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415 lines
16 KiB
HTML
415 lines
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<title>Chi-Square Test</title>
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<meta name="description" content="The Chi-Square Test gives a way to help you decide if something is just random chance or not." />
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 class="center">Chi-Square Test</h1>
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<div style="float:right; margin: 0 0 5px 10px; background-color: #def;">
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<table border="0">
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<tr>
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<td align="center"><img src="images/chi-square-group-a.jpg" width="300" height="218" alt="chi square groups" /></td>
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</tr>
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<tr>
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<td align="center">Single: <span class="large">47</span> Married: <span class="large">71</span> Divorced: <span class="large">35</span></td>
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</tr>
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<tr>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center"><img src="images/chi-square-group-b.jpg" width="300" height="207" alt="chi square groups" /></td>
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</tr>
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<tr>
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<td align="center">Single: <span class="large">44</span> Married: <span class="large">85</span> Divorced: <span class="large">40</span></td>
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</tr>
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</table>
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</div>
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<h3> </h3>
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<h3>Groups and Numbers</h3>
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<p>You research two groups and put them in categories of single, married or divorced: </p>
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<p> </p>
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<p>The numbers are definitely different, but ... </p>
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<ul>
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<li>Is that just random chance?</li>
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<li>Or have you found something interesting? </li>
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</ul>
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<p> </p>
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<p class="larger">The <b>Chi-Square Test</b> gives a "p" value to help you decide! </p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: "Which holiday do you prefer?"</h3>
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<table border="0" align="center">
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<tr>
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<td width="90"> </td>
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<td width="90">Beach</td>
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<td width="90">Cruise</td>
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</tr>
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<tr>
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<td>Men</td>
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<td>209</td>
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<td>280</td>
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</tr>
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<tr>
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<td>Women</td>
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<td>225</td>
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<td>248</td>
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</tr>
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</table>
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<h3>Does Gender affect Preferred Holiday? </h3>
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<p>If Gender (Man or Woman) <b>does</b> affect Preferred Holiday we say they are <b>dependent</b>.</p>
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<p>By doing some special calculations (explained later), we come up with a "p" value:</p>
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<p class="center large">p value is 0.132</p>
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<p>Now, <b>p < 0.05</b> is the usual test for <b>dependence</b>. </p>
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<p>In this case <b>p is greater than 0.05</b>, so we believe the variables are <b>independent</b> (ie not linked together).</p>
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<p>In other words Men and Women probably do <b>not</b> have a different preference for Beach Holidays or Cruises.</p>
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<p>It was just random differences which we expect when collecting data.</p>
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</div>
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<h2>Understanding "p" Value</h2>
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<p> "p" is the <a href="probability.html">probability</a> the variables are <b>independent</b>.</p>
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<div class="center80">
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<p>Imagine that the previous example was in fact two random samples of <b>Men</b> each time:</p>
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<table border="0" align="center">
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<tr>
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<td width="180" align="center"><img src="images/chi-square1.gif" width="140" height="119" alt="chi square group 1" /></td>
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<td width="180" align="center"><img src="images/chi-square2.gif" width="140" height="119" alt="chi square group 2" /></td>
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</tr>
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<tr>
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<td align="center"><b>Men:</b><br>
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Beach 209, Cruise 280 </td>
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<td align="center"><b>Men:</b><br>
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Beach 225, Cruise 248 </td>
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</tr>
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</table>
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<p>Is it <b>likely</b> you would get such different results surveying Men each time?</p>
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</div>
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<p>Well the "p" value of <b>0.132</b> says that it really could happen every so often.</p>
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<p><a href="survey-conducting.html">Surveys</a> are random after all. We expect slightly different results each time, right?</p>
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<p>So most people want to see a <b>p</b> value less than <b>0.05</b> before they are happy to say the results show the groups have a different response. </p>
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<p>Let's see another example:</p>
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<div class="example">
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<h3>Example: "Which pet do you prefer?"</h3>
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<table border="0" align="center">
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<tr>
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<td width="90"> </td>
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<td width="90">Cat</td>
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<td width="90">Dog</td>
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</tr>
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<tr>
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<td>Men</td>
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<td>207</td>
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<td>282</td>
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</tr>
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<tr>
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<td>Women</td>
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<td>231</td>
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<td>242</td>
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</tr>
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</table>
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<p>By doing the calculations (shown later), we come up with:</p>
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<p class="center large">P value is 0.043</p>
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<p>In this case <b>p < 0.05</b>, so this result is thought of as being "significant" meaning we think the variables are <b>not</b> independent. </p>
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<p>In other words, because <b>0.043 < 0.05</b> we think that Gender is linked to Pet Preference (Men and Women have different preferences for Cats and Dogs).</p>
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</div>
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<p><i>Just out of interest, notice that the numbers in our two examples are similar, but the resulting p-values are very different: <b>0.132</b> and <b>0.043</b>. This shows how sensitive the test is!</i></p>
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<h2>Why p<0.05 ?</h2>
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<p>It is just a choice! <b>Using p<0.05 is common</b>, but we could have chosen p<0.01 to be even more sure that the groups behave differently, or any value really.</p>
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<h2>Calculating P-Value</h2>
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<p>So how do we calculate this p-value? We use the Chi-Square Test!</p>
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<h2>Chi-Square Test</h2>
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<div class="words">
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<p>Note: <b>Chi</b> Sounds like "Hi" but with a <b>K</b>, so it sounds like "<b>Ki</b> square"</p>
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<p>And Chi is the greek letter <span style="font: bold italic 20px 'Times New Roman', Times, serif;">Χ</span>, so we can also write it <span style="font: bold italic 20px 'Times New Roman', Times, serif;">Χ<sup>2</sup></span></p>
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</div>
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<p>Important points before we get started:</p>
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<ul>
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<li>This test only works for <b>categorical</b> data (data in categories), such as Gender {Men, Women} or color {Red, Yellow, Green, Blue} etc, but <b>not numerical</b> data such as height or weight.</li>
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<li>The numbers must be large enough. Each entry must be <b>5</b> or more. In our example we have values such as 209, 282, etc, so we are good to go.</li>
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</ul>
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<h3>Our first step is to state our <b>hypotheses</b>:</h3>
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<div class="words">
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<p><b>Hypothesis</b>: A statement that might be true, which can then be tested.</p>
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</div>
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<p>The two <b>hypotheses</b> are.</p>
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<ul>
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<li>Gender and preference for cats or dogs are <b>independent</b>. </li>
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<li>Gender and preference for cats or dogs are <b>not independent</b>.</li>
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</ul>
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<h3>Lay the data out in a table:</h3>
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<table border="0" align="center">
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<tr>
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<td width="90"> </td>
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<td width="90" align="center">Cat</td>
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<td width="90" align="center">Dog</td>
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</tr>
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<tr>
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<td>Men</td>
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<td align="center">207</td>
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<td align="center">282</td>
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</tr>
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<tr>
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<td>Women</td>
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<td align="center">231</td>
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<td align="center">242</td>
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</tr>
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</table>
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<h3>Add up rows and columns:</h3>
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<table border="0" align="center">
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<tr>
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<td width="90"> </td>
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<td width="90" align="center">Cat</td>
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<td width="90" align="center">Dog</td>
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<td width="90" align="center"> </td>
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</tr>
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<tr>
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<td>Men</td>
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<td align="center">207</td>
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<td align="center">282</td>
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<td align="center" class="hilite">489</td>
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</tr>
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<tr>
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<td>Women</td>
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<td align="center">231</td>
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<td align="center">242</td>
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<td align="center" class="hilite">473</td>
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</tr>
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<tr>
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<td> </td>
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<td align="center" class="hilite">438</td>
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<td align="center" class="hilite">524</td>
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<td align="center" class="hilite">962</td>
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</tr>
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</table>
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<h3>Calculate "Expected Value" for each entry: </h3>
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<p>Multiply each row total by each column total and divide by the overall total:</p>
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<table border="0" align="center">
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<tr>
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<td width="90"> </td>
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<td width="120" align="center">Cat</td>
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<td width="120" align="center">Dog</td>
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<td width="90" align="center"> </td>
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</tr>
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<tr>
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<td>Men</td>
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<td align="center" class="hilite"><span class="intbl"><em>489×438</em><strong>962</strong></span><!-- 489*438/962 --></td>
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<td align="center" class="hilite"><span class="intbl"><em>489×524</em><strong>962</strong></span><!-- 489*524/962 --></td>
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<td align="center">489</td>
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</tr>
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<tr>
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<td>Women</td>
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<td align="center" class="hilite"><span class="intbl"><em>473×438</em><strong>962</strong></span><!-- 473*438/962 --></td>
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<td align="center" class="hilite"><span class="intbl"><em>473×524</em><strong>962</strong></span><!-- 473*524/962 --></td>
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<td align="center">473</td>
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</tr>
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<tr>
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<td> </td>
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<td align="center">438</td>
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<td align="center">524</td>
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<td align="center">962</td>
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</tr>
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</table>
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<p>Which gives us:</p>
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<table border="0" align="center">
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<tr>
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<td width="90"> </td>
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<td width="120" align="center">Cat</td>
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<td width="120" align="center">Dog</td>
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<td width="90" align="center"> </td>
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</tr>
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<tr>
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<td>Men</td>
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<td align="center" class="hilite">222.64</td>
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<td align="center" class="hilite">266.36</td>
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<td align="center">489</td>
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</tr>
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<tr>
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<td>Women</td>
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<td align="center" class="hilite">215.36</td>
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<td align="center" class="hilite">257.64</td>
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<td align="center">473</td>
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</tr>
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<tr>
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<td> </td>
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<td align="center">438</td>
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<td align="center">524</td>
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<td align="center">962</td>
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</tr>
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</table>
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<h3>Subtract expected from observed, square it, then divide by expected:</h3>
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<p>In other words, use formula <span class="intbl"><em>(O−E)<sup>2</sup></em><strong>E</strong></span> where </p>
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<ul>
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<li>O = <b>Observed</b> (actual) value</li>
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<li>E = <b>Expected</b> value </li>
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</ul>
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<table border="0" align="center">
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<tr>
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<td width="90"> </td>
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<td width="120" align="center">Cat</td>
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<td width="120" align="center">Dog</td>
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<td width="90" align="center"> </td>
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</tr>
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<tr>
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<td>Men</td>
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<td align="center" class="hilite"><span class="intbl">
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<em>(207−222.64)<sup>2</sup></em>
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<strong>222.64</strong>
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</span></td>
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<td align="center" class="hilite"><span class="intbl">
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<em>(282−266.36)<sup>2</sup></em>
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<strong>266.36</strong>
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</span></td>
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<td align="center">489</td>
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</tr>
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<tr>
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<td>Women</td>
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<td align="center" class="hilite"><span class="intbl">
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<em>(231−215.36)<sup>2</sup></em>
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<strong>215.36</strong>
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</span></td>
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<td align="center" class="hilite"><span class="intbl">
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<em>(242−257.64)<sup>2</sup></em>
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<strong>257.64</strong>
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</span></td>
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<td align="center">473</td>
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</tr>
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<tr>
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<td> </td>
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<td align="center">438</td>
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<td align="center">524</td>
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<td align="center">962</td>
|
||
</tr>
|
||
</table>
|
||
<p>Which gets us:</p>
|
||
<table border="0" align="center">
|
||
<tr>
|
||
<td width="90"> </td>
|
||
<td width="120" align="center">Cat</td>
|
||
<td width="120" align="center">Dog</td>
|
||
<td width="90" align="center"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td>Men</td>
|
||
<td align="center" class="hilite">1.099</td>
|
||
<td align="center" class="hilite">0.918</td>
|
||
<td align="center">489</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Women</td>
|
||
<td align="center" class="hilite">1.136</td>
|
||
<td align="center" class="hilite">0.949</td>
|
||
<td align="center">473</td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td align="center">438</td>
|
||
<td align="center">524</td>
|
||
<td align="center">962</td>
|
||
</tr>
|
||
</table>
|
||
<h3>Now add up those calculated values:</h3>
|
||
<p class="center">1.099 + 0.918 + 1.136 + 0.949 = 4.102</p>
|
||
<p class="center large">Chi-Square is 4.102</p>
|
||
<h2>From Chi-Square to p</h2>
|
||
|
||
<h3>Degrees of Freedom</h3>
|
||
<p>First we need a "Degree of Freedom"</p>
|
||
<p class="center large">Degree of Freedom = (rows − 1) × (columns − 1)</p>
|
||
<div class="example">
|
||
<p>For our example we have 2 rows and 2 columns:</p>
|
||
<p class="center"> <b>DF</b> = (2 − 1)(2 − 1) = 1×1 = <b>1 </b></p>
|
||
</div>
|
||
<h3>p-value</h3>
|
||
<p>The rest of the calculation is difficult, so either look it up in a <a href="chi-square-table.html">table</a> or use the <a href="chi-square-calculator.html">Chi-Square Calculator</a>.</p>
|
||
<p>The result is:</p>
|
||
<p class="center large"> p = 0.04283</p>
|
||
<p>Done!</p>
|
||
<h2>Chi-Square Formula</h2>
|
||
<p>This is the formula for Chi-Square:</p>
|
||
<p class="center large"><span class="times"><i>Χ<sup>2</sup></i></span> = <span class="times" style="font-size:250%; vertical-align:middle;">Σ</span><span class="intbl"><em>(O − E)<sup>2</sup></em><strong>E</strong></span></p>
|
||
<!-- (O-E)^2/E -->
|
||
|
||
<ul>
|
||
<li><span class="times">Σ</span> means to sum up (see <a href="../algebra/sigma-notation.html">Sigma Notation</a>)</li>
|
||
<li>O = each <b>Observed</b> (actual) value</li>
|
||
<li>E = each <b>Expected</b> value </li>
|
||
</ul>
|
||
<p>So we calculate <span class="intbl"><em>(O−E)<sup>2</sup></em><strong>E</strong></span> for each pair of observed and expected values then sum them all up.</p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script type="text/javascript">getQ(11372, 11373, 11374, 11375, 11376, 11377, 11378, 11379, 11380, 11381);</script>
|
||
</div>
|
||
|
||
<div class="related">
|
||
<a href="chi-square-calculator.html">Chi-Square Calculator</a> <a href="index.html">Data Index</a>
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