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<title>Finding a Central Value</title>
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<meta name="description" content="When you have two or more numbers it is nice to find a value for the center" />
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<article id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Finding a Central Value</h1>
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<p align="center">When you have two or more numbers it is nice to find a value for the "center".</p>
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<h2>2 Numbers</h2>
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<p>With just 2 numbers the answer is easy: go half-way between.</p>
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<div class="example">
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<h3>Example: what is the central value for 3 and 7?</h3>
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<p>Answer: Half-way between, which is 5.</p>
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<p align="center"><img src="images/mean-3-7.svg" alt="Mean of 3 and 7" width= /></p>
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<p>We can calculate it by adding 3 and 7 and then dividing the result by 2:</p>
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<div class="so">(3+7) / 2 = 10/2 = 5</div>
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</div>
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<h2>3 or More Numbers</h2>
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<p>We can use that idea of "adding then dividing" when we have 3 or more numbers: </p>
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<div class="example">
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<h3>Example: what is the central value of 3, 7 and 8?</h3>
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<p>Answer: We calculate it by adding 3, 7 and 8 and then dividing the results by 3 (because there are 3 numbers):</p>
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<div class="so">(3+7+8) / 3 = 18/3 = 6</div>
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<p align="center"><img src="images/mean-3-7-8.svg" alt="Mean of 3, 7 and 8" /></p>
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</div>
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<p>Notice that we divide by 3 because we have 3 numbers ... very important!</p>
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<h2>The Mean</h2>
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<p>So far we have been calculating the <a href="../mean.html">Mean</a> (or the Average):
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</p>
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<div class="def">
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<p align="center" class="larger">Mean: Add up the numbers and divide by how many numbers. </p>
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</div>
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<p>But sometimes the Mean can let you down:</p>
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<div class="example">
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<h3>Example: Birthday Activities</h3>
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<p>Uncle Bob wants to know the average age at the party, to choose an activity.</p>
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<p>There will be 6 kids aged 13, and also 5 babies aged 1.</p>
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<p>Add up all the ages, and divide by 11 (because there are 11 numbers):</p>
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<div class="so">(13+13+13+13+13+13+1+1+1+1+1) / 11 = <b>7.5...</b></div>
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<table border="0">
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<tr>
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<td><img src="images/bouncy-castle.jpg" width="129" height="126" alt="bouncy castle" /></td>
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<td><p>The mean age is about <b>7½</b>, so he gets a <b>Jumping Castle</b>!</p>
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<p align="center" class="larger">The 13 year olds are embarrassed, <br />
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and the 1 year olds can't jump!</p></td>
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</tr>
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</table>
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</div>
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<p>The Mean was <b>accurate</b>, but in this case it was <b>not useful</b>.</p>
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<h2>The Median</h2>
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<p>But you could also use the <a href="../median.html">Median</a>: simply list all numbers in order and choose the middle one:</p>
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<div class="example">
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<h3>Example: Birthday Activities (continued)</h3>
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<p>List the ages in order:</p>
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<p align="center" class="larger">1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13</p>
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<p>Choose the middle number:</p>
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<p align="center" class="larger">1, 1, 1, 1, 1, <span class="hilite"><b>13</b></span>, 13, 13, 13, 13, 13</p>
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<p>The Median age is <b>13</b> ... so let's have a <b>Disco</b>!</p>
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</div>
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<p>Sometimes there are <b>two</b> middle numbers. Just average those two:</p>
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<div class="example">
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<h3>Example: What is the Median of 3, 4, 7, 9, 12, 15</h3>
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<p>There are two numbers in the middle:</p>
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<p align="center" class="larger">3, 4, <span class="hilite">7, 9</span>, 12, 15</p>
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<p>So we average them:</p>
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<div class="so">(7+9) / 2 = 16/2 = 8</div>
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<p>The Median is <b>8</b></p>
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</div>
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<h2>The Mode</h2>
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<p>The <a href="../mode.html">Mode</a> is the value that occurs most often:</p>
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<div class="example">
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<h3>Example: Birthday Activities (continued)</h3>
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<p>Group the numbers so we can count them:</p>
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<p align="center" class="larger">1, 1, 1, 1, 1, <span class="hilite">13, 13, 13, 13, 13, 13</span></p>
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<p>"13" occurs 6 times, "1" occurs only 5 times, so the mode is <b>13</b>. </p>
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</div>
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<div class="center80">
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<p class="center"><i>How to remember? Think "mode is most"</i></p>
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</div>
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<p>But Mode can be tricky, there can sometimes be more than one Mode.</p>
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<div class="example">
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<h3>Example: What is the Mode of 3, 4, 4, 5, 6, 6, 7</h3>
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<p>Well ... 4 occurs twice but 6 <b>also</b> occurs twice.</p>
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<p>So <b>both 4 and 6</b> are modes.</p>
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</div>
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<p>When there are two modes it is called "bimodal", when there are three or more modes we call it "multimodal". </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/outlier.svg" alt="outlier" /></p><h2>Outliers</h2>
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<p><a href="outliers.html">Outliers</a> are values that "<b>lie</b> <b>out</b>side" the other values. </p>
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<p>They can change the mean a lot, so we can either not use them (and say so) or use the median or mode instead. </p>
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<div class="example">
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<h3>Example: 3, 4, 4, 5 and 104</h3>
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<p><b>Mean</b>: Add them up, and divide by 5 (as there are 5 numbers):</p>
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<div class="so">(3+4+4+5+104) / 5 = <b>24</b></div>
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<p>24 does not represent those numbers well at all!</p>
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<p>Without the 104 the mean is:</p>
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<div class="so">(3+4+4+5) / 4 = <b>4</b></div>
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<p>But please tell people you are not including the outlier.</p>
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<p><b>Median</b>: They are in order, so just choose the middle number, which is <b>4</b>:</p>
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<p align="center" class="larger">3, 4, <span class="hilite"><b>4</b></span>, 5, 104 </p>
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<p><b>Mode</b>: 4 occurs most often, so the Mode is <b>4</b></p>
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<p align="center" class="larger">3, <span class="hilite"><b>4, 4</b></span>, 5, 104 </p>
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</div>
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<h2>Other Means</h2>
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<p>The mean (average) we have been looking at is more correctly called the <b>Arithmetic Mean</b>. </p>
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<p>There are other types of mean! Here are two <b>examples</b>:</p>
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<div class="dotpoint">
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<p>The <b>Geometric Mean</b> multiplies the numbers together, then does a square root or cube root etc depending on how many numbers, like in this example:</p>
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<div class="example">
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<h3>Example: The <span class="center">Geometric Mean</span> of <b>2 and 18</b></h3>
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<ul>
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<li>First we multiply them: 2 × 18 = 36</li>
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<li>Then (as there are two numbers) take the square root: √36 = <b>6</b> </li>
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</ul>
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</div>
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<p>Learn more at <a href="../numbers/geometric-mean.html">Geometric Mean</a>.</p>
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</div>
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<div class="dotpoint">
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<p>The <b>Harmonic Mean</b> adds up "1 divided by number" then flips it like this:</p>
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<div class="example">
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<h3>Example: The <span class="center">Harmonic Mean</span> of <b>2, 4, 5 and 100</b></h3>
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<p>With <b>4</b> numbers we get:</p>
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<table border="0" align="center">
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<tr>
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<td align="center" style="border-bottom: 1px solid black;">4</td>
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<td width="40" rowspan="2" align="center">=</td>
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<td align="center" style="border-bottom: 1px solid black;">4</td>
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<td rowspan="2"> = <b>4.17</b> (to 2 places)</td>
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</tr>
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<tr>
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<td><span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> + <span class="intbl"><em>1</em><strong>5</strong></span> + <span class="intbl"><em>1</em><strong>100</strong></span></td>
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<td>0.96</td>
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</tr>
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</table>
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</div>
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<p>Learn more at <a href="../numbers/harmonic-mean.html">Harmonic Mean</a>.</p>
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</div>
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<h2>Conclusion</h2>
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<p><b>Mean, Median and Mode</b> are the most common ways of measuring central value, but there are other ways. </p>
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<p>Use the one that best suits your data. Or better still, use all three!</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(1876, 1877, 1878, 1879, 1880, 1881, 3050, 3051, 3052);</script>
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</div>
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<div class="related"> <a href="../mean.html">Mean</a> <a href="../median.html">Median</a> <a href="../mode.html">Mode</a> <a href="index.html"> Data Index</a> </div>
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