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  25 <div class="section math_toolkit_dist_stat_tut_weg_cs_eg_chi_sq_test">
  26 <div class="titlepage"><div><div><h6 class="title">
  27 <a name="math_toolkit.dist.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 </h6></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
  34               process 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
  67               freedom (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
  74               N-1 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
 277               the sample is less than 10: this hypothesis is rejected in the analysis
 278               above, 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; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
 284       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 285         Distributed under the Boost Software License, Version 1.0. (See accompanying
 286         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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