<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Chi-Square Test for the Standard Deviation</title> <link rel="stylesheet" href="../../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../../index.html" title="Math Toolkit"> <link rel="up" href="../cs_eg.html" title="Chi Squared Distribution Examples"> <link rel="prev" href="chi_sq_intervals.html" title="Confidence Intervals on the Standard Deviation"> <link rel="next" href="chi_sq_size.html" title="Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="chi_sq_intervals.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../cs_eg.html"><img src="../../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../../index.html"><img src="../../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="chi_sq_size.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h6 class="title"> <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 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> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="chi_sq_intervals.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../cs_eg.html"><img src="../../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../../index.html"><img src="../../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="chi_sq_size.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>