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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h6 class="title"> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc"></a><a class="link" href="normal_misc.html" title="Some Miscellaneous Examples of the Normal (Gaussian) Distribution"> |
| Some Miscellaneous Examples of the Normal (Gaussian) Distribution</a> |
| </h6></div></div></div> |
| <p> |
| The sample program <a href="../../../../../../../../example/normal_misc_examples.cpp" target="_top">normal_misc_examples.cpp</a> |
| illustrates their use. |
| </p> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.traditional_tables"></a><h5> |
| <a name="id972622"></a> |
| <a class="link" href="normal_misc.html#math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.traditional_tables">Traditional |
| Tables</a> |
| </h5> |
| <p> |
| </p> |
| <p> |
| First we need some includes to access the normal distribution (and |
| some std output of course). |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// for normal_distribution |
| </span> <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double. |
| </span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</span><span class="special">;</span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span> |
| |
| <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Example: Normal distribution, Miscellaneous Applications."</span><span class="special">;</span> |
| |
| <span class="keyword">try</span> |
| <span class="special">{</span> |
| <span class="special">{</span> <span class="comment">// Traditional tables and values.</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Let's start by printing some traditional tables. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">step</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> <span class="comment">// in z |
| </span><span class="keyword">double</span> <span class="identifier">range</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="comment">// min and max z = -range to +range. |
| </span><span class="keyword">int</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="number">17</span><span class="special">;</span> <span class="comment">// traditional tables are only computed to much lower precision. |
| </span> |
| <span class="comment">// Construct a standard normal distribution s |
| </span> <span class="identifier">normal</span> <span class="identifier">s</span><span class="special">;</span> <span class="comment">// (default mean = zero, and standard deviation = unity) |
| </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard normal distribution, mean = "</span><span class="special"><<</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> |
| <span class="special"><<</span> <span class="string">", standard deviation = "</span> <span class="special"><<</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| First the probability distribution function (pdf). |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability distribution function values"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" z "</span> <span class="string">" pdf "</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span> |
| <span class="keyword">for</span> <span class="special">(</span><span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">range</span><span class="special">;</span> <span class="identifier">z</span> <span class="special"><</span> <span class="identifier">range</span> <span class="special">+</span> <span class="identifier">step</span><span class="special">;</span> <span class="identifier">z</span> <span class="special">+=</span> <span class="identifier">step</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">left</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">setw</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">z</span> <span class="special"><<</span> <span class="string">" "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">precision</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">12</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">6</span><span class="special">);</span> <span class="comment">// default</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| And the area under the normal curve from -∞ up to z, the cumulative |
| distribution function (cdf). |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="comment">// For a standard normal distribution |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard normal mean = "</span><span class="special"><<</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> |
| <span class="special"><<</span> <span class="string">", standard deviation = "</span> <span class="special"><<</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Integral (area under the curve) from - infinity up to z "</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" z "</span> <span class="string">" cdf "</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="keyword">for</span> <span class="special">(</span><span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">range</span><span class="special">;</span> <span class="identifier">z</span> <span class="special"><</span> <span class="identifier">range</span> <span class="special">+</span> <span class="identifier">step</span><span class="special">;</span> <span class="identifier">z</span> <span class="special">+=</span> <span class="identifier">step</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">left</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">setw</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">z</span> <span class="special"><<</span> <span class="string">" "</span> |
| <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">precision</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">12</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">6</span><span class="special">);</span> <span class="comment">// default</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| And all this you can do with a nanoscopic amount of work compared |
| to the team of <span class="bold"><strong>human computers</strong></span> toiling |
| with Milton Abramovitz and Irene Stegen at the US National Bureau |
| of Standards (now <a href="http://www.nist.gov" target="_top">NIST</a>). |
| Starting in 1938, their "Handbook of Mathematical Functions |
| with Formulas, Graphs and Mathematical Tables", was eventually |
| published in 1964, and has been reprinted numerous times since. (A |
| major replacement is planned at <a href="http://dlmf.nist.gov" target="_top">Digital |
| Library of Mathematical Functions</a>). |
| </p> |
| <p> |
| </p> |
| <p> |
| Pretty-printing a traditional 2-dimensional table is left as an exercise |
| for the student, but why bother now that the Math Toolkit lets you |
| write |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">z</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="string">"Area for z = "</span> <span class="special"><<</span> <span class="identifier">z</span> <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">z</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// to get the area for z.</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Correspondingly, we can obtain the traditional 'critical' values |
| for significance levels. For the 95% confidence level, the significance |
| level usually called alpha, is 0.05 = 1 - 0.95 (for a one-sided test), |
| so we can write |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"95% of area has a z below "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.95</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="comment">// 95% of area has a z below 1.64485</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| and a two-sided test (a comparison between two levels, rather than |
| a one-sided test) |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"95% of area has a z between "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="special">-</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.975</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="comment">// 95% of area has a z between 1.95996 and -1.95996</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| First, define a table of significance levels: these are the probabilities |
| that the true occurrence frequency lies outside the calculated interval. |
| </p> |
| <p> |
| </p> |
| <p> |
| It is convenient to have an alpha level for the probability that |
| z lies outside just one standard deviation. This will not be some |
| nice neat number like 0.05, but we can easily calculate it, |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha1</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="number">2</span><span class="special">;</span> <span class="comment">// 0.3173105078629142 |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">17</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"Significance level for z == 1 is "</span> <span class="special"><<</span> <span class="identifier">alpha1</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| and place in our array of favorite alpha values. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.3173105078629142</span><span class="special">,</span> <span class="comment">// z for 1 standard deviation. |
| </span> <span class="number">0.20</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Confidence value as % is (1 - alpha) * 100 (so alpha 0.05 == 95% |
| confidence) that the true occurrence frequency lies <span class="bold"><strong>inside</strong></span> |
| the calculated interval. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"level of significance (alpha)"</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">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2-sided 1 -sided z(alpha) "</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</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">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</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">s</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="comment">// Use quantile(complement(s, alpha[i]/2)) to avoid potential loss of accuracy from quantile(s, 1 - alpha[i]/2) |
| </span><span class="special">}</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Notice the distinction between one-sided (also called one-tailed) |
| where we are using a > <span class="bold"><strong>or</strong></span> < |
| test (and not both) and considering the area of the tail (integral) |
| from z up to +∞, and a two-sided test where we are using two > |
| <span class="bold"><strong>and</strong></span> < tests, and thus considering |
| two tails, from -∞ up to z low and z high up to +∞. |
| </p> |
| <p> |
| </p> |
| <p> |
| So the 2-sided values alpha[i] are calculated using alpha[i]/2. |
| </p> |
| <p> |
| </p> |
| <p> |
| If we consider a simple example of alpha = 0.05, then for a two-sided |
| test, the lower tail area from -∞ up to -1.96 is 0.025 (alpha/2) and |
| the upper tail area from +z up to +1.96 is also 0.025 (alpha/2), |
| and the area between -1.96 up to 12.96 is alpha = 0.95. and the sum |
| of the two tails is 0.025 + 0.025 = 0.05, |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.standard_deviations_either_side_of_the_mean"></a><h5> |
| <a name="id976060"></a> |
| <a class="link" href="normal_misc.html#math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.standard_deviations_either_side_of_the_mean">Standard |
| deviations either side of the Mean</a> |
| </h5> |
| <p> |
| </p> |
| <p> |
| Armed with the cumulative distribution function, we can easily calculate |
| the easy to remember proportion of values that lie within 1, 2 and |
| 3 standard deviations from the mean. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">3</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">showpoint</span> <span class="special"><<</span> <span class="string">"cdf(s, s.standard_deviation()) = "</span> |
| <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">())</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// from -infinity to 1 sd |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cdf(complement(s, s.standard_deviation())) = "</span> |
| <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction 1 standard deviation within either side of mean is "</span> |
| <span class="special"><<</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special">*</span> <span class="number">2</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction 2 standard deviations within either side of mean is "</span> |
| <span class="special"><<</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special">*</span> <span class="number">2</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction 3 standard deviations within either side of mean is "</span> |
| <span class="special"><<</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">3</span> <span class="special">*</span> <span class="identifier">s</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()))</span> <span class="special">*</span> <span class="number">2</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| To a useful precision, the 1, 2 & 3 percentages are 68, 95 and |
| 99.7, and these are worth memorising as useful 'rules of thumb', |
| as, for example, in <a href="http://en.wikipedia.org/wiki/Standard_deviation" target="_top">standard |
| deviation</a>: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Fraction 1 standard deviation within either side of mean is 0.683 |
| Fraction 2 standard deviations within either side of mean is 0.954 |
| Fraction 3 standard deviations within either side of mean is 0.997 |
| </pre> |
| <p> |
| </p> |
| <p> |
| We could of course get some really accurate values for these <a href="http://en.wikipedia.org/wiki/Confidence_interval" target="_top">confidence |
| intervals</a> by using cout.precision(15); |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Fraction 1 standard deviation within either side of mean is 0.682689492137086 |
| Fraction 2 standard deviations within either side of mean is 0.954499736103642 |
| Fraction 3 standard deviations within either side of mean is 0.997300203936740 |
| </pre> |
| <p> |
| </p> |
| <p> |
| But before you get too excited about this impressive precision, don't |
| forget that the <span class="bold"><strong>confidence intervals of the |
| standard deviation</strong></span> are surprisingly wide, especially if |
| you have estimated the standard deviation from only a few measurements. |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.some_simple_examples"></a><h5> |
| <a name="id976885"></a> |
| <a class="link" href="normal_misc.html#math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.some_simple_examples">Some |
| simple examples</a> |
| </h5> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.life_of_light_bulbs"></a><h5> |
| <a name="id976898"></a> |
| <a class="link" href="normal_misc.html#math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.life_of_light_bulbs">Life |
| of light bulbs</a> |
| </h5> |
| <p> |
| </p> |
| <p> |
| Examples from K. Krishnamoorthy, Handbook of Statistical Distributions |
| with Applications, ISBN 1 58488 635 8, page 125... implemented using |
| the Math Toolkit library. |
| </p> |
| <p> |
| </p> |
| <p> |
| A few very simple examples are shown here: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="comment">// K. Krishnamoorthy, Handbook of Statistical Distributions with Applications, |
| </span> <span class="comment">// ISBN 1 58488 635 8, page 125, example 10.3.5</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Mean lifespan of 100 W bulbs is 1100 h with standard deviation of |
| 100 h. Assuming, perhaps with little evidence and much faith, that |
| the distribution is normal, we construct a normal distribution called |
| <span class="emphasis"><em>bulbs</em></span> with these values: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">mean_life</span> <span class="special">=</span> <span class="number">1100.</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">life_standard_deviation</span> <span class="special">=</span> <span class="number">100.</span><span class="special">;</span> |
| <span class="identifier">normal</span> <span class="identifier">bulbs</span><span class="special">(</span><span class="identifier">mean_life</span><span class="special">,</span> <span class="identifier">life_standard_deviation</span><span class="special">);</span> |
| <span class="keyword">double</span> <span class="identifier">expected_life</span> <span class="special">=</span> <span class="number">1000.</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| The we can use the Cumulative distribution function to predict fractions |
| (or percentages, if * 100) that will last various lifetimes. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction of bulbs that will last at best (<=) "</span> <span class="comment">// P(X <= 1000) |
| </span> <span class="special"><<</span> <span class="identifier">expected_life</span> <span class="special"><<</span> <span class="string">" is "</span><span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">expected_life</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction of bulbs that will last at least (>) "</span> <span class="comment">// P(X > 1000) |
| </span> <span class="special"><<</span> <span class="identifier">expected_life</span> <span class="special"><<</span> <span class="string">" is "</span><span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">expected_life</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">min_life</span> <span class="special">=</span> <span class="number">900</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">max_life</span> <span class="special">=</span> <span class="number">1200</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction of bulbs that will last between "</span> |
| <span class="special"><<</span> <span class="identifier">min_life</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">max_life</span> <span class="special"><<</span> <span class="string">" is "</span> |
| <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">max_life</span><span class="special">)</span> <span class="comment">// P(X <= 1200) |
| </span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bulbs</span><span class="special">,</span> <span class="identifier">min_life</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X <= 900)</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| Real-life failures are often very ab-normal, with a significant |
| number that 'dead-on-arrival' or suffer failure very early in their |
| life: the lifetime of the survivors of 'early mortality' may be |
| well described by the normal distribution. |
| </p></td></tr> |
| </table></div> |
| <p> |
| </p> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.how_many_onions_"></a><h5> |
| <a name="id977409"></a> |
| <a class="link" href="normal_misc.html#math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.how_many_onions_">How |
| many onions?</a> |
| </h5> |
| <p> |
| </p> |
| <p> |
| Weekly demand for 5 lb sacks of onions at a store is normally distributed |
| with mean 140 sacks and standard deviation 10. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">140.</span><span class="special">;</span> <span class="comment">// sacks per week. |
| </span><span class="keyword">double</span> <span class="identifier">standard_deviation</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> |
| <span class="identifier">normal</span> <span class="identifier">sacks</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span> |
| |
| <span class="keyword">double</span> <span class="identifier">stock</span> <span class="special">=</span> <span class="number">160.</span><span class="special">;</span> <span class="comment">// per week. |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Percentage of weeks overstocked "</span> |
| <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">sacks</span><span class="special">,</span> <span class="identifier">stock</span><span class="special">)</span> <span class="special">*</span> <span class="number">100.</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X <=160) |
| </span><span class="comment">// Percentage of weeks overstocked 97.7</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| So there will be lots of mouldy onions! So we should be able to say |
| what stock level will meet demand 95% of the weeks. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">stock_95</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">sacks</span><span class="special">,</span> <span class="number">0.95</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Store should stock "</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">stock_95</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" sacks to meet 95% of demands."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| And it is easy to estimate how to meet 80% of demand, and waste even |
| less. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">stock_80</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">sacks</span><span class="special">,</span> <span class="number">0.80</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Store should stock "</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">stock_80</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" sacks to meet 8 out of 10 demands."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.packing_beef"></a><h5> |
| <a name="id977827"></a> |
| <a class="link" href="normal_misc.html#math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.packing_beef">Packing |
| beef</a> |
| </h5> |
| <p> |
| </p> |
| <p> |
| A machine is set to pack 3 kg of ground beef per pack. Over a long |
| period of time it is found that the average packed was 3 kg with |
| a standard deviation of 0.1 kg. Assuming the packing is normally |
| distributed, we can find the fraction (or %) of packages that weigh |
| more than 3.1 kg. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">3.</span><span class="special">;</span> <span class="comment">// kg |
| </span><span class="keyword">double</span> <span class="identifier">standard_deviation</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> <span class="comment">// kg |
| </span><span class="identifier">normal</span> <span class="identifier">packs</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span> |
| |
| <span class="keyword">double</span> <span class="identifier">max_weight</span> <span class="special">=</span> <span class="number">3.1</span><span class="special">;</span> <span class="comment">// kg |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Percentage of packs > "</span> <span class="special"><<</span> <span class="identifier">max_weight</span> <span class="special"><<</span> <span class="string">" is "</span> |
| <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">max_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X > 3.1) |
| </span> |
| <span class="keyword">double</span> <span class="identifier">under_weight</span> <span class="special">=</span> <span class="number">2.9</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span><span class="string">"fraction of packs <= "</span> <span class="special"><<</span> <span class="identifier">under_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> |
| <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="comment">// fraction of packs <= 2.9 with a mean of 3 is 0.841345 |
| </span><span class="comment">// This is 0.84 - more than the target 0.95 |
| </span><span class="comment">// Want 95% to be over this weight, so what should we set the mean weight to be? |
| </span><span class="comment">// KK StatCalc says: |
| </span><span class="keyword">double</span> <span class="identifier">over_mean</span> <span class="special">=</span> <span class="number">3.0664</span><span class="special">;</span> |
| <span class="identifier">normal</span> <span class="identifier">xpacks</span><span class="special">(</span><span class="identifier">over_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">under_weight</span> |
| <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">xpacks</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> |
| <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">xpacks</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="comment">// fraction of packs >= 2.9 with a mean of 3.06449 is 0.950005 |
| </span><span class="keyword">double</span> <span class="identifier">under_fraction</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> <span class="comment">// so 95% are above the minimum weight mean - sd = 2.9 |
| </span><span class="keyword">double</span> <span class="identifier">low_limit</span> <span class="special">=</span> <span class="identifier">standard_deviation</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">offset</span> <span class="special">=</span> <span class="identifier">mean</span> <span class="special">-</span> <span class="identifier">low_limit</span> <span class="special">-</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">);</span> |
| <span class="keyword">double</span> <span class="identifier">nominal_mean</span> <span class="special">=</span> <span class="identifier">mean</span> <span class="special">+</span> <span class="identifier">offset</span><span class="special">;</span> |
| |
| <span class="identifier">normal</span> <span class="identifier">nominal_packs</span><span class="special">(</span><span class="identifier">nominal_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Setting the packer to "</span> <span class="special"><<</span> <span class="identifier">nominal_mean</span> <span class="special"><<</span> <span class="string">" will mean that "</span> |
| <span class="special"><<</span> <span class="string">"fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">under_weight</span> |
| <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">nominal_packs</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Setting the packer to 3.06449 will mean that fraction of packs >= |
| 2.9 is 0.95. |
| </p> |
| <p> |
| </p> |
| <p> |
| Setting the packer to 3.13263 will mean that fraction of packs >= |
| 2.9 is 0.99, but will more than double the mean loss from 0.0644 |
| to 0.133. |
| </p> |
| <p> |
| </p> |
| <p> |
| Alternatively, we could invest in a better (more precise) packer |
| with a lower standard deviation. |
| </p> |
| <p> |
| </p> |
| <p> |
| To estimate how much better (how much smaller standard deviation) |
| it would have to be, we need to get the 5% quantile to be located |
| at the under_weight limit, 2.9 |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> <span class="comment">// wanted p th quantile. |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quantile of "</span> <span class="special"><<</span> <span class="identifier">p</span> <span class="special"><<</span> <span class="string">" = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">packs</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="string">", mean = "</span> <span class="special"><<</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special"><<</span> <span class="string">", sd = "</span> <span class="special"><<</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">//</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1 |
| </p> |
| <p> |
| </p> |
| <p> |
| With the current packer (mean = 3, sd = 0.1), the 5% quantile is |
| at 2.8551 kg, a little below our target of 2.9 kg. So we know that |
| the standard deviation is going to have to be smaller. |
| </p> |
| <p> |
| </p> |
| <p> |
| Let's start by guessing that it (now 0.1) needs to be halved, to |
| a standard deviation of 0.05 |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack05</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quantile of "</span> <span class="special"><<</span> <span class="identifier">p</span> <span class="special"><<</span> <span class="string">" = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">pack05</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="string">", mean = "</span> <span class="special"><<</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special"><<</span> <span class="string">", sd = "</span> <span class="special"><<</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span><span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">under_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> |
| <span class="special"><<</span> <span class="string">" and standard deviation of "</span> <span class="special"><<</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> |
| <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack05</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="comment">//</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Fraction of packs >= 2.9 with a mean of 3 and standard deviation |
| of 0.05 is 0.9772 |
| </p> |
| <p> |
| </p> |
| <p> |
| So 0.05 was quite a good guess, but we are a little over the 2.9 |
| target, so the standard deviation could be a tiny bit more. So we |
| could do some more guessing to get closer, say by increasing to 0.06 |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack06</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="number">0.06</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quantile of "</span> <span class="special"><<</span> <span class="identifier">p</span> <span class="special"><<</span> <span class="string">" = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">pack06</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="string">", mean = "</span> <span class="special"><<</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special"><<</span> <span class="string">", sd = "</span> <span class="special"><<</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span><span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">under_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> |
| <span class="special"><<</span> <span class="string">" and standard deviation of "</span> <span class="special"><<</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> |
| <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack06</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Fraction of packs >= 2.9 with a mean of 3 and standard deviation |
| of 0.06 is 0.9522 |
| </p> |
| <p> |
| </p> |
| <p> |
| Now we are getting really close, but to do the job properly, we could |
| use root finding method, for example the tools provided, and used |
| elsewhere, in the Math Toolkit, see <a class="link" href="../../../../toolkit/internals1/roots2.html" title="Root Finding Without Derivatives">Root |
| Finding Without Derivatives</a>. |
| </p> |
| <p> |
| </p> |
| <p> |
| But in this normal distribution case, we could be even smarter and |
| make a direct calculation. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">s</span><span class="special">;</span> <span class="comment">// For standard normal distribution, |
| </span><span class="keyword">double</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">2.9</span><span class="special">;</span> <span class="comment">// Our required limit. |
| </span><span class="comment">// then probability p = N((x - mean) / sd) |
| </span><span class="comment">// So if we want to find the standard deviation that would be required to meet this limit, |
| </span><span class="comment">// so that the p th quantile is located at x, |
| </span><span class="comment">// in this case the 0.95 (95%) quantile at 2.9 kg pack weight, when the mean is 3 kg. |
| </span> |
| <span class="keyword">double</span> <span class="identifier">prob</span> <span class="special">=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">sd</span><span class="special">);</span> |
| <span class="keyword">double</span> <span class="identifier">qp</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="number">0.95</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"prob = "</span> <span class="special"><<</span> <span class="identifier">prob</span> <span class="special"><<</span> <span class="string">", quantile(p) "</span> <span class="special"><<</span> <span class="identifier">qp</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// p = 0.241971, quantile(p) 1.64485 |
| </span><span class="comment">// Rearranging, we can directly calculate the required standard deviation: |
| </span><span class="keyword">double</span> <span class="identifier">sd95</span> <span class="special">=</span> <span class="identifier">abs</span><span class="special">((</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">qp</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If we want the "</span><span class="special"><<</span> <span class="identifier">p</span> <span class="special"><<</span> <span class="string">" th quantile to be located at "</span> |
| <span class="special"><<</span> <span class="identifier">x</span> <span class="special"><<</span> <span class="string">", would need a standard deviation of "</span> <span class="special"><<</span> <span class="identifier">sd95</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">normal</span> <span class="identifier">pack95</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">sd95</span><span class="special">);</span> <span class="comment">// Distribution of the 'ideal better' packer. |
| </span><span class="identifier">cout</span> <span class="special"><<</span><span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">under_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> |
| <span class="special"><<</span> <span class="string">" and standard deviation of "</span> <span class="special"><<</span> <span class="identifier">pack95</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> |
| <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack95</span><span class="special">,</span> <span class="identifier">under_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Fraction of packs >= 2.9 with a mean of 3 and standard deviation of 0.0608 is 0.95</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Notice that these two deceptively simple questions (do we over-fill |
| or measure better) are actually very common. The weight of beef might |
| be replaced by a measurement of more or less anything. But the calculations |
| rely on the accuracy of the standard deviation - something that is |
| almost always less good than we might wish, especially if based on |
| a few measurements. |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.length_of_bolts"></a><h5> |
| <a name="id980594"></a> |
| <a class="link" href="normal_misc.html#math_toolkit.dist.stat_tut.weg.normal_example.normal_misc.length_of_bolts">Length |
| of bolts</a> |
| </h5> |
| <p> |
| </p> |
| <p> |
| A bolt is usable if between 3.9 and 4.1 long. From a large batch |
| of bolts, a sample of 50 show a mean length of 3.95 with standard |
| deviation 0.1. Assuming a normal distribution, what proportion is |
| usable? The true sample mean is unknown, but we can use the sample |
| mean and standard deviation to find approximate solutions. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"> <span class="identifier">normal</span> <span class="identifier">bolts</span><span class="special">(</span><span class="number">3.95</span><span class="special">,</span> <span class="number">0.1</span><span class="special">);</span> |
| <span class="keyword">double</span> <span class="identifier">top</span> <span class="special">=</span> <span class="number">4.1</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">bottom</span> <span class="special">=</span> <span class="number">3.9</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction long enough [ P(X <= "</span> <span class="special"><<</span> <span class="identifier">top</span> <span class="special"><<</span> <span class="string">") ] is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">top</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction too short [ P(X <= "</span> <span class="special"><<</span> <span class="identifier">bottom</span> <span class="special"><<</span> <span class="string">") ] is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">bottom</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction OK -between "</span> <span class="special"><<</span> <span class="identifier">bottom</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">top</span> |
| <span class="special"><<</span> <span class="string">"[ P(X <= "</span> <span class="special"><<</span> <span class="identifier">top</span> <span class="special"><<</span> <span class="string">") - P(X<= "</span> <span class="special"><<</span> <span class="identifier">bottom</span> <span class="special"><<</span> <span class="string">" ) ] is "</span> |
| <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">top</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">bottom</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Fraction too long [ P(X > "</span> <span class="special"><<</span> <span class="identifier">top</span> <span class="special"><<</span> <span class="string">") ] is "</span> |
| <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="identifier">top</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"95% of bolts are shorter than "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">bolts</span><span class="special">,</span> <span class="number">0.95</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| </pre> |
| <p> |
| </p> |
| <p> |
| </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> |
| </tr></table> |
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