which means to sum up</li>
<li>|| (the vertical bars) mean <a href="../numbers/absolute-value.html">Absolute Value</a>, basically to ignore minus signs</li>
<li><b>x</b> is each value (such as 3 or 16)</li>
<li><b>&mu;</b> is the mean (in our example <b>&mu; = 9</b>)</li>
<li><b>N</b> is the number of values (in our example <b>N = 8</b>)</li>
</ul>
<p>Let's look at those in more detail:</p>
<h2>Absolute Deviation </h2>
<p>Each distance we calculate is called an <b>Absolute Deviation</b>, because it is the <a href="../numbers/absolute-value.html">Absolute Value</a> of the deviation (how far from the mean).</p>
<table border="0" align="center">
<tr>
<td><img src="images/absolute-deviation.svg" alt="absolute deviation" /></td>
<td><p>To show &quot;Absolute Value&quot; we put &quot;|&quot; marks either side like this:</p>
<p class="center"> <b>|-3| = 3</b></p>
<p>For any value <b>x</b>:</p>
<p class="center larger">Absolute Deviation = |x - &mu;|</p></td>
</tr>
</table>
<p>From our example, the value <b>16</b> has <b>Absolute Deviation = |x - &mu;| = |16 - 9| = |7| = 7</b></p>
<p>And now let's add them all up ...</p>
<h2>Sigma</h2>
<p>The symbol for &quot;Sum Up&quot; is <b>&Sigma;</b> (called <a href="../algebra/sigma-notation.html">Sigma Notation</a>), so we have: </p>
<p class="center larger">Sum of Absolute Deviations = &Sigma;|x - &mu;|</p>
<p>Divide by how many values <b>N</b> and we have:</p>
<div class="def">
<p class="center large">Mean Deviation = <span class="intbl">
<em>&Sigma;|x &minus; &mu;|</em>
<strong>N</strong>
</span></p>
</div>
<p>&nbsp;</p>
<p>Let's do our example again, using the proper symbols: </p>
<div class="example">
<h3>Example: the <span class="center">Mean Deviation</span> of 3, 6, 6, 7, 8, 11, 15, 16</h3>
<p>Step 1: Find the <b>mean</b>:</p>
<p class="center larger"><b>&mu;</b> = &nbsp;<span class="intbl"><em>3 + 6 + 6 + 7 + 8 + 11 + 15 + 16</em><strong>8</strong></span>&nbsp; = &nbsp;<span class="intbl"><em>72</em><strong>8</strong></span>&nbsp; = 9</p>
<p>Step 2: Find the <b>Absolute Deviations</b>:</p>
<div class="beach">
<table border="0" align="center">
<tr>
<th width="120" align="center">x</th>
<th width="120" align="center">|x - &mu;|</th>
</tr>
<tr>
<td align="center">3</td>
<td align="center">6</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">7</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">8</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">11</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">15</td>
<td align="center">6</td>
</tr>
<tr>
<td align="center">16</td>
<td align="center">7</td>
</tr>
<tr>
<td align="center">&nbsp;</td>
<td align="center"><span class="center larger">&Sigma;|x - &mu;| = 30</span></td>
</tr>
</table>
</div>
<p>&nbsp;</p>
<p>Step 3. Find the <b>Mean Deviation</b>:</p>
<p class="center larger">Mean Deviation = &nbsp;<span class="intbl"><em><span class="center larger">&Sigma;|x - &mu;|</span></em><strong>N</strong></span>&nbsp; <span class="larger">=</span> &nbsp;<span class="intbl"><em>30</em><strong>8</strong></span>&nbsp; <span class="larger">= 3.75</span></p>
</div>
<div class="words">
<p><i>Note: the mean deviation is sometimes called the Mean Absolute Deviation (MAD) because it is the mean of the absolute deviations.</i></p>
</div>
<h2>What Does It &quot;Mean&quot; ?</h2>
<p>Mean Deviation tells us how far, on average, all values are from the middle.</p>
<p>Here is an example (using the same <b>data</b> as on the <a href="standard-deviation.html">Standard Deviation</a> page):</p>
<div class="example">
<h3>Example: You and your friends have just measured the heights of your dogs
(in millimeters):</h3>
<p class="center">
<img src="images/statistics-dogs-graph.gif" width="612" height="204" alt="statistics dogs graph" /></p>
<p>The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm
and 300mm.</p>
<p>&nbsp;</p>
<p>Step 1: Find the <b>mean</b>:</p>
<p class="center larger"><b>&mu;</b> = &nbsp;<span class="intbl"><em>600 + 470 + 170 + 430 + 300</em><strong>5</strong></span>&nbsp; = &nbsp;<span class="intbl"><em>1970</em><strong>5</strong></span>&nbsp; = <b>394</b></p>
<p>Step 2: Find the <b>Absolute Deviations</b>:</p>
<div class="beach">
<table border="0" align="center">
<tr>
<th width="120" align="center">x</th>
<th width="130" align="center">|x - &mu;|</th>
</tr>
<tr>
<td align="center">600</td>
<td align="center">206</td>
</tr>
<tr>
<td align="center">470</td>
<td align="center">76</td>
</tr>
<tr>
<td align="center">170</td>
<td align="center">224</td>
</tr>
<tr>
<td align="center">430</td>
<td align="center">36</td>
</tr>
<tr>
<td align="center">300</td>
<td align="center">94</td>
</tr>
<tr>
<td align="center">&nbsp;</td>
<td align="center"><span class="center larger">&Sigma;|x - &mu;| = 636</span></td>
</tr>
</table>
</div>
<p>&nbsp;</p>
<p>Step 3. Find the <b>Mean Deviation</b>:</p>
<p class="center larger">Mean Deviation = &nbsp;<span class="intbl"><em><span class="center larger">&Sigma;|x - &mu;|</span></em><strong>N</strong></span>&nbsp; <span class="larger">=</span> &nbsp;<span class="intbl"><em>636</em><strong>5</strong></span>&nbsp; <span class="larger">= 127.2</span></p>
<p>&nbsp;</p>
<p>So, on average, the dogs' heights are <b>127.2 mm from the mean</b>.</p>
<p>(Compare that with the <a href="standard-deviation.html">Standard Deviation</a> of <b>147 mm</b>)</p>
</div>
<h2>A Useful Check</h2>
<p>The deviations on <b>one side</b> of the mean should equal the deviations on the <b>other side</b>.</p>
<p>From our first example:</p>
<div class="example">
<h3>Example: 3, 6, 6, 7, 8, 11, 15, 16</h3>
<p>The deviations are:</p>
<p class="center"><img src="images/mean-deviation.svg" alt="mean deviation" /></p>
<table border="0" align="center">
<tr>
<td width="160" align="right"><span class="center larger">6 + 3 + 3 + 2 + 1</span></td>
<td><span class="center larger">&nbsp; = &nbsp;</span></td>
<td width="160"><span class="center larger">2 + 6 + 7</span></td>
</tr>
<tr>
<td width="160" align="right"><span class="center larger">15</span></td>
<td><span class="center larger">&nbsp; = &nbsp;</span></td>
<td width="160"><span class="center larger">15</span></td>
</tr>
</table>
</div>
<p>Likewise:</p>
<div class="example">
<h3>Example: Dogs</h3>
<p>Deviations left of mean: 224 + 94 = <b>318</b></p>
<p>Deviations right of mean: 206 + 76 + 36 = <b>318</b></p>
</div>
<p>If they are not equal ... you may have made a msitake!</p>
<p>&nbsp;</p>
<div class="questions">
<script type="text/javascript">getQ(8838, 8839, 8840, 8841, 8842, 8843, 8844, 8845, 8846, 8847);</script>&nbsp;
</div>
<div class="related"> <a href="standard-deviation.html">Standard Deviation</a>
<a href="index.html">Data Index</a>
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