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340 lines
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<h1 class="center">Standard Deviation and Variance</h1>
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<p class="center"><i>Deviation just means how far from the normal</i></p>
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<h2><b><a name="Top" id="Top"></a></b>Standard Deviation</h2>
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<p>The Standard Deviation is a measure of how spread
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out numbers are.</p>
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<p>Its symbol is <b class="larger">σ</b> (the greek letter sigma)</p>
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<p>The formula is easy: it is the<b> square root</b> of the <b>Variance.</b> So now you ask, "What is the Variance?"</p>
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<h2>Variance</h2>
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<p>The Variance is defined as:</p>
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<div class="def">
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<p class="center larger">The average of the <b>squared</b> differences from the Mean.</p>
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</div>
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<p>To calculate the variance follow these steps:</p>
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<ul>
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<div class="bigul">
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<li>Work out the <a href="../mean.html">Mean</a> (the simple average
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of the numbers)</li>
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<li>Then for each number: subtract the Mean and square the result
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(the <i>squared difference</i>).</li>
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<li>Then work out the average of those squared differences. (<a href="#WhySquare">Why
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Square?</a>)</li>
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</div>
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</ul>
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<h2>Example</h2>
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<p>You and your friends have just measured the heights of your dogs
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(in millimeters):</p>
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<p class="center"><img src="images/statistics-dogs-graph.gif" alt="dogs on graph shoulder heights" style="max-width:100%" height="204" width="612"></p>
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<p>The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm
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and 300mm.</p>
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<p>Find out the Mean, the Variance, and the Standard Deviation.</p>
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<p>Your first step is to find the Mean:</p>
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<h3>Answer:</h3>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><span class="larger">Mean</span></td>
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<td style="text-align:center; width:40px;"><span class="larger">=</span></td>
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<td class="larger"><span class="intbl"><em>600 + 470 + 170 + 430 + 300</em><strong>5</strong></span></td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center; width:40px;"><span class="larger">=</span></td>
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<td class="larger"><span class="intbl"><em>1970</em><strong>5</strong></span></td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center; width:40px;"><span class="larger">=</span></td>
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<td class="larger">394</td>
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</tr>
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</tbody></table>
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<p>so the mean (average) height is 394 mm. Let's plot this on the chart:</p>
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<p class="center"><img src="images/statistics-dogs-mean.gif" alt="dogs on graph: mean" style="max-width:100%" height="207" width="619"></p>
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<p>Now we calculate each dog's difference from the Mean:</p>
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<p class="center"><img src="images/statistics-dogs-deviation.gif" alt="dogs on graph: deviation" style="max-width:100%" height="165" width="618"></p>
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<p>To calculate the Variance, take each difference, square it, and
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then average the result:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td colspan="3" class="larger" nowrap="nowrap"><b>Variance</b></td>
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</tr>
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<tr>
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<td class="larger" nowrap="nowrap">σ<sup>2</sup></td>
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<td class="larger" align="center" width="40">=</td>
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<td class="larger"><span class="intbl"><em>206<sup>2</sup> + 76<sup>2</sup> + (−224)<sup>2</sup> + 36<sup>2</sup> + (−94)<sup>2</sup></em><strong>5</strong></span></td>
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</tr>
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<tr>
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<td class="larger"> </td>
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<td class="larger" align="center">=</td>
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<td class="larger"><span class="intbl"><em>42436 + 5776 + 50176 + 1296 + 8836</em><strong>5</strong></span></td>
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</tr>
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<tr>
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<td class="larger"> </td>
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<td class="larger" align="center">=</td>
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<td class="larger"><span class="intbl"><em>108520</em><strong>5</strong></span></td>
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</tr>
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<tr>
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<td class="larger"> </td>
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<td class="larger" align="center">=</td>
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<td class="larger">21704</td>
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</tr>
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</tbody></table>
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<p>So the Variance is <b>21,704</b></p>
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<p>And the Standard Deviation is just the square root of Variance,
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so:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td colspan="3"><span class="center larger"><b>Standard Deviation</b></span></td>
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</tr>
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<tr>
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<td><span class="center larger">σ </span></td>
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<td class="larger" align="center" width="40">=</td>
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<td><span class="center larger">√21704 </span></td>
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</tr>
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<tr>
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<td> </td>
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<td class="larger" align="center">=</td>
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<td><span class="center larger"> 147.32...</span></td>
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</tr>
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<tr>
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<td> </td>
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<td class="larger" align="center">=</td>
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<td><span class="center larger"><b>147</b> <i>(to the nearest mm)</i></span></td>
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</tr>
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</tbody></table>
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<p> </p>
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<p>And the good thing about the Standard Deviation is that it is useful.
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Now we can show which heights are within one Standard Deviation
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(147mm) of the Mean:</p>
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<p class="center"><img src="images/statistics-standard-deviation.gif" alt="dogs on graph: standard deviation" style="max-width:100%" height="208" width="621"></p>
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<p>So, using the Standard Deviation we have a "standard"
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way of knowing what is normal, and what is extra large or extra
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small.</p>
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<p>Rottweilers <b>are</b> tall dogs. And Dachshunds <b>are</b> a bit
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short, right?</p>
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<h2>Using</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/normal-distrubution-1sd.svg" alt="normal distrubution 1 sd = 68%" height="144" width="300"></p>
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<p>We can expect about 68% of values to be within plus-or-minus
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1 standard deviation.</p>
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<p>Read <a href="standard-normal-distribution.html">Standard Normal Distribution</a> to learn more.</p>
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<p>Also try the <a href="standard-deviation-calculator.html">Standard Deviation Calculator</a>.</p>
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<h2>But ... there is a small change with <b>Sample</b> Data</h2>
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<p>Our example has been for a <b>Population</b> (the 5 dogs are the only dogs we are interested in).</p>
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<p>But if the data is a <b>Sample</b> (a selection taken from a bigger Population), then the calculation changes!</p>
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<div class="def">
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<p class="larger">When you have "N" data values that are:</p>
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<div class="bigul">
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<ul>
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<li><b>The Population</b>: divide by <b>N</b> when calculating Variance (like we did)</li>
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<li><b>A Sample</b>: divide by <b>N-1</b> when calculating Variance</li>
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</ul>
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</div>
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</div>
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<p><span class="center">All other calculations stay the same, including how we calculated the mean.</span></p>
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<div class="example">
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<p>Example: if our 5 dogs are just a <b>sample</b> of a bigger population of dogs, we divide by <b>4 instead of 5</b> like this:</p>
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<div class="so">Sample Variance = 108,520 / <span class="hilite"><b>4</b></span> = <b>27,130</b></div>
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<div class="so">Sample Standard Deviation = √27,130 = <b>165</b> (to the nearest mm) </div>
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</div>
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<p>Think of it as a "correction" when your data is only a sample.</p>
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<h2>Formulas</h2>
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<p>Here are the two formulas, explained at <a href="standard-deviation-formulas.html">Standard Deviation Formulas</a> if you want to know more:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>The "<b>Population</b> Standard Deviation":</p></td>
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<td style="text-align:right;"> </td>
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<td><img src="images/standard-deviation-formula.svg" alt="square root of [ (1/N) times Sigma i=1 to N of (xi - mu)^2 ]" height="59" width="200"></td>
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</tr>
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<tr>
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<td style="text-align:right;">The "<b>Sample</b> Standard Deviation<b>":</b></td>
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<td style="text-align:right;"> </td>
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<td><img src="images/standard-deviation-sample.svg" alt="square root of [ (1/(N-1)) times Sigma i=1 to N of (xi - xbar)^2 ]" height="58" width="210"></td>
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</tr>
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</tbody></table>
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<p class="center">Looks complicated, but the important change is to<br>
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divide by <b>N-1</b> (instead of <b>N</b>) when calculating a Sample Standard Deviation.</p>
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<p class="center larger"> </p>
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<p> </p>
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<div class="center80">
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<h3><b><a name="WhySquare" id="WhySquare"></a>*Footnote: Why <i>square</i> the differences?</b></h3>
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<p>If we just add up the differences from the mean ... the negatives cancel the positives:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/standard-deviation-why-a.gif" alt="standard deviation why a" height="92" width="118"></td>
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<td> </td>
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<td><span class="intbl"><em>4 + 4 − 4 − 4</em><strong>4</strong></span> = <span class="hilite">0</span></td>
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</tr>
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</tbody></table>
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<p>So that won't work. How about we use <a href="../numbers/absolute-value.html">absolute values</a>?</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/standard-deviation-why-a.gif" alt="standard deviation why a" height="92" width="118"></td>
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<td> </td>
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<td><span class="intbl">
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<em>|4| + |4| + |−4| + |−4|</em><strong>4</strong></span> = <span class="intbl">
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<em>4 + 4 + 4 + 4</em>
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<strong>4</strong>
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</span> = <span class="hilite">4</span></td>
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</tr>
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</tbody></table>
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<p>That looks good (and is the <a href="mean-deviation.html">Mean Deviation</a>), but what about this case:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/standard-deviation-why-b.gif" alt="standard deviation why b" height="125" width="118"></td>
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<td> </td>
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<td><span class="intbl">
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<em>|7| + |1| + |−6| + |−2|</em><strong>4</strong></span> = <span class="intbl">
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<em>7 + 1 + 6 + 2</em>
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<strong>4</strong>
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</span> = <span class="hilite">4</span></td>
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</tr>
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</tbody></table>
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<p>Oh No! It also gives a value of 4,
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Even though the differences are more spread out.</p>
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<p>So let us try squaring each difference (and taking the square root at the end):</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/standard-deviation-why-a.gif" alt="standard deviation why a" height="92" width="118"></td>
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<td> </td>
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<td>
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<span class="lg">√(</span><span class="intbl">
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<em>4<sup>2</sup> + 4<sup>2</sup> + (-4)<sup>2</sup> + (-4)<sup>2</sup></em><strong>4</strong></span><span class="lg">)</span> = <span class="lg">√(</span><span class="intbl">
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<em>64</em>
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<strong>4</strong>
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</span><span class="lg">)</span> = <span class="hilite">4</span></td>
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</tr>
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<tr>
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<td><img src="images/standard-deviation-why-b.gif" alt="standard deviation why b" height="125" width="118"></td>
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<td> </td>
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<td><span class="lg">√(</span><span class="intbl">
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<em>7<sup>2</sup> + 1<sup>2</sup> + (-6)<sup>2</sup> + (-2)<sup>2</sup></em>
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<strong>4</strong></span><span class="lg">)</span> = <span class="lg">√(</span><span class="intbl">
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<em>90</em>
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<strong>4</strong>
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</span><span class="lg">)</span> = <span class="hilite">4.74...</span></td>
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</tr>
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</tbody></table>
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<p>That is nice! The Standard Deviation is bigger when the differences are more spread out ... just what we want.</p>
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<p>In fact this method is a similar idea to <a href="../algebra/distance-2-points.html">distance between points</a>, just applied in a different way.</p>
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<p>And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics.</p>
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