Yates's correction for continuity

<h2 id="correction-for-approximation-error">Correction for approximation error</h2>
<p>Using the <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a> to interpret <a href="/facts/Pearson%2527s_chi-squared_test/rGtoKuU6">Pearson's chi-squared statistic</a> requires one to assume that the <a href="/facts/Discrete_probability_distribution/EpsKKVRu">discrete</a> probability of observed <a href="/facts/Binomial_distribution/UMoFMjDj">binomial frequencies</a> in the table can be approximated by the continuous <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a>. This assumption is not quite correct, and introduces some error.
</p><p>To reduce the error in approximation, <a href="/facts/Frank_Yates/qNsf6uJQ">Frank Yates</a>, an <a href="/facts/England/U9mE5bGL">English</a> <a href="/facts/Statistician/9FBnXfeN">statistician</a>, suggested a correction for continuity that adjusts the formula for <a href="/facts/Pearson%2527s_chi-squared_test/rGtoKuU6">Pearson's chi-squared test</a> by subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> This reduces the chi-squared value obtained and thus increases its <a href="/facts/P-value/InnYd1bz">p-value</a>.
</p><p>The effect of Yates's correction is to prevent overestimation of statistical significance for small data. This formula is chiefly used when at least one cell of the table has an expected count smaller than 5.
</p>

∑
          
            i
            =
            1
          
          
            N
          
        
        
          O
          
            i
          
        
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        20
        
      
    
    {\displaystyle \sum _{i=1}^{N}O_{i}=20\,}

<p>The following is Yates's corrected version of <a href="/facts/Pearson%2527s_chi-squared_test/rGtoKuU6">Pearson's chi-squared statistics</a>:
</p>

χ
          
            Yates
          
          
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          ∑
          
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                E
                
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              E
              
                i
              
            
          
        
      
    
    {\displaystyle \chi _{\text{Yates}}^{2}=\sum _{i=1}^{N}{(|O_{i}-E_{i}|-0.5)^{2} \over E_{i}}}

<p>where:
</p>
<i>Oi</i> = an observed frequency
<i>Ei</i> = an expected (theoretical) frequency, asserted by the null hypothesis
<i>N</i> = number of distinct events
<h2 id="2--2-table">2 × 2 table</h2>
<p>As a short-cut, for a 2 × 2 table with the following entries:
</p>
<table><tbody><tr><th> </th><th>S</th><th>F</th><th> </th></tr><tr><th>A</th><td><i>a</i></td><td><i>b</i></td><td>a+b</td></tr><tr><th>B</th><td><i>c</i></td><td><i>d</i></td><td>c+d</td></tr><tr><th> </th><td>a+c</td><td>b+d</td><td><i>N</i></td></tr></tbody></table>

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            Yates
          
          
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    {\displaystyle \chi _{\text{Yates}}^{2}={\frac {N(|ad-bc|-N/2)^{2}}{(a+b)(c+d)(a+c)(b+d)}}.}

<p>In some cases, this is better.
</p>

χ
          
            Yates
          
          
            2
          
        
        =
        
          
            
              N
              (
              max
              (
              0
              ,
              
                |
              
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              N
              
                /
              
              2
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                N
                
                  S
                
              
              
                N
                
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                N
                
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    {\displaystyle \chi _{\text{Yates}}^{2}={\frac {N(\max(0,|ad-bc|-N/2))^{2}}{N_{S}N_{F}N_{A}N_{B}}}.}

<p>Yates's correction should always be applied, as it will tend to improve the accuracy of the p-value obtained. However, in situations with large sample sizes, using the correction will have little effect on the value of the test statistic, and hence the p-value.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Continuity_correction/qkfEECMd">Continuity correction</a></li>
<li><a href="/facts/Binomial_proportion_confidence_interval/EJxbztl5">Wilson score interval with continuity correction</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Yates, F (1934). "Contingency table involving small numbers and the χ2 test". Supplement to the Journal of the Royal Statistical Society 1(2): 217–235. JSTOR 2983604 <a href="/wiki/Frank_Yates" target="_blank">/wiki/Frank_Yates</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Yates's correction for continuity open-in-new

Yates's correction for continuity