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    24. 

  24. </div>
    25. 

  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
    27. 

  27. <a name="math_toolkit.t_test"></a><a class="link" href="t_test.html" title="t-tests"><span class="emphasis"><em>t</em></span>-tests</a>
    28. 

  28. </h2></div></div></div>
    29. 

  29. <h4>
    30. 

  30. <a name="math_toolkit.t_test.h0"></a>
    31. 

  31.       <span class="phrase"><a name="math_toolkit.t_test.synopsis"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.synopsis">Synopsis</a>
    32. 

  32.     </h4>
    33. 

  33. <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    34. 

  34. 

  35. <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="identifier">statistics</span> <span class="special">{</span>
    36. 

  36. 

  37. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
    38. 

  38. <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">sample_mean</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">sample_variance</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">num_samples</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
    39. 

  39. 

  40. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
    41. 

  41. <span class="keyword">auto</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">begin</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">end</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">iterator_traits</span><span class="special">&lt;</span><span class="identifier">ForwardIterator</span><span class="special">&gt;::</span><span class="identifier">value_type</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
    42. 

  42. 

  43. <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</span>
    44. 

  44. <span class="keyword">auto</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
    45. 

  45. 

  46. <span class="special">}}}</span>
    47. 

  47. </pre>
    48. 

  48. <h4>
    49. 

  49. <a name="math_toolkit.t_test.h1"></a>
    50. 

  50.       <span class="phrase"><a name="math_toolkit.t_test.background"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.background">Background</a>
    51. 

  51.     </h4>
    52. 

  52. <p>
    53. 

  53.       A one-sample <span class="emphasis"><em>t</em></span>-test attempts to answer the question "given
    54. 

  54.       a sample mean, is it likely that the population mean of my data is a certain
    55. 

  55.       value?" The test statistic is
    56. 

  56.     </p>
    57. 

  57. <p>
    58. 

  58.       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/one_sample_t_test_statistic.svg" width="97" height="59"></object></span>
    59. 

  59.     </p>
    60. 

  60. <p>
    61. 

  61.       where &#181;<sub>0</sub> is the assumed mean, <span class="emphasis"><em>s</em></span><sup>2</sup> is the sample variance,
    62. 

  62.       and <span class="emphasis"><em>n</em></span> is the number of samples. If the absolute value
    63. 

  63.       of the test statistic is large, then we have low confidence that the population
    64. 

  64.       mean is equal to &#181;<sub>0</sub>, and if the absolute value of the test statistic is small,
    65. 

  65.       we have high confidence. We now ask the question "what constitutes large
    66. 

  66.       and small in this context?"
    67. 

  67.     </p>
    68. 

  68. <p>
    69. 

  69.       Under reasonable assumptions, the test statistic <span class="emphasis"><em>t</em></span> can
    70. 

  70.       be assumed to come from a Student's <span class="emphasis"><em>t</em></span>-distribution. Since
    71. 

  71.       we wish to know if the sample mean deviates from the true mean in either direction,
    72. 

  72.       the test is two-tailed. Hence the <span class="emphasis"><em>p</em></span>-value is straightforward
    73. 

  73.       to calculate from the Student's <span class="emphasis"><em>t</em></span>-distribution on <span class="emphasis"><em>n</em></span>
    74. 

  74.       - 1 degrees of freedom, but nonetheless it is convenient to have it computed
    75. 

  75.       here.
    76. 

  76.     </p>
    77. 

  77. <p>
    78. 

  78.       An example usage is as follows:
    79. 

  79.     </p>
    80. 

  80. <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
    81. 

  81. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
    82. 

  82. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
    83. 

  83. 

  84. <span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
    85. 

  85. <span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">gen</span><span class="special">{</span><span class="identifier">rd</span><span class="special">()};</span>
    86. 

  86. <span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">dis</span><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
    87. 

  87. <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">(</span><span class="number">1024</span><span class="special">);</span>
    88. 

  88. <span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&amp;</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span>
    89. 

  89.   <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">gen</span><span class="special">);</span>
    90. 

  90. <span class="special">}</span>
    91. 

  91. 

  92. <span class="keyword">auto</span> <span class="special">[</span><span class="identifier">t</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="number">0.0</span><span class="special">);</span>
    93. 

  93. </pre>
    94. 

  94. <p>
    95. 

  95.       The test statistic is the first element of the pair, and the <span class="emphasis"><em>p</em></span>-value
    96. 

  96.       is the second element.
    97. 

  97.     </p>
    98. 

  98. <h4>
    99. 

  99. <a name="math_toolkit.t_test.h2"></a>
    100. 

  100.       <span class="phrase"><a name="math_toolkit.t_test.performance"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.performance">Performance</a>
    101. 

  101.     </h4>
    102. 

  102. <p>
    103. 

  103.       There are two cases: Where the mean and sample variance have already been computed,
    104. 

  104.       and the case where the mean and sample variance must be computed on the fly.
    105. 

  105.     </p>
    106. 

  106. <pre class="programlisting"><span class="special">----------------------------------------------</span>
    107. 

  107. <span class="identifier">Benchmark</span>                                <span class="identifier">Time</span>
    108. 

  108. <span class="special">----------------------------------------------</span>
    109. 

  109. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">291</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">210.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    110. 

  110. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>             <span class="number">1064</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">114.697</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    111. 

  111. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>              <span class="number">407</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">599.213</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    112. 

  112. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">595</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">821.086</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    113. 

  113. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>            <span class="number">1475</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">662.071</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    114. 

  114. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>            <span class="number">1746</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1118.85</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
    115. 

  115. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>            <span class="number">3303</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.15492</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    116. 

  116. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>           <span class="number">6404</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.19139</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    117. 

  117. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>          <span class="number">12461</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.2245</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    118. 

  118. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>          <span class="number">24805</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23029</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    119. 

  119. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>          <span class="number">49639</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.22956</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    120. 

  120. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>         <span class="number">98685</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23698</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    121. 

  121. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>        <span class="number">197434</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23656</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    122. 

  122. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>        <span class="number">393929</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23952</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    123. 

  123. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>       <span class="number">790967</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23466</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    124. 

  124. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>      <span class="number">1582366</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23434</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    125. 

  125. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>      <span class="number">3141112</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24358</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    126. 

  126. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>     <span class="number">6260407</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24792</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    127. 

  127. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>    <span class="number">12521811</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24784</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    128. 

  128. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>    <span class="number">25076257</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24619</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    129. 

  129. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>    <span class="number">50226183</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.2444</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    130. 

  130. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>  <span class="number">100522789</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24353</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
    131. 

  131. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>           <span class="number">5.99</span> <span class="identifier">N</span>
    132. 

  132. <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>               <span class="number">0</span> <span class="special">%</span>
    133. 

  133. <span class="identifier">OneSampleTTestKnownMeanAndVariance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span>        <span class="number">207</span> <span class="identifier">ns</span>
    134. 

  134. </pre>
    135. 

  135. </div>
    136. 

  136. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    137. 

  137. <td align="left"></td>
    138. 

  138. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    139. 

  139.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    140. 

  140.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    141. 

  141.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    142. 

  142.       Daryle Walker and Xiaogang Zhang<p>
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  143.         Distributed under the Boost Software License, Version 1.0. (See accompanying
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