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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg"></a><a class="link" href="find_mean_and_sd_eg.html" title="Find mean and standard deviation example">Find
  28           mean and standard deviation example</a>
  29 </h5></div></div></div>
  30 <p>
  31             First we need some includes to access the normal distribution, the algorithms
  32             to find location and scale (and some std output of course).
  33           </p>
  34 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for normal_distribution</span>
  35   <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>
  36 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">cauchy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span> <span class="comment">// for cauchy_distribution</span>
  37   <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">cauchy</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span>
  38 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_location</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  39   <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">find_location</span><span class="special">;</span>
  40 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_scale</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  41   <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">find_scale</span><span class="special">;</span>
  42   <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">complement</span><span class="special">;</span>
  43   <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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span>
  44
  45 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
  46   <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>
  47 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
  48   <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>
  49 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">limits</span><span class="special">&gt;</span>
  50   <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span>
  51 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">stdexcept</span><span class="special">&gt;</span>
  52 </pre>
  53 <h5>
  54 <a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.h0"></a>
  55             <span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_find_location_and_find_sca"></a></span><a class="link" href="find_mean_and_sd_eg.html#math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_find_location_and_find_sca">Using
  56             find_location and find_scale to meet dispensing and measurement specifications</a>
  57           </h5>
  58 <p>
  59             Consider an example from K Krishnamoorthy, Handbook of Statistical Distributions
  60             with Applications, ISBN 1-58488-635-8, (2006) p 126, example 10.3.7.
  61           </p>
  62 <p>
  63             "A machine is set to pack 3 kg of ground beef per pack. Over a long
  64             period of time it is found that the average packed was 3 kg with a standard
  65             deviation of 0.1 kg. Assume the packing is normally distributed."
  66           </p>
  67 <p>
  68             We start by constructing a normal distribution with the given parameters:
  69           </p>
  70 <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>
  71 <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>
  72 <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>
  73 </pre>
  74 <p>
  75             We can then find the fraction (or %) of packages that weigh more than
  76             3.1 kg.
  77           </p>
  78 <pre class="programlisting"><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>
  79 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Percentage of packs &gt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">max_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" is "</span>
  80 <span class="special">&lt;&lt;</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="number">100.</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X &gt; 3.1)</span>
  81 </pre>
  82 <p>
  83             We might want to ensure that 95% of packs are over a minimum weight specification,
  84             then we want the value of the mean such that P(X &lt; 2.9) = 0.05.
  85           </p>
  86 <p>
  87             Using the mean of 3 kg, we can estimate the fraction of packs that fail
  88             to meet the specification of 2.9 kg.
  89           </p>
  90 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">minimum_weight</span> <span class="special">=</span> <span class="number">2.9</span><span class="special">;</span>
  91 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &lt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
  92   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</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">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  93 <span class="comment">// fraction of packs &lt;= 2.9 with a mean of 3 is 0.841345</span>
  94 </pre>
  95 <p>
  96             This is 0.84 - more than the target fraction of 0.95. If we want 95%
  97             to be over the minimum weight, what should we set the mean weight to
  98             be?
  99           </p>
 100 <p>
 101             Using the KK StatCalc program supplied with the book and the method given
 102             on page 126 gives 3.06449.
 103           </p>
 104 <p>
 105             We can confirm this by constructing a new distribution which we call
 106             'xpacks' with a safety margin mean of 3.06449 thus:
 107           </p>
 108 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">over_mean</span> <span class="special">=</span> <span class="number">3.06449</span><span class="special">;</span>
 109 <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>
 110 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
 111 <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">xpacks</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span>
 112   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</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">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 113 <span class="comment">// fraction of packs &gt;= 2.9 with a mean of 3.06449 is 0.950005</span>
 114 </pre>
 115 <p>
 116             Using this Math Toolkit, we can calculate the required mean directly
 117             thus:
 118           </p>
 119 <pre class="programlisting"><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>
 120 <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>
 121 <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>
 122 <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>
 123 <span class="comment">// mean + (mean - low_limit - quantile(packs, under_fraction));</span>
 124
 125 <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>
 126 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nominal_mean</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
 127   <span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
 128   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</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">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 129 <span class="comment">// Setting the packer to 3.06449 will mean that fraction of packs &gt;= 2.9 is 0.95</span>
 130 </pre>
 131 <p>
 132             This calculation is generalized as the free function called <code class="computeroutput"><span class="identifier">find_location</span></code>, see <a class="link" href="../../../dist_ref/dist_algorithms.html" title="Distribution Algorithms">algorithms</a>.
 133           </p>
 134 <p>
 135             To use this we will need to
 136           </p>
 137 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">find_location</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 138   <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">find_location</span><span class="special">;</span>
 139 </pre>
 140 <p>
 141             and then use find_location function to find safe_mean, &amp; construct
 142             a new normal distribution called 'goodpacks'.
 143           </p>
 144 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">safe_mean</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
 145 <span class="identifier">normal</span> <span class="identifier">good_packs</span><span class="special">(</span><span class="identifier">safe_mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
 146 </pre>
 147 <p>
 148             with the same confirmation as before:
 149           </p>
 150 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">nominal_mean</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
 151   <span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
 152   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">good_packs</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 153 <span class="comment">// Setting the packer to 3.06449 will mean that fraction of packs &gt;= 2.9 is 0.95</span>
 154 </pre>
 155 <h5>
 156 <a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.h1"></a>
 157             <span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_cauchy_lorentz_instead_of_"></a></span><a class="link" href="find_mean_and_sd_eg.html#math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.using_cauchy_lorentz_instead_of_">Using
 158             Cauchy-Lorentz instead of normal distribution</a>
 159           </h5>
 160 <p>
 161             After examining the weight distribution of a large number of packs, we
 162             might decide that, after all, the assumption of a normal distribution
 163             is not really justified. We might find that the fit is better to a <a class="link" href="../../../dist_ref/dists/cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy Distribution</a>.
 164             This distribution has wider 'wings', so that whereas most of the values
 165             are closer to the mean than the normal, there are also more values than
 166             'normal' that lie further from the mean than the normal.
 167           </p>
 168 <p>
 169             This might happen because a larger than normal lump of meat is either
 170             included or excluded.
 171           </p>
 172 <p>
 173             We first create a <a class="link" href="../../../dist_ref/dists/cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy
 174             Distribution</a> with the original mean and standard deviation, and
 175             estimate the fraction that lie below our minimum weight specification.
 176           </p>
 177 <pre class="programlisting"><span class="identifier">cauchy</span> <span class="identifier">cpacks</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
 178 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Cauchy Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
 179   <span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
 180   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">cpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 181 <span class="comment">// Cauchy Setting the packer to 3 will mean that fraction of packs &gt;= 2.9 is 0.75</span>
 182 </pre>
 183 <p>
 184             Note that far fewer of the packs meet the specification, only 75% instead
 185             of 95%. Now we can repeat the find_location, using the cauchy distribution
 186             as template parameter, in place of the normal used above.
 187           </p>
 188 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">lc</span> <span class="special">=</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">cauchy</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
 189 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_location&lt;cauchy&gt;(minimum_weight, over fraction, standard_deviation); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">lc</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 190 <span class="comment">// find_location&lt;cauchy&gt;(minimum_weight, over fraction, packs.standard_deviation()); 3.53138</span>
 191 </pre>
 192 <p>
 193             Note that the safe_mean setting needs to be much higher, 3.53138 instead
 194             of 3.06449, so we will make rather less profit.
 195           </p>
 196 <p>
 197             And again confirm that the fraction meeting specification is as expected.
 198           </p>
 199 <pre class="programlisting"><span class="identifier">cauchy</span> <span class="identifier">goodcpacks</span><span class="special">(</span><span class="identifier">lc</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">);</span>
 200 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Cauchy Setting the packer to "</span> <span class="special">&lt;&lt;</span> <span class="identifier">lc</span> <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
 201   <span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
 202   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">goodcpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 203 <span class="comment">// Cauchy Setting the packer to 3.53138 will mean that fraction of packs &gt;= 2.9 is 0.95</span>
 204 </pre>
 205 <p>
 206             Finally we could estimate the effect of a much tighter specification,
 207             that 99% of packs met the specification.
 208           </p>
 209 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Cauchy Setting the packer to "</span>
 210   <span class="special">&lt;&lt;</span> <span class="identifier">find_location</span><span class="special">&lt;</span><span class="identifier">cauchy</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="number">0.99</span><span class="special">,</span> <span class="identifier">standard_deviation</span><span class="special">)</span>
 211   <span class="special">&lt;&lt;</span> <span class="string">" will mean that "</span>
 212   <span class="special">&lt;&lt;</span> <span class="string">"fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span>
 213   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">goodcpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 214 </pre>
 215 <p>
 216             Setting the packer to 3.13263 will mean that fraction of packs &gt;=
 217             2.9 is 0.99, but will more than double the mean loss from 0.0644 to 0.133
 218             kg per pack.
 219           </p>
 220 <p>
 221             Of course, this calculation is not limited to packs of meat, it applies
 222             to dispensing anything, and it also applies to a 'virtual' material like
 223             any measurement.
 224           </p>
 225 <p>
 226             The only caveat is that the calculation assumes that the standard deviation
 227             (scale) is known with a reasonably low uncertainty, something that is
 228             not so easy to ensure in practice. And that the distribution is well
 229             defined, <a class="link" href="../../../dist_ref/dists/normal_dist.html" title="Normal (Gaussian) Distribution">Normal
 230             Distribution</a> or <a class="link" href="../../../dist_ref/dists/cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy
 231             Distribution</a>, or some other.
 232           </p>
 233 <p>
 234             If one is simply dispensing a very large number of packs, then it may
 235             be feasible to measure the weight of hundreds or thousands of packs.
 236             With a healthy 'degrees of freedom', the confidence intervals for the
 237             standard deviation are not too wide, typically about + and - 10% for
 238             hundreds of observations.
 239           </p>
 240 <p>
 241             For other applications, where it is more difficult or expensive to make
 242             many observations, the confidence intervals are depressingly wide.
 243           </p>
 244 <p>
 245             See <a class="link" href="../cs_eg/chi_sq_intervals.html" title="Confidence Intervals on the Standard Deviation">Confidence
 246             Intervals on the standard deviation</a> for a worked example <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>
 247             of estimating these intervals.
 248           </p>
 249 <h5>
 250 <a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.h2"></a>
 251             <span class="phrase"><a name="math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.changing_the_scale_or_standard_d"></a></span><a class="link" href="find_mean_and_sd_eg.html#math_toolkit.stat_tut.weg.find_eg.find_mean_and_sd_eg.changing_the_scale_or_standard_d">Changing
 252             the scale or standard deviation</a>
 253           </h5>
 254 <p>
 255             Alternatively, we could invest in a better (more precise) packer (or
 256             measuring device) with a lower standard deviation, or scale.
 257           </p>
 258 <p>
 259             This might cost more, but would reduce the amount we have to 'give away'
 260             in order to meet the specification.
 261           </p>
 262 <p>
 263             To estimate how much better (how much smaller standard deviation) it
 264             would have to be, we need to get the 5% quantile to be located at the
 265             under_weight limit, 2.9
 266           </p>
 267 <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>
 268 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</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>
 269   <span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 270 </pre>
 271 <p>
 272             Quantile of 0.05 = 2.83551, mean = 3, sd = 0.1
 273           </p>
 274 <p>
 275             With the current packer (mean = 3, sd = 0.1), the 5% quantile is at 2.8551
 276             kg, a little below our target of 2.9 kg. So we know that the standard
 277             deviation is going to have to be smaller.
 278           </p>
 279 <p>
 280             Let's start by guessing that it (now 0.1) needs to be halved, to a standard
 281             deviation of 0.05 kg.
 282           </p>
 283 <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>
 284 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</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>
 285   <span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 286 <span class="comment">// Quantile of 0.05 = 2.91776, mean = 3, sd = 0.05</span>
 287
 288 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
 289   <span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack05</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
 290   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</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">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 291 <span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.05 is 0.97725</span>
 292 </pre>
 293 <p>
 294             So 0.05 was quite a good guess, but we are a little over the 2.9 target,
 295             so the standard deviation could be a tiny bit more. So we could do some
 296             more guessing to get closer, say by increasing standard deviation to
 297             0.06 kg, constructing another new distribution called pack06.
 298           </p>
 299 <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>
 300 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">" = "</span> <span class="special">&lt;&lt;</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>
 301   <span class="special">&lt;&lt;</span> <span class="string">", mean = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">", sd = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 302 <span class="comment">// Quantile of 0.05 = 2.90131, mean = 3, sd = 0.06</span>
 303
 304 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
 305   <span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack06</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
 306   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</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">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 307 <span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.06 is 0.95221</span>
 308 </pre>
 309 <p>
 310             Now we are getting really close, but to do the job properly, we might
 311             need to use root finding method, for example the tools provided, and
 312             used elsewhere, in the Math Toolkit, see <a class="link" href="../../../roots_noderiv.html" title="Root Finding Without Derivatives">root-finding
 313             without derivatives</a>
 314           </p>
 315 <p>
 316             But in this (normal) distribution case, we can and should be even smarter
 317             and make a direct calculation.
 318           </p>
 319 <p>
 320             Our required limit is minimum_weight = 2.9 kg, often called the random
 321             variate z. For a standard normal distribution, then probability p = N((minimum_weight
 322             - mean) / sd).
 323           </p>
 324 <p>
 325             We want to find the standard deviation that would be required to meet
 326             this limit, so that the p th quantile is located at z (minimum_weight).
 327             In this case, the 0.05 (5%) quantile is at 2.9 kg pack weight, when the
 328             mean is 3 kg, ensuring that 0.95 (95%) of packs are above the minimum
 329             weight.
 330           </p>
 331 <p>
 332             Rearranging, we can directly calculate the required standard deviation:
 333           </p>
 334 <pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">N01</span><span class="special">;</span> <span class="comment">// standard normal distribution with mean zero and unit standard deviation.</span>
 335 <span class="identifier">p</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span>
 336 <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">N01</span><span class="special">,</span> <span class="identifier">p</span><span class="special">);</span>
 337 <span class="keyword">double</span> <span class="identifier">sd95</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">minimum_weight</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>
 338
 339 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"For the "</span><span class="special">&lt;&lt;</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">"th quantile to be located at "</span>
 340   <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">", would need a standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">sd95</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 341 <span class="comment">// For the 0.05th quantile to be located at 2.9, would need a standard deviation of 0.0607957</span>
 342 </pre>
 343 <p>
 344             We can now construct a new (normal) distribution pack95 for the 'better'
 345             packer, and check that our distribution will meet the specification.
 346           </p>
 347 <pre class="programlisting"><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>
 348 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
 349   <span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack95</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
 350   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</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">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 351 <span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95</span>
 352 </pre>
 353 <p>
 354             This calculation is generalized in the free function find_scale, as shown
 355             below, giving the same standard deviation.
 356           </p>
 357 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ss</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">under_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">());</span>
 358 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ss</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 359 <span class="comment">// find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); 0.0607957</span>
 360 </pre>
 361 <p>
 362             If we had defined an over_fraction, or percentage that must pass specification
 363           </p>
 364 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">over_fraction</span> <span class="special">=</span> <span class="number">0.95</span><span class="special">;</span>
 365 </pre>
 366 <p>
 367             And (wrongly) written
 368           </p>
 369 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">sso</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">over_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">());</span>
 370 </pre>
 371 <p>
 372             With the default policy, we would get a message like
 373           </p>
 374 <pre class="programlisting">Message from thrown exception was:
 375    Error in function boost::math::find_scale&lt;Dist, Policy&gt;(double, double, double, Policy):
 376    Computed scale (-0.060795683191176959) is &lt;= 0! Was the complement intended?
 377 </pre>
 378 <p>
 379             But this would return a <span class="bold"><strong>negative</strong></span> standard
 380             deviation - obviously impossible. The probability should be 1 - over_fraction,
 381             not over_fraction, thus:
 382           </p>
 383 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ss1o</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">over_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">());</span>
 384 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ss1o</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 385 <span class="comment">// find_scale&lt;normal&gt;(minimum_weight, under_fraction, packs.mean()); 0.0607957</span>
 386 </pre>
 387 <p>
 388             But notice that using '1 - over_fraction' - will lead to a loss of accuracy,
 389             especially if over_fraction was close to unity. (See <a class="link" href="../../overview/complements.html#why_complements">why
 390             complements?</a>). In this (very common) case, we should instead use
 391             the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a>,
 392             giving the most accurate result.
 393           </p>
 394 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ssc</span> <span class="special">=</span> <span class="identifier">find_scale</span><span class="special">&lt;</span><span class="identifier">normal</span><span class="special">&gt;(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">minimum_weight</span><span class="special">,</span> <span class="identifier">over_fraction</span><span class="special">,</span> <span class="identifier">packs</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()));</span>
 395 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"find_scale&lt;normal&gt;(complement(minimum_weight, over_fraction, packs.mean())); "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ssc</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 396 <span class="comment">// find_scale&lt;normal&gt;(complement(minimum_weight, over_fraction, packs.mean())); 0.0607957</span>
 397 </pre>
 398 <p>
 399             Note that our guess of 0.06 was close to the accurate value of 0.060795683191176959.
 400           </p>
 401 <p>
 402             We can again confirm our prediction thus:
 403           </p>
 404 <pre class="programlisting"><span class="identifier">normal</span> <span class="identifier">pack95c</span><span class="special">(</span><span class="identifier">mean</span><span class="special">,</span> <span class="identifier">ssc</span><span class="special">);</span>
 405 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span><span class="string">"Fraction of packs &gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">minimum_weight</span> <span class="special">&lt;&lt;</span> <span class="string">" with a mean of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span>
 406   <span class="special">&lt;&lt;</span> <span class="string">" and standard deviation of "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pack95c</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span>
 407   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">pack95c</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 408 <span class="comment">// Fraction of packs &gt;= 2.9 with a mean of 3 and standard deviation of 0.0607957 is 0.95</span>
 409 </pre>
 410 <p>
 411             Notice that these two deceptively simple questions:
 412           </p>
 413 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 414                 Do we over-fill to make sure we meet a minimum specification (or
 415                 under-fill to avoid an overdose)?
 416               </li></ul></div>
 417 <p>
 418             and/or
 419           </p>
 420 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 421                 Do we measure better?
 422               </li></ul></div>
 423 <p>
 424             are actually extremely common.
 425           </p>
 426 <p>
 427             The weight of beef might be replaced by a measurement of more or less
 428             anything, from drug tablet content, Apollo landing rocket firing, X-ray
 429             treatment doses...
 430           </p>
 431 <p>
 432             The scale can be variation in dispensing or uncertainty in measurement.
 433           </p>
 434 <p>
 435             See <a href="../../../../../../example/find_mean_and_sd_normal.cpp" target="_top">find_mean_and_sd_normal.cpp</a>
 436             for full source code &amp; appended program output.
 437           </p>
 438 </div>
 439 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 440 <td align="left"></td>
 441 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 442       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 443       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 444       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 445       Daryle Walker and Xiaogang Zhang<p>
 446         Distributed under the Boost Software License, Version 1.0. (See accompanying
 447         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>)
 448       </p>
 449 </div></td>
 450 </tr></table>
 451 <hr>
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