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  27 <a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_test"></a><a class="link" href="chi_sq_test.html" title="Chi-Square Test for the Standard Deviation">Chi-Square
  28           Test for the Standard Deviation</a>
  29 </h5></div></div></div>
  30 <p>
  31             We use this test to determine whether the standard deviation of a sample
  32             differs from a specified value. Typically this occurs in process change
  33             situations where we wish to compare the standard deviation of a new process
  34             to an established one.
  35           </p>
  36 <p>
  37             The code for this example is contained in <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>,
  38             and we'll begin by defining the procedure that will print out the test
  39             statistics:
  40           </p>
  41 <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">chi_squared_test</span><span class="special">(</span>
  42     <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span>     <span class="comment">// Sample std deviation</span>
  43     <span class="keyword">double</span> <span class="identifier">D</span><span class="special">,</span>      <span class="comment">// True std deviation</span>
  44     <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">,</span>    <span class="comment">// Sample size</span>
  45     <span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span>  <span class="comment">// Significance level</span>
  46 <span class="special">{</span>
  47 </pre>
  48 <p>
  49             The procedure begins by printing a summary of the input data:
  50           </p>
  51 <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
  52 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
  53
  54 <span class="comment">// Print header:</span>
  55 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
  56    <span class="string">"______________________________________________\n"</span>
  57    <span class="string">"Chi Squared test for sample standard deviation\n"</span>
  58    <span class="string">"______________________________________________\n\n"</span><span class="special">;</span>
  59 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
  60 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  61 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Sample Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sd</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  62 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Expected True Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
  63 </pre>
  64 <p>
  65             The test statistic (T) is simply the ratio of the sample and "true"
  66             standard deviations squared, multiplied by the number of degrees of freedom
  67             (the sample size less one):
  68           </p>
  69 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">t_stat</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">D</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">D</span><span class="special">);</span>
  70 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Test Statistic"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t_stat</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  71 </pre>
  72 <p>
  73             The distribution we need to use, is a Chi Squared distribution with N-1
  74             degrees of freedom:
  75           </p>
  76 <pre class="programlisting"><span class="identifier">chi_squared</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
  77 </pre>
  78 <p>
  79             The various hypothesis that can be tested are summarised in the following
  80             table:
  81           </p>
  82 <div class="informaltable"><table class="table">
  83 <colgroup>
  84 <col>
  85 <col>
  86 </colgroup>
  87 <thead><tr>
  88 <th>
  89                     <p>
  90                       Hypothesis
  91                     </p>
  92                   </th>
  93 <th>
  94                     <p>
  95                       Test
  96                     </p>
  97                   </th>
  98 </tr></thead>
  99 <tbody>
 100 <tr>
 101 <td>
 102                     <p>
 103                       The null-hypothesis: there is no difference in standard deviation
 104                       from the specified value
 105                     </p>
 106                   </td>
 107 <td>
 108                     <p>
 109                       Reject if T &lt; &#967;<sup>2</sup><sub>(1-alpha/2; N-1)</sub> or T &gt; &#967;<sup>2</sup><sub>(alpha/2; N-1)</sub>
 110                     </p>
 111                   </td>
 112 </tr>
 113 <tr>
 114 <td>
 115                     <p>
 116                       The alternative hypothesis: there is a difference in standard
 117                       deviation from the specified value
 118                     </p>
 119                   </td>
 120 <td>
 121                     <p>
 122                       Reject if &#967;<sup>2</sup><sub>(1-alpha/2; N-1)</sub> &gt;= T &gt;= &#967;<sup>2</sup><sub>(alpha/2; N-1)</sub>
 123                     </p>
 124                   </td>
 125 </tr>
 126 <tr>
 127 <td>
 128                     <p>
 129                       The alternative hypothesis: the standard deviation is less
 130                       than the specified value
 131                     </p>
 132                   </td>
 133 <td>
 134                     <p>
 135                       Reject if &#967;<sup>2</sup><sub>(1-alpha; N-1)</sub> &lt;= T
 136                     </p>
 137                   </td>
 138 </tr>
 139 <tr>
 140 <td>
 141                     <p>
 142                       The alternative hypothesis: the standard deviation is greater
 143                       than the specified value
 144                     </p>
 145                   </td>
 146 <td>
 147                     <p>
 148                       Reject if &#967;<sup>2</sup><sub>(alpha; N-1)</sub> &gt;= T
 149                     </p>
 150                   </td>
 151 </tr>
 152 </tbody>
 153 </table></div>
 154 <p>
 155             Where &#967;<sup>2</sup><sub>(alpha; N-1)</sub> is the upper critical value of the Chi Squared distribution,
 156             and &#967;<sup>2</sup><sub>(1-alpha; N-1)</sub> is the lower critical value.
 157           </p>
 158 <p>
 159             Recall that the lower critical value is the same as the quantile, and
 160             the upper critical value is the same as the quantile from the complement
 161             of the probability, that gives us the following code to calculate the
 162             critical values:
 163           </p>
 164 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ucv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">));</span>
 165 <span class="keyword">double</span> <span class="identifier">ucv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
 166 <span class="keyword">double</span> <span class="identifier">lcv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">);</span>
 167 <span class="keyword">double</span> <span class="identifier">lcv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">);</span>
 168 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 169    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 170 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Upper Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 171    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">ucv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 172 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 173    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 174 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Lower Critical Value at alpha/2: "</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 175    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">scientific</span> <span class="special">&lt;&lt;</span> <span class="identifier">lcv2</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
 176 </pre>
 177 <p>
 178             Now that we have the critical values, we can compare these to our test
 179             statistic, and print out the result of each hypothesis and test:
 180           </p>
 181 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span>
 182    <span class="string">"Results for Alternative Hypothesis and alpha"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span>
 183    <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">alpha</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span><span class="special">;</span>
 184 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Alternative Hypothesis              Conclusion\n"</span><span class="special">;</span>
 185
 186 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation != "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">"            "</span><span class="special">;</span>
 187 <span class="keyword">if</span><span class="special">((</span><span class="identifier">ucv2</span> <span class="special">&lt;</span> <span class="identifier">t_stat</span><span class="special">)</span> <span class="special">||</span> <span class="special">(</span><span class="identifier">lcv2</span> <span class="special">&gt;</span> <span class="identifier">t_stat</span><span class="special">))</span>
 188    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
 189 <span class="keyword">else</span>
 190    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
 191
 192 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation  &lt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">"            "</span><span class="special">;</span>
 193 <span class="keyword">if</span><span class="special">(</span><span class="identifier">lcv</span> <span class="special">&gt;</span> <span class="identifier">t_stat</span><span class="special">)</span>
 194    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
 195 <span class="keyword">else</span>
 196    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
 197
 198 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation  &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">D</span> <span class="special">&lt;&lt;</span> <span class="string">"            "</span><span class="special">;</span>
 199 <span class="keyword">if</span><span class="special">(</span><span class="identifier">ucv</span> <span class="special">&lt;</span> <span class="identifier">t_stat</span><span class="special">)</span>
 200    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
 201 <span class="keyword">else</span>
 202    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
 203 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 204 </pre>
 205 <p>
 206             To see some example output we'll use the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm" target="_top">gear
 207             data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
 208             e-Handbook of Statistical Methods.</a>. The data represents measurements
 209             of gear diameter from a manufacturing process. The program output is
 210             deliberately designed to mirror the DATAPLOT output shown in the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm" target="_top">NIST
 211             Handbook Example</a>.
 212           </p>
 213 <pre class="programlisting">______________________________________________
 214 Chi Squared test for sample standard deviation
 215 ______________________________________________
 216
 217 Number of Observations                                 =  100
 218 Sample Standard Deviation                              =  0.00628
 219 Expected True Standard Deviation                       =  0.10000
 220
 221 Test Statistic                                         =  0.39030
 222 CDF of test statistic:                                 =  1.438e-099
 223 Upper Critical Value at alpha:                         =  1.232e+002
 224 Upper Critical Value at alpha/2:                       =  1.284e+002
 225 Lower Critical Value at alpha:                         =  7.705e+001
 226 Lower Critical Value at alpha/2:                       =  7.336e+001
 227
 228 Results for Alternative Hypothesis and alpha           =  0.0500
 229
 230 Alternative Hypothesis              Conclusion
 231 Standard Deviation != 0.100            ACCEPTED
 232 Standard Deviation  &lt; 0.100            ACCEPTED
 233 Standard Deviation  &gt; 0.100            REJECTED
 234 </pre>
 235 <p>
 236             In this case we are testing whether the sample standard deviation is
 237             0.1, and the null-hypothesis is rejected, so we conclude that the standard
 238             deviation <span class="emphasis"><em>is not</em></span> 0.1.
 239           </p>
 240 <p>
 241             For an alternative example, consider the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm" target="_top">silicon
 242             wafer data</a> again from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
 243             e-Handbook of Statistical Methods.</a>. In this scenario a supplier
 244             of 100 ohm.cm silicon wafers claims that his fabrication process can
 245             produce wafers with sufficient consistency so that the standard deviation
 246             of resistivity for the lot does not exceed 10 ohm.cm. A sample of N =
 247             10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm,
 248             and the question we ask ourselves is "Is the suppliers claim correct?".
 249           </p>
 250 <p>
 251             The program output now looks like this:
 252           </p>
 253 <pre class="programlisting">______________________________________________
 254 Chi Squared test for sample standard deviation
 255 ______________________________________________
 256
 257 Number of Observations                                 =  10
 258 Sample Standard Deviation                              =  13.97000
 259 Expected True Standard Deviation                       =  10.00000
 260
 261 Test Statistic                                         =  17.56448
 262 CDF of test statistic:                                 =  9.594e-001
 263 Upper Critical Value at alpha:                         =  1.692e+001
 264 Upper Critical Value at alpha/2:                       =  1.902e+001
 265 Lower Critical Value at alpha:                         =  3.325e+000
 266 Lower Critical Value at alpha/2:                       =  2.700e+000
 267
 268 Results for Alternative Hypothesis and alpha           =  0.0500
 269
 270 Alternative Hypothesis              Conclusion
 271 Standard Deviation != 10.000            REJECTED
 272 Standard Deviation  &lt; 10.000            REJECTED
 273 Standard Deviation  &gt; 10.000            ACCEPTED
 274 </pre>
 275 <p>
 276             In this case, our null-hypothesis is that the standard deviation of the
 277             sample is less than 10: this hypothesis is rejected in the analysis above,
 278             and so we reject the manufacturers claim.
 279           </p>
 280 </div>
 281 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 282 <td align="left"></td>
 283 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 284       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 285       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 286       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 287       Daryle Walker and Xiaogang Zhang<p>
 288         Distributed under the Boost Software License, Version 1.0. (See accompanying
 289         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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