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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<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>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.t_test.h0"></a>
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<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>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</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">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
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<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"><</span><span class="identifier">ForwardIterator</span><span class="special">>::</span><span class="identifier">value_type</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<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">&</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>
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<span class="special">}}}</span>
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</pre>
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<h4>
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<a name="math_toolkit.t_test.h1"></a>
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<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>
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</h4>
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<p>
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A one-sample <span class="emphasis"><em>t</em></span>-test attempts to answer the question "given
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a sample mean, is it likely that the population mean of my data is a certain
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value?" The test statistic is
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/one_sample_t_test_statistic.svg" width="97" height="59"></object></span>
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</p>
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<p>
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where µ<sub>0</sub> is the assumed mean, <span class="emphasis"><em>s</em></span><sup>2</sup> is the sample variance,
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and <span class="emphasis"><em>n</em></span> is the number of samples. If the absolute value
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of the test statistic is large, then we have low confidence that the population
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mean is equal to µ<sub>0</sub>, and if the absolute value of the test statistic is small,
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we have high confidence. We now ask the question "what constitutes large
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and small in this context?"
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</p>
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<p>
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Under reasonable assumptions, the test statistic <span class="emphasis"><em>t</em></span> can
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be assumed to come from a Student's <span class="emphasis"><em>t</em></span>-distribution. Since
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we wish to know if the sample mean deviates from the true mean in either direction,
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the test is two-tailed. Hence the <span class="emphasis"><em>p</em></span>-value is straightforward
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to calculate from the Student's <span class="emphasis"><em>t</em></span>-distribution on <span class="emphasis"><em>n</em></span>
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- 1 degrees of freedom, but nonetheless it is convenient to have it computed
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here.
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</p>
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<p>
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An example usage is as follows:
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</p>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span>
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<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">random</span><span class="special">></span>
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<span class="preprocessor">#include</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">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
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<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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">(</span><span class="number">1024</span><span class="special">);</span>
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<span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span>
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<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>
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<span class="special">}</span>
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<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>
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</pre>
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<p>
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The test statistic is the first element of the pair, and the <span class="emphasis"><em>p</em></span>-value
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is the second element.
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</p>
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<h4>
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<a name="math_toolkit.t_test.h2"></a>
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<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>
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</h4>
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<p>
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There are two cases: Where the mean and sample variance have already been computed,
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and the case where the mean and sample variance must be computed on the fly.
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</p>
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<pre class="programlisting"><span class="special">----------------------------------------------</span>
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<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
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<span class="special">----------------------------------------------</span>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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|
<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</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>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">5.99</span> <span class="identifier">N</span>
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<span class="identifier">OneSampleTTest</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_RMS</span> <span class="number">0</span> <span class="special">%</span>
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<span class="identifier">OneSampleTTestKnownMeanAndVariance</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="number">207</span> <span class="identifier">ns</span>
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</pre>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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