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| <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 Test for the Standard Deviation</a> |
| </h6></div></div></div> |
| <p> |
| We use this test to determine whether the standard deviation of a sample |
| differs from a specified value. Typically this occurs in process change |
| situations where we wish to compare the standard deviation of a new |
| process to an established one. |
| </p> |
| <p> |
| 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>, |
| and we'll begin by defining the procedure that will print out the test |
| statistics: |
| </p> |
| <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">chi_squared_test</span><span class="special">(</span> |
| <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span> <span class="comment">// Sample std deviation |
| </span> <span class="keyword">double</span> <span class="identifier">D</span><span class="special">,</span> <span class="comment">// True std deviation |
| </span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">,</span> <span class="comment">// Sample size |
| </span> <span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span> <span class="comment">// Significance level |
| </span><span class="special">{</span> |
| </pre> |
| <p> |
| The procedure begins by printing a summary of the input data: |
| </p> |
| <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> |
| <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> |
| |
| <span class="comment">// Print header: |
| </span><span class="identifier">cout</span> <span class="special"><<</span> |
| <span class="string">"______________________________________________\n"</span> |
| <span class="string">"Chi Squared test for sample standard deviation\n"</span> |
| <span class="string">"______________________________________________\n\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Number of Observations"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">N</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Sample Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">Sd</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Expected True Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">D</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| The test statistic (T) is simply the ratio of the sample and "true" |
| standard deviations squared, multiplied by the number of degrees of |
| freedom (the sample size less one): |
| </p> |
| <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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Test Statistic"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">t_stat</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| The distribution we need to use, is a Chi Squared distribution with |
| N-1 degrees of freedom: |
| </p> |
| <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> |
| </pre> |
| <p> |
| The various hypothesis that can be tested are summarised in the following |
| table: |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Hypothesis |
| </p> |
| </th> |
| <th> |
| <p> |
| Test |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| The null-hypothesis: there is no difference in standard deviation |
| from the specified value |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if T < χ<sup>2</sup><sub>(1-alpha/2; N-1)</sub> or T > χ<sup>2</sup><sub>(alpha/2; N-1)</sub> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| The alternative hypothesis: there is a difference in standard |
| deviation from the specified value |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if χ<sup>2</sup><sub>(1-alpha/2; N-1)</sub> >= T >= χ<sup>2</sup><sub>(alpha/2; N-1)</sub> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| The alternative hypothesis: the standard deviation is less |
| than the specified value |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if χ<sup>2</sup><sub>(1-alpha; N-1)</sub> <= T |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| The alternative hypothesis: the standard deviation is greater |
| than the specified value |
| </p> |
| </td> |
| <td> |
| <p> |
| Reject if χ<sup>2</sup><sub>(alpha; N-1)</sub> >= T |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <p> |
| Where χ<sup>2</sup><sub>(alpha; N-1)</sub> is the upper critical value of the Chi Squared distribution, |
| and χ<sup>2</sup><sub>(1-alpha; N-1)</sub> is the lower critical value. |
| </p> |
| <p> |
| Recall that the lower critical value is the same as the quantile, and |
| the upper critical value is the same as the quantile from the complement |
| of the probability, that gives us the following code to calculate the |
| critical values: |
| </p> |
| <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> |
| <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> |
| <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> |
| <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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Upper Critical Value at alpha: "</span> <span class="special"><<</span> <span class="string">"= "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">ucv</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Upper Critical Value at alpha/2: "</span> <span class="special"><<</span> <span class="string">"= "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">ucv2</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Lower Critical Value at alpha: "</span> <span class="special"><<</span> <span class="string">"= "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">lcv</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Lower Critical Value at alpha/2: "</span> <span class="special"><<</span> <span class="string">"= "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">lcv2</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span> |
| </pre> |
| <p> |
| Now that we have the critical values, we can compare these to our test |
| statistic, and print out the result of each hypothesis and test: |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> |
| <span class="string">"Results for Alternative Hypothesis and alpha"</span> <span class="special"><<</span> <span class="string">"= "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">alpha</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Alternative Hypothesis Conclusion\n"</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard Deviation != "</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">D</span> <span class="special"><<</span> <span class="string">" "</span><span class="special">;</span> |
| <span class="keyword">if</span><span class="special">((</span><span class="identifier">ucv2</span> <span class="special"><</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">></span> <span class="identifier">t_stat</span><span class="special">))</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span> |
| <span class="keyword">else</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard Deviation < "</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">D</span> <span class="special"><<</span> <span class="string">" "</span><span class="special">;</span> |
| <span class="keyword">if</span><span class="special">(</span><span class="identifier">lcv</span> <span class="special">></span> <span class="identifier">t_stat</span><span class="special">)</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span> |
| <span class="keyword">else</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard Deviation > "</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">D</span> <span class="special"><<</span> <span class="string">" "</span><span class="special">;</span> |
| <span class="keyword">if</span><span class="special">(</span><span class="identifier">ucv</span> <span class="special"><</span> <span class="identifier">t_stat</span><span class="special">)</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span> |
| <span class="keyword">else</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| 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 |
| data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH |
| e-Handbook of Statistical Methods.</a>. The data represents measurements |
| of gear diameter from a manufacturing process. The program output is |
| 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 |
| Handbook Example</a>. |
| </p> |
| <pre class="programlisting">______________________________________________ |
| Chi Squared test for sample standard deviation |
| ______________________________________________ |
| |
| Number of Observations = 100 |
| Sample Standard Deviation = 0.00628 |
| Expected True Standard Deviation = 0.10000 |
| |
| Test Statistic = 0.39030 |
| CDF of test statistic: = 1.438e-099 |
| Upper Critical Value at alpha: = 1.232e+002 |
| Upper Critical Value at alpha/2: = 1.284e+002 |
| Lower Critical Value at alpha: = 7.705e+001 |
| Lower Critical Value at alpha/2: = 7.336e+001 |
| |
| Results for Alternative Hypothesis and alpha = 0.0500 |
| |
| Alternative Hypothesis Conclusion |
| Standard Deviation != 0.100 ACCEPTED |
| Standard Deviation < 0.100 ACCEPTED |
| Standard Deviation > 0.100 REJECTED |
| </pre> |
| <p> |
| In this case we are testing whether the sample standard deviation is |
| 0.1, and the null-hypothesis is rejected, so we conclude that the standard |
| deviation <span class="emphasis"><em>is not</em></span> 0.1. |
| </p> |
| <p> |
| For an alternative example, consider the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm" target="_top">silicon |
| wafer data</a> again from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH |
| e-Handbook of Statistical Methods.</a>. In this scenario a supplier |
| of 100 ohm.cm silicon wafers claims that his fabrication process can |
| produce wafers with sufficient consistency so that the standard deviation |
| of resistivity for the lot does not exceed 10 ohm.cm. A sample of N |
| = 10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm, |
| and the question we ask ourselves is "Is the suppliers claim correct?". |
| </p> |
| <p> |
| The program output now looks like this: |
| </p> |
| <pre class="programlisting">______________________________________________ |
| Chi Squared test for sample standard deviation |
| ______________________________________________ |
| |
| Number of Observations = 10 |
| Sample Standard Deviation = 13.97000 |
| Expected True Standard Deviation = 10.00000 |
| |
| Test Statistic = 17.56448 |
| CDF of test statistic: = 9.594e-001 |
| Upper Critical Value at alpha: = 1.692e+001 |
| Upper Critical Value at alpha/2: = 1.902e+001 |
| Lower Critical Value at alpha: = 3.325e+000 |
| Lower Critical Value at alpha/2: = 2.700e+000 |
| |
| Results for Alternative Hypothesis and alpha = 0.0500 |
| |
| Alternative Hypothesis Conclusion |
| Standard Deviation != 10.000 REJECTED |
| Standard Deviation < 10.000 REJECTED |
| Standard Deviation > 10.000 ACCEPTED |
| </pre> |
| <p> |
| In this case, our null-hypothesis is that the standard deviation of |
| the sample is less than 10: this hypothesis is rejected in the analysis |
| above, and so we reject the manufacturers claim. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
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