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451 lines
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HTML
451 lines
19 KiB
HTML
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<title>Normal Distribution</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Normal Distribution</h1>
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<p>Data can be "distributed" (spread out) in different ways.</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:center;">It can be spread out<br>
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more on the left</td>
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<td style="width:50px;"> </td>
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<td style="text-align:center;"><span class="right"><br>
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Or more on the right</span></td>
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</tr>
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<tr>
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<td style="text-align:center;"><img src="images/normal-distribution-skew-left.svg" alt="data skewed left" height="177" width="202" ></td>
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<td> </td>
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<td style="text-align:center;"><img src="images/normal-distribution-skew-right.svg" alt="data skewed right" height="177" width="202" ></td>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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<td> </td>
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<td style="text-align:center;"> </td>
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</tr>
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</tbody></table>
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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:center;">Or it can be all jumbled up</td>
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</tr>
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<tr>
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<td style="text-align:center;"><img src="images/normal-distribution-random.svg" alt="data random" height="177" width="206" ></td>
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</tr>
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</tbody></table>
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<p>But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this:</p>
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<p class="center"><img src="images/normal-distribution-1.svg" alt="bell curve" height="174" width="384" ></p>
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<p class="center">The blue curve is a Normal Distribution.<br>
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The yellow <a href="histograms.html">histogram</a> shows
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some data that<br>
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follows it closely,
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but not perfectly (which is usual).</p>
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<table class="center">
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<tbody>
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<tr>
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<td><img src="images/bell.jpg" alt="bell" height="93" width="100" ></td>
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<td>It is often called a "Bell Curve"<br>
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because it looks like a bell.</td>
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</tr>
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</tbody></table>
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<p>Many things closely follow a Normal Distribution:</p>
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<ul>
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<li>heights of people</li>
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<li>size of things produced by machines</li>
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<li>errors in measurements</li>
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<li>blood pressure</li>
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<li>marks on a test</li>
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</ul>
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<p>We say the data is "normally distributed":</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/normal-distribution-2.svg" alt="normal distribution with mean median mode at center" height="162" width="299" ></p>
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<p>The <b>Normal Distribution</b> has:</p>
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<div class="bigul">
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<ul>
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<li><a href="../mean.html">mean</a> = <a href="../median.html">median</a> = <a href="../mode.html">mode</a></li>
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<li>symmetry about the center</li>
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<li>50% of values less than the mean<br>
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and 50% greater than the mean</li>
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</ul>
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</div>
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<h2>Quincunx</h2>
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<table align="center">
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<tbody>
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<tr>
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<td>
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<p>You can see a normal distribution being created by random chance!</p>
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<p>It is called the <a href="quincunx.html">Quincunx</a> and it is an amazing machine.</p>
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<p>Have a play with it!</p></td>
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<td> </td>
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<td><a href="quincunx.html"><img src="images/quincunx.jpg" alt="quincunx" height="172" width="129" ></a></td>
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</tr>
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</tbody></table>
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<h2>Standard Deviations</h2>
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<p>The <a href="standard-deviation.html">Standard Deviation</a> is a measure of how spread
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out numbers are (read that page for details on how to calculate it).</p>
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<p>When we <a href="standard-deviation-calculator.html">calculate the standard deviation</a> we find that <b>generally</b>:</p>
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<table class="center">
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<tbody>
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<tr>
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<td><img src="images/normal-distrubution-3sds.svg" alt="normal distrubution 68%, 95%, 99.7%" height="480" width="299" ></td>
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<td>
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<p> </p>
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<p><b>68%</b> of values are within<br>
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<b>1 standard deviation</b> of the mean</p>
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<p> </p>
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<p> </p>
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<p><b>95%</b> of values are within<br>
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<b>2 standard deviations</b> of the mean</p>
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<p> </p>
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<p> </p>
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<p><b>99.7%</b> of values are within <b><br>
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3 standard deviations</b> of the mean</p></td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="example">
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<h3>Example: 95% of students at school are between <b>1.1m and 1.7m</b> tall.</h3>
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<p>Assuming this data is <b>normally distributed</b> can you calculate the mean and standard deviation?</p>
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<p>The mean is halfway between 1.1m and 1.7m:</p>
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<p class="center larger">Mean = (1.1m + 1.7m) / 2 = <b>1.4m</b></p>
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<p>95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) 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><span class="center larger">1 standard deviation</span></td>
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<td><span class="center larger">= (1.7m-1.1m) / 4</span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="center larger">= 0.6m / 4</span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="center larger">= <b>0.15m</b></span></td>
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</tr>
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</tbody></table>
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<p class="center"><span class="right">And this is the result:</span><br>
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<img src="images/normal-distribution-exb1.svg" alt="normal distribution 95%" height="142" width="260" ></p>
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</div>
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<p>It is good to know the standard deviation, because we can say that any value is:</p>
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<ul>
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<li><b>likely</b> to be within 1 standard deviation (68 out of 100 should be)</li>
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<li><b>very likely</b> to be within 2 standard deviations (95 out of 100 should be)</li>
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<li><b>almost certainly</b> within 3 standard deviations (997 out of 1000 should be)</li>
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</ul>
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<h2>Standard Scores</h2>
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<div class="words">
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<p>The number of <b>standard deviations from the mean</b> is also called the "Standard Score", "sigma" or "z-score". Get used to those words!</p>
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</div>
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<div class="example">
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<h3>Example: In that same school one of your friends is 1.85m tall</h3>
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<p style="float:right; margin: 0 0 5px 10px;"> </p>
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<span style="float:right; margin: 0 0 5px 10px;"><img src="images/normal-distribution-exb1.svg" alt="normal distribution 95%" height="142" width="260" ></span>
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<p>You can see on the bell curve that 1.85m is <b>3 standard deviations</b> from the mean of 1.4, so:</p>
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<p class="larger">Your friend's height has a "z-score" of 3.0</p>
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<div style="clear:both"></div>
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<p>It is also possible to <b>calculate</b> how many standard deviations 1.85 is from the mean</p>
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<p><i>How far is 1.85 from the mean?</i></p>
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<p class="center">It is 1.85 - 1.4 =<b> 0.45m from the mean</b></p>
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<p><i>How many standard deviations is that?</i> The standard deviation is 0.15m, so:</p>
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<p class="center">0.45m / 0.15m = <b>3 standard deviations</b></p>
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</div>
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<p>So to convert a value to a Standard Score ("z-score"):</p>
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<ul>
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<li>first subtract the mean,</li>
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<li>then divide by the Standard Deviation</li>
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</ul>
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<p>And doing that is called "Standardizing":</p>
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<p class="center"><img src="images/standardizing.svg" alt="standardizing" style="max-width:100%" height="169" width="609" ></p>
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<p>We can take any Normal Distribution and convert it to The Standard Normal Distribution.</p>
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<div class="example">
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<h3>Example: Travel Time</h3>
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<p>A survey of daily travel time had these results (in minutes):</p>
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<p class="center">26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34</p>
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<p>The <b>Mean is 38.8 minutes</b>, and the <b>Standard Deviation is 11.4 minutes</b> (you can copy and paste the values into the <a href="standard-deviation-calculator.html">Standard Deviation Calculator</a> if you want).</p>
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<p>Convert the values to z-scores ("standard scores").</p>
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<p> </p>
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<p>To convert <b>26</b>:</p>
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<div class="so"> first subtract the mean: 26 − 38.8 = −12.8, </div>
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<div class="so">then divide by the Standard Deviation: −12.8/11.4 = <span class="hilite">−1.12</span></div>
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<p>So <b>26</b> is <b>−1.12 Standard Deviations</b> from the Mean</p>
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<p> </p>
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<p>Here are the first three conversions</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:right;">
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<td style="text-align:center;">Original Value</td>
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<td style="text-align:center;">Calculation</td>
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<td style="text-align:center;">Standard Score<br>
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(z-score)</td>
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</tr>
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<tr style="text-align:right;">
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<td style="text-align:center;"><b>26</b></td>
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<td style="text-align:center;">(26-38.8) / 11.4 =</td>
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<td style="text-align:center;"><b>−1.12</b></td>
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</tr>
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<tr style="text-align:right;">
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<td style="text-align:center;"><b>33</b></td>
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<td style="text-align:center;">(33-38.8) / 11.4 =</td>
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<td style="text-align:center;"><b>−0.51</b></td>
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</tr>
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<tr style="text-align:right;">
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<td style="text-align:center;"><b>65</b></td>
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<td style="text-align:center;">(65-38.8) / 11.4 =</td>
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<td style="text-align:center;"><b>+2.30</b></td>
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</tr>
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<tr style="text-align:right;">
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<td style="text-align:center;">...</td>
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<td style="text-align:center;">...</td>
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<td style="text-align:center;">...</td>
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</tr>
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</tbody></table>
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<p> </p>
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<p>And here they are graphically:</p>
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<p class="center"><img src="images/standard-normal-distribution-scores.gif" alt="standard normal distribution scores" height="163" width="263" ></p>
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<p>You can calculate the rest of the z-scores yourself!</p>
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</div>
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<p> </p>
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<p>The <b>z-score formula</b> that we have been using is:</p>
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<div class="def">
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<p class="center large">z = <span class="intbl"><em>x − μ</em><strong>σ</strong></span></p>
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</div>
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<ul>
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<li><b>z</b> is the "z-score" (Standard Score)</li>
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<li><b>x</b> is the value to be standardized</li>
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<li><b>μ</b> ('mu") is the mean</li>
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<li><b>σ</b> ("sigma") is the standard deviation</li>
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</ul>
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<p>And this is how to use it:</p>
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<div class="example">
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<h3>Example: Travel Time (continued)</h3>
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<p>Here are the first three conversions using the "z-score formula":</p>
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<p class="center large">z = <span class="intbl"><em>x − μ</em><strong>σ</strong></span></p>
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<ul>
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<li>μ = 38.8</li>
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<li>σ = 11.4</li>
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</ul>
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<table align="center" border="1">
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<tbody>
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<tr style="text-align:right;">
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<th align="center" width="50">x</th>
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<th align="center" width="110"><span class="intbl"><em>x − μ</em><strong>σ</strong></span></th>
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<th align="center" width="80">z<br>
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(z-score)</th>
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</tr>
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<tr style="text-align:right;">
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<td style="text-align:center;">26</td>
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<td style="text-align:center;"><span class="intbl"><em>26 − 38.8</em><strong>11.4</strong></span></td>
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<td style="text-align:center;">= −1.12</td>
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</tr>
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<tr style="text-align:right;">
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<td style="text-align:center;">33</td>
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<td style="text-align:center;"><span class="intbl"><em>33 − 38.8</em><strong>11.4</strong></span></td>
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<td style="text-align:center;">= −0.51</td>
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</tr>
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<tr style="text-align:right;">
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<td style="text-align:center;">65</td>
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||
<td style="text-align:center;"><span class="intbl"><em>65 − 38.8</em><strong>11.4</strong></span></td>
|
||
<td style="text-align:center;">= +2.30</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="text-align:center;">...</td>
|
||
<td style="text-align:center;">...</td>
|
||
<td style="text-align:center;">...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>The exact calculations we did before, just following the formula.</p>
|
||
</div>
|
||
|
||
|
||
<h2>Why Standardize ... ?</h2>
|
||
|
||
<p>It can help us make decisions about our data.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Professor Willoughby is marking a test.</h3>
|
||
<p>Here are the students' results (out of 60 points):</p>
|
||
<p class="center">20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17</p>
|
||
<p>Most students didn't even get 30 out of 60, and<b> most will fail</b>.</p>
|
||
<p>The test must have been really hard, so the Prof decides to Standardize all the scores and only fail people more than 1 standard deviation below the mean.</p>
|
||
<p>The <b>Mean is 23</b>, and the <b>Standard Deviation is 6.6</b>, and these are the Standard Scores:</p>
|
||
<p class="center">-0.45, <span class="hilite">-1.21</span>, 0.45, 1.36, -0.76, 0.76, 1.82, <span class="hilite">-1.36</span>, 0.45, -0.15, -0.91</p>
|
||
<p>Now only 2 students will <span class="hilite">fail</span> (the ones lower than −1 standard deviation)</p>
|
||
<p>Much fairer!</p>
|
||
</div>
|
||
<p>It also makes life easier because we only need one table (the <a href="standard-normal-distribution-table.html">Standard Normal Distribution Table</a>), rather than doing calculations individually for each value of mean and standard deviation.</p>
|
||
|
||
|
||
<h2>In More Detail</h2>
|
||
|
||
<p>Here is the Standard Normal Distribution with percentages for every <b>half of a standard deviation</b>, and cumulative percentages:</p>
|
||
<p class="center"><img src="images/normal-distrubution-large.svg" alt="normal distrubution large bell curve" style="max-width:100%" height="329" width="650" ></p>
|
||
|
||
<div class="example">
|
||
<p>Example: Your score in a recent test was <b>0.5 standard deviations</b> above the average, how many people scored<b> lower</b> than you did?</p>
|
||
<ul>
|
||
<li>Between 0 and 0.5 is <b>19.1%</b></li>
|
||
<li>Less than 0 is <b>50%</b> (left half of the curve)</li>
|
||
</ul>
|
||
<p>So the total less than you is:</p>
|
||
<p class="center larger">50% + 19.1% = 69.1%</p>
|
||
<p>In theory <b>69.1% scored less</b> than you did (but with real data the percentage may be different)</p>
|
||
</div>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="../measure/images/measuring-1kg-2.jpg" alt="measuring 1kg " height="240" width="200" ></p>
|
||
|
||
|
||
<h2>A Practical Example: Your company packages sugar in 1 kg bags.</h2>
|
||
|
||
<p>When you weigh a sample of bags you get these results:</p>
|
||
<ul>
|
||
<li>1007g, 1032g, 1002g, 983g, 1004g, ... (a hundred measurements)</li>
|
||
<li>Mean = 1010g</li>
|
||
<li>Standard Deviation = 20g</li>
|
||
</ul>
|
||
<p><b>Some values are less than 1000g ... can you fix that?</b></p>
|
||
<p>The normal distribution of your measurements looks like this:</p>
|
||
<p class="center"><img src="images/normal-distribution-ex1.gif" alt="normal distribution ex1" height="150" width="258" ></p>
|
||
<p class="center larger">31% of the bags are less than 1000g,<br>
|
||
which is cheating the customer!</p>
|
||
<p>It is a random thing, so we can't <b>stop</b> bags having less than 1000g, but we can try to <b>reduce it</b> a lot.</p>
|
||
<p>Let's adjust the machine so that 1000g is:</p>
|
||
<ul>
|
||
<div class="bigul">
|
||
<li>at −3 standard deviations:</li>
|
||
<div class="center80">From the big bell curve above we see that <b>0.1%</b> are less. But maybe that is too small.</div>
|
||
<li>at −2.5 standard deviations:</li>
|
||
<div class="center80">Below 3 is 0.1% and between 3 and 2.5 standard deviations is 0.5%, together that is 0.1% + 0.5% = <b>0.6%</b> (a good choice I think)</div>
|
||
</div>
|
||
</ul>
|
||
<p>So let us adjust the machine to have <b>1000g at −2.5 standard deviations</b> from the mean.</p>
|
||
<p>Now, we can adjust it to:</p>
|
||
<ul>
|
||
<li>increase the amount of sugar in each bag (which changes the mean), or</li>
|
||
<li>make it more accurate (which reduces the standard deviation)</li>
|
||
</ul>
|
||
<p>Let us try both.</p>
|
||
<h4>Adjust the mean amount in each bag</h4>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/normal-distribution-ex2.gif" alt="normal distribution ex2" height="145" width="330" ></p>
|
||
<p>The standard deviation is 20g, and we need 2.5 of them:</p>
|
||
<p class="center larger">2.5 × 20g = 50g</p>
|
||
<p>So the machine should average <b>1050g</b>, like this:</p>
|
||
<p> </p>
|
||
<h4>Adjust the accuracy of the machine</h4>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/normal-distribution-ex3.gif" alt="normal distribution ex3" height="150" width="258" ></p>
|
||
<p>Or we can keep the same mean (of 1010g), but then we need 2.5 standard
|
||
deviations to be equal to 10g:</p>
|
||
<p class="center larger">10g / 2.5 = 4g</p>
|
||
<p>So the standard deviation should be <b>4g</b>, like this:</p>
|
||
<p>(We hope the machine is that accurate!)</p>
|
||
<p>Or perhaps we could have some combination of better accuracy and slightly larger average size, I will leave that up to you!</p>
|
||
|
||
|
||
<h2>More Accurate Values ...</h2>
|
||
|
||
<p>Use the <a href="standard-normal-distribution-table.html">Standard Normal Distribution Table</a> when you want more accurate values.</p>
|
||
<p> </p>
|
||
<div class="questions">2619, 2620, 2621, 2622, 2623, 2624, 2625, 2626, 3844, 3845</div>
|
||
|
||
<div class="related">
|
||
<a href="standard-deviation.html">Standard Deviation</a>
|
||
<a href="standard-deviation-calculator.html">Standard Deviation Calculator</a>
|
||
<a href="standard-normal-distribution-table.html">Standard Normal Distribution Table</a>
|
||
<a href="quincunx.html">Quincunx</a>
|
||
<a href="index.html">Probability and Statistics Index</a>
|
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