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326 lines
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HTML
326 lines
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<title>First Digits Rule! (Benford's Law)</title>
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<h1 align="center">First Digits Rule!<br>
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(Benford's Law)
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</h1>
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<p class="center larger">Don't cheat with numbers,
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they can give you away.
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<br>
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So says <i>Benford's Law.</i><br>
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<img src="images/number-smile.gif" width="177" height="98" alt="numbers smile" /></p>
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<h2>First Digits</h2>
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<p>How often would you expect a <b>"1"</b> to be the first digit in a set of numbers?</p>
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<div class="example">
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<h3>Example: you are looking at a list of expenses, with numbers like:</h3>
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<ul>
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<li>$65.20 (first digit is <b>6</b>)</li>
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<li>$35.00 (first digit is <b>3</b>)</li>
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<li>$7.50 (first digit is <b>7</b>)</li>
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<li>$12.50 (first digit is <b>1</b>)</li>
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</ul>
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<p>Would there be as many <span class="larger"><b>1</b></span>'s as <span class="larger"><b>2</b></span>'s for the first digit?</p>
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</div>
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<p>Well <span class="larger"><b>1</b></span> is just a number like <span class="larger"><b>2</b></span> to <span class="larger"><b>9</b></span>, right? </p>
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<p>So it seems like it <b>should</b> be the first digit <b>1 out of 9 times (about 11%)</b>:</p>
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<table border="0" align="center">
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<tr align="center">
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<td width="45">1</td>
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<td width="45">2</td>
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<td width="45">3</td>
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<td width="45">4</td>
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<td width="45">5</td>
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<td width="45">6</td>
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<td width="45">7</td>
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<td width="45">8</td>
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<td width="45">9</td>
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</tr>
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<tr align="center">
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<td width="45">11%</td>
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<td width="45">11%</td>
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<td width="45">11%</td>
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<td width="45">11%</td>
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<td width="45">11%</td>
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<td width="45">11%</td>
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<td width="45">11%</td>
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<td width="45">11%</td>
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<td width="45">11%</td>
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</tr>
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</table>
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<p class="large"><b>But no!</b> </p>
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<p>A man called Dr. Frank Benford discovered that in many cases, the number <span class="larger"><b>1</b></span> is the first digit <b>about 30% of the time</b>. </p>
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<p>And the poor old number <span class="larger"><b>9</b></span> is the first digit <b>only 5%</b> of the time.</p>
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<div class="fun">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../algebra/images/logarithm-book.jpg" width="91" height="90" alt="logarithm book" /></p>
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<p>The story is that a man called Simon Newcomb noticed a book of <a href="../algebra/logarithms.html">logarithms</a> was <b>very worn at the start</b> but not at the end. </p>
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<p><i>"Why are people more interested in 1's and 2's than 8's and 9's?"</i> </p>
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<p>He decided to investigate! (Would you investigate something odd?)</p>
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</div> <p>Dr. Benford found this amazing thing also happened with baseball statistics, areas of rivers, population sizes, street addresses and many more cases.</p>
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<h2>Why is this?</h2>
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<p>Well, let's think about street addresses:</p>
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<div class="example">
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<h3>What are the first digits of house numbers?</h3>
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<ul>
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<li>some streets are short: 1,2,3,4,5,6 </li>
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<li>some streets are longer: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 (notice how many, have 1 as the first digit?).</li>
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<li>other streets are a bit longer, with numbers from 1 to 30 (many "1"s and "2"s)</li>
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<li>And when the streets are very long we have lots of them starting at 100.</li>
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</ul>
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<p>The result is that numbers starting with 1 are more common, 2 is also fairly common and 9 least of all.</p>
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</div>
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<div class="example">
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<h3> Example: Stock Prices</h3>
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<p>Let us say a price starts at 1.00, and goes up 10% each time:</p>
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<table border="0" align="center">
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<tr align="center">
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<th width="100">Price</th>
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<th width="100">First Digit</th>
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</tr>
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<tr align="center"> <td>1.00</td> <td>1</td> </tr>
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<tr align="center"> <td>1.10</td> <td>1</td> </tr>
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<tr align="center"> <td>1.21</td> <td>1</td> </tr>
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<tr align="center"> <td>1.33</td> <td>1</td> </tr>
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<tr align="center"> <td>1.46</td> <td>1</td> </tr>
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<tr align="center"> <td>1.61</td> <td>1</td> </tr>
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<tr align="center"> <td>1.77</td> <td>1</td> </tr>
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<tr align="center"> <td>1.95</td> <td>1</td> </tr>
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<tr align="center"> <td>2.14</td> <td>2</td> </tr>
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<tr align="center"> <td>2.36</td> <td>2</td> </tr>
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<tr align="center"> <td>2.59</td> <td>2</td> </tr>
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<tr align="center"> <td>2.85</td> <td>2</td> </tr>
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<tr align="center"> <td>3.14</td> <td>3</td> </tr>
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<tr align="center"> <td>3.45</td> <td>3</td> </tr>
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<tr align="center"> <td>3.80</td> <td>3</td> </tr>
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<tr align="center"> <td>4.18</td> <td>4</td> </tr>
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<tr align="center"> <td>4.59</td> <td>4</td> </tr>
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<tr align="center"> <td>5.05</td> <td>5</td> </tr>
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<tr align="center"> <td>5.56</td> <td>5</td> </tr>
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<tr align="center"> <td>6.12</td> <td>6</td> </tr>
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<tr align="center"> <td>6.73</td> <td>6</td> </tr>
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<tr align="center"> <td>7.40</td> <td>7</td> </tr>
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<tr align="center"> <td>8.14</td> <td>8</td> </tr>
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<tr align="center"> <td>8.95</td> <td>8</td> </tr>
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<tr align="center"> <td>9.85</td> <td>9</td> </tr>
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</table>
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<p>Lots of <b>1</b>'s, quite a few <b>2</b>'s, less <b>3</b>'s, etc</p>
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</div>
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<h2>The Result</h2>
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<p>In fact Benford figured that the probability of a first digit being <b>d</b> is:</p>
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<p class="center large">P(d) = log<sub>10</sub>(1 + 1/d)</p>
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<div class="example">
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<h3>Example: the probability of a first digit of 2:</h3>
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<div class="tbl">
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<div class="row"><span class="left"><span class="center larger">P(2) </span></span><span class="right"><span class="center larger">= log<sub>10</sub>(1 + 1/2)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center larger">= log<sub>10</sub>(1.5)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center larger">= 0.17609...</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="center larger">= 17.6% (rounded)</span></span></div>
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</div>
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</div>
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<p>And these are the probabilities:</p>
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<table border="0" align="center">
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<tr align="center">
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<td width="55">1</td>
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<td width="55">2</td>
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<td width="55">3</td>
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<td width="55">4</td>
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<td width="55">5</td>
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<td width="55">6</td>
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<td width="55">7</td>
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<td width="55">8</td>
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<td width="55">9</td>
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</tr>
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<tr align="center">
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<td width="55">30.1%</td>
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<td width="55">17.6%</td>
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<td width="55">12.5%</td>
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<td width="55">9.7%</td>
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<td width="55">7.9%</td>
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<td width="55">6.7%</td>
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<td width="55">5.8%</td>
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<td width="55">5.1%</td>
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<td width="55">4.6%</td>
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</tr>
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</table>
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<div class="example">
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<h3>Example: Sam went through a list of 100 work expenses for the year.</h3>
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<p>There was $1.95 for a pen, $4.95 for a marker, etc. Here are the counts of the <b>first digits</b>:</p>
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<table border="0" align="center">
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<tr>
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<td align="right">First Digit:</td>
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<td width="35" align="right">1</td>
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<td width="35" align="right">2</td>
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<td width="35" align="right">3</td>
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<td width="35" align="right">4</td>
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<td width="35" align="right">5</td>
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<td width="35" align="right">6</td>
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<td width="35" align="right">7</td>
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<td width="35" align="right">8</td>
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<td width="35" align="right">9</td>
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</tr>
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<tr>
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<td align="right">Count:</td>
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<td width="35" align="right">26</td>
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<td width="35" align="right">19</td>
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<td width="35" align="right">10</td>
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<td width="35" align="right">11</td>
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<td width="35" align="right">9</td>
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<td width="35" align="right">15</td>
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<td width="35" align="right">2</td>
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<td width="35" align="right">5</td>
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<td width="35" align="right">4</td>
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</tr>
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</table>
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<p>It follows Benford's Law pretty well. </p>
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<p>Except there are a lot of "6"s, because printer paper costs $6 and they buy a lot of it.</p>
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</div>
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<h2>Lotteries</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../data/images/lottery-ticket.jpg" width="200" height="82" alt="lottery ticket" /></p>
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<p><a href="../data/lottery.html">Lottery</a> numbers <b>don't</b> follow this rule, because they aren't the size or amount of anything, they are really just symbols (and a lottery would work as well using letters or pictures).</p>
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<h2>Finding Cheaters</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/number-surprise.gif" width="80" height="118" alt="number surprise" /></p> <p>When people try to fake numbers they often choose the first digit randomly and end up with as many "9"s as "1"s.</p>
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<p>But a computer program can go through all the numbers and count first digits to see how often a "1" appears compared to a "5" or "9". If it looks suspicious ... watch out!</p>
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<p>This can help uncover tax cheats, election rigging and more.</p>
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<p> </p>
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<div class="fun">
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<h3>Your Turn</h3>
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<p>Gather a list of 100 numbers from a category of your choosing. Make sure the numbers count or measure something (and aren't just symbols). </p>
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<p>Here are some suggestions:</p>
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<ul>
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<li>House Numbers</li>
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<li>City populations</li>
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<li>Supermarket prices</li>
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<li>Used car prices</li>
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</ul>
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<p>Find their first digits and complete this table:</p>
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<table border="2" align="center">
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<tr>
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<td height="40" align="right"><b>First Digit:</b></td>
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<td width="35" align="right"><b>1</b></td>
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<td width="35" align="right"><b>2</b></td>
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<td width="35" align="right"><b>3</b></td>
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<td width="35" align="right"><b>4</b></td>
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<td width="35" align="right"><b>5</b></td>
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<td width="35" align="right"><b>6</b></td>
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<td width="35" align="right"><b>7</b></td>
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<td width="35" align="right"><b>8</b></td>
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<td width="35" align="right"><b>9</b></td>
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</tr>
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<tr>
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<td height="40" align="right"><b>Count:</b></td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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</tr>
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</table>
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<p>What did you find?</p>
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<p> </p>
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<h3>Bonus Activity</h3>
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<p>Get some friends to make up pretend shopping lists with how much each item costs. Find the first digits and put them in a table:</p>
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<table border="2" align="center">
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<tr>
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<td height="40" align="right"><b>First Digit:</b></td>
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<td width="35" align="right"><b>1</b></td>
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<td width="35" align="right"><b>2</b></td>
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<td width="35" align="right"><b>3</b></td>
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<td width="35" align="right"><b>4</b></td>
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<td width="35" align="right"><b>5</b></td>
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<td width="35" align="right"><b>6</b></td>
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<td width="35" align="right"><b>7</b></td>
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<td width="35" align="right"><b>8</b></td>
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<td width="35" align="right"><b>9</b></td>
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</tr>
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|
<tr>
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<td height="40" align="right"><b>Count:</b></td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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|
<td width="35" align="right"> </td>
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<td width="35" align="right"> </td>
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</tr>
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</table>
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<p>What did you find?</p>
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</div>
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<p> </p>
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