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| 24 | </div> | |
| 25 | <div class="section"> | |
| 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"><</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> | |
| 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"><</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">></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"><</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">></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"><</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">></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"><</span><span class="identifier">iostream</span><span class="special">></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"><</span><span class="identifier">iomanip</span><span class="special">></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"><</span><span class="identifier">limits</span><span class="special">></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"><</span><span class="identifier">stdexcept</span><span class="special">></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"><<</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> | |
| 80 | <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="number">100.</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// P(X > 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 < 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"><<</span><span class="string">"Fraction of packs <= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> | |
| 92 | <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">minimum_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 93 | <span class="comment">// fraction of packs <= 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"><<</span> <span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> | |
| 111 | <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> | |
| 112 | <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">minimum_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 113 | <span class="comment">// fraction of packs >= 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"><<</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> | |
| 127 | <span class="special"><<</span> <span class="string">"fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> | |
| 128 | <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">minimum_weight</span><span class="special">))</span> <span class="special"><<</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 >= 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"><</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">></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, & 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"><</span><span class="identifier">normal</span><span class="special">>(</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"><<</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> | |
| 151 | <span class="special"><<</span> <span class="string">"fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> | |
| 152 | <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">good_packs</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special"><<</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 >= 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"><<</span> <span class="string">"Cauchy Setting the packer to "</span> <span class="special"><<</span> <span class="identifier">mean</span> <span class="special"><<</span> <span class="string">" will mean that "</span> | |
| 179 | <span class="special"><<</span> <span class="string">"fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> | |
| 180 | <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">cpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special"><<</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 >= 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"><</span><span class="identifier">cauchy</span><span class="special">>(</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"><<</span> <span class="string">"find_location<cauchy>(minimum_weight, over fraction, standard_deviation); "</span> <span class="special"><<</span> <span class="identifier">lc</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 190 | <span class="comment">// find_location<cauchy>(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"><<</span> <span class="string">"Cauchy Setting the packer to "</span> <span class="special"><<</span> <span class="identifier">lc</span> <span class="special"><<</span> <span class="string">" will mean that "</span> | |
| 201 | <span class="special"><<</span> <span class="string">"fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> | |
| 202 | <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">goodcpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special"><<</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 >= 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"><<</span> <span class="string">"Cauchy Setting the packer to "</span> | |
| 210 | <span class="special"><<</span> <span class="identifier">find_location</span><span class="special"><</span><span class="identifier">cauchy</span><span class="special">>(</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"><<</span> <span class="string">" will mean that "</span> | |
| 212 | <span class="special"><<</span> <span class="string">"fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> | |
| 213 | <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">goodcpacks</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special"><<</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 >= | |
| 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"><<</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> | |
| 269 | <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> | |
| 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"><<</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> | |
| 285 | <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> | |
| 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"><<</span><span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> | |
| 289 | <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> | |
| 290 | <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">minimum_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 291 | <span class="comment">// Fraction of packs >= 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"><<</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> | |
| 301 | <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> | |
| 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"><<</span><span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> | |
| 305 | <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> | |
| 306 | <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">minimum_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 307 | <span class="comment">// Fraction of packs >= 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/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"><<</span> <span class="string">"For 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> | |
| 340 | <span class="special"><<</span> <span class="identifier">minimum_weight</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> | |
| 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"><<</span><span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> | |
| 349 | <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> | |
| 350 | <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">minimum_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 351 | <span class="comment">// Fraction of packs >= 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"><</span><span class="identifier">normal</span><span class="special">>(</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"><<</span> <span class="string">"find_scale<normal>(minimum_weight, under_fraction, packs.mean()); "</span> <span class="special"><<</span> <span class="identifier">ss</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 359 | <span class="comment">// find_scale<normal>(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"><</span><span class="identifier">normal</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> | |
| 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<Dist, Policy>(double, double, double, Policy): | |
| 376 | Computed scale (-0.060795683191176959) is <= 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"><</span><span class="identifier">normal</span><span class="special">>(</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"><<</span> <span class="string">"find_scale<normal>(minimum_weight, under_fraction, packs.mean()); "</span> <span class="special"><<</span> <span class="identifier">ss1o</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 385 | <span class="comment">// find_scale<normal>(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"><</span><span class="identifier">normal</span><span class="special">>(</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"><<</span> <span class="string">"find_scale<normal>(complement(minimum_weight, over_fraction, packs.mean())); "</span> <span class="special"><<</span> <span class="identifier">ssc</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 396 | <span class="comment">// find_scale<normal>(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"><<</span><span class="string">"Fraction of packs >= "</span> <span class="special"><<</span> <span class="identifier">minimum_weight</span> <span class="special"><<</span> <span class="string">" with a mean of "</span> <span class="special"><<</span> <span class="identifier">mean</span> | |
| 406 | <span class="special"><<</span> <span class="string">" and standard deviation of "</span> <span class="special"><<</span> <span class="identifier">pack95c</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span> | |
| 407 | <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">pack95c</span><span class="special">,</span> <span class="identifier">minimum_weight</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 408 | <span class="comment">// Fraction of packs >= 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 & 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 © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 442 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 443 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 444 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 445 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 446 | 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>) | |
| 447 | </p> | |
| 448 | </div></td> | |
| 449 | </tr></table> | |
| 450 | <hr> | |
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