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  27 <a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals"></a><a class="link" href="chi_sq_intervals.html" title="Confidence Intervals on the Standard Deviation">Confidence
  28           Intervals on the Standard Deviation</a>
  29 </h5></div></div></div>
  30 <p>
  31             Once you have calculated the standard deviation for your data, a legitimate
  32             question to ask is "How reliable is the calculated standard deviation?".
  33             For this situation the Chi Squared distribution can be used to calculate
  34             confidence intervals for the standard deviation.
  35           </p>
  36 <p>
  37             The full example code &amp; sample output is in <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>.
  38           </p>
  39 <p>
  40             We'll begin by defining the procedure that will calculate and print out
  41             the confidence intervals:
  42           </p>
  43 <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">confidence_limits_on_std_deviation</span><span class="special">(</span>
  44      <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span>    <span class="comment">// Sample Standard Deviation</span>
  45      <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">)</span>   <span class="comment">// Sample size</span>
  46 <span class="special">{</span>
  47 </pre>
  48 <p>
  49             We'll begin by printing out some general information:
  50           </p>
  51 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
  52    <span class="string">"________________________________________________\n"</span>
  53    <span class="string">"2-Sided Confidence Limits For Standard Deviation\n"</span>
  54    <span class="string">"________________________________________________\n\n"</span><span class="special">;</span>
  55 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">7</span><span class="special">);</span>
  56 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">N</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  57 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sd</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  58 </pre>
  59 <p>
  60             and then define a table of significance levels for which we'll calculate
  61             intervals:
  62           </p>
  63 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
  64 </pre>
  65 <p>
  66             The distribution we'll need to calculate the confidence intervals is
  67             a Chi Squared distribution, with N-1 degrees of freedom:
  68           </p>
  69 <pre class="programlisting"><span class="identifier">chi_squared</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
  70 </pre>
  71 <p>
  72             For each value of alpha, the formula for the confidence interval is given
  73             by:
  74           </p>
  75 <p>
  76             <span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_tut1.svg"></span>
  77           </p>
  78 <p>
  79             Where <span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_tut2.svg"></span> is the upper critical value, and <span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_tut3.svg"></span> is
  80             the lower critical value of the Chi Squared distribution.
  81           </p>
  82 <p>
  83             In code we begin by printing out a table header:
  84           </p>
  85 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
  86         <span class="string">"_____________________________________________\n"</span>
  87         <span class="string">"Confidence          Lower          Upper\n"</span>
  88         <span class="string">" Value (%)          Limit          Limit\n"</span>
  89         <span class="string">"_____________________________________________\n"</span><span class="special">;</span>
  90 </pre>
  91 <p>
  92             and then loop over the values of alpha and calculate the intervals for
  93             each: remember that the lower critical value is the same as the quantile,
  94             and the upper critical value is the same as the quantile from the complement
  95             of the probability:
  96           </p>
  97 <pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
  98 <span class="special">{</span>
  99    <span class="comment">// Confidence value:</span>
 100    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
 101    <span class="comment">// Calculate limits:</span>
 102    <span class="keyword">double</span> <span class="identifier">lower_limit</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">)));</span>
 103    <span class="keyword">double</span> <span class="identifier">upper_limit</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">N</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
 104    <span class="comment">// Print Limits:</span>
 105    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">lower_limit</span><span class="special">;</span>
 106    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">upper_limit</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 107 <span class="special">}</span>
 108 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 109 </pre>
 110 <p>
 111             To see some example output we'll use the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm" target="_top">gear
 112             data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
 113             e-Handbook of Statistical Methods.</a>. The data represents measurements
 114             of gear diameter from a manufacturing process.
 115           </p>
 116 <pre class="programlisting">________________________________________________
 117 2-Sided Confidence Limits For Standard Deviation
 118 ________________________________________________
 119
 120 Number of Observations                  =  100
 121 Standard Deviation                      =  0.006278908
 122
 123
 124 _____________________________________________
 125 Confidence          Lower          Upper
 126  Value (%)          Limit          Limit
 127 _____________________________________________
 128     50.000        0.00601        0.00662
 129     75.000        0.00582        0.00685
 130     90.000        0.00563        0.00712
 131     95.000        0.00551        0.00729
 132     99.000        0.00530        0.00766
 133     99.900        0.00507        0.00812
 134     99.990        0.00489        0.00855
 135     99.999        0.00474        0.00895
 136 </pre>
 137 <p>
 138             So at the 95% confidence level we conclude that the standard deviation
 139             is between 0.00551 and 0.00729.
 140           </p>
 141 <h5>
 142 <a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals.h0"></a>
 143             <span class="phrase"><a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals.confidence_intervals_as_a_functi"></a></span><a class="link" href="chi_sq_intervals.html#math_toolkit.stat_tut.weg.cs_eg.chi_sq_intervals.confidence_intervals_as_a_functi">Confidence
 144             intervals as a function of the number of observations</a>
 145           </h5>
 146 <p>
 147             Similarly, we can also list the confidence intervals for the standard
 148             deviation for the common confidence levels 95%, for increasing numbers
 149             of observations.
 150           </p>
 151 <p>
 152             The standard deviation used to compute these values is unity, so the
 153             limits listed are <span class="bold"><strong>multipliers</strong></span> for any
 154             particular standard deviation. For example, given a standard deviation
 155             of 0.0062789 as in the example above; for 100 observations the multiplier
 156             is 0.8780 giving the lower confidence limit of 0.8780 * 0.006728 = 0.00551.
 157           </p>
 158 <pre class="programlisting">____________________________________________________
 159 Confidence level (two-sided)            =  0.0500000
 160 Standard Deviation                      =  1.0000000
 161 ________________________________________
 162 Observations        Lower          Upper
 163                     Limit          Limit
 164 ________________________________________
 165          2         0.4461        31.9102
 166          3         0.5207         6.2847
 167          4         0.5665         3.7285
 168          5         0.5991         2.8736
 169          6         0.6242         2.4526
 170          7         0.6444         2.2021
 171          8         0.6612         2.0353
 172          9         0.6755         1.9158
 173         10         0.6878         1.8256
 174         15         0.7321         1.5771
 175         20         0.7605         1.4606
 176         30         0.7964         1.3443
 177         40         0.8192         1.2840
 178         50         0.8353         1.2461
 179         60         0.8476         1.2197
 180        100         0.8780         1.1617
 181        120         0.8875         1.1454
 182       1000         0.9580         1.0459
 183      10000         0.9863         1.0141
 184      50000         0.9938         1.0062
 185     100000         0.9956         1.0044
 186    1000000         0.9986         1.0014
 187 </pre>
 188 <p>
 189             With just 2 observations the limits are from <span class="bold"><strong>0.445</strong></span>
 190             up to to <span class="bold"><strong>31.9</strong></span>, so the standard deviation
 191             might be about <span class="bold"><strong>half</strong></span> the observed value
 192             up to <span class="bold"><strong>30 times</strong></span> the observed value!
 193           </p>
 194 <p>
 195             Estimating a standard deviation with just a handful of values leaves
 196             a very great uncertainty, especially the upper limit. Note especially
 197             how far the upper limit is skewed from the most likely standard deviation.
 198           </p>
 199 <p>
 200             Even for 10 observations, normally considered a reasonable number, the
 201             range is still from 0.69 to 1.8, about a range of 0.7 to 2, and is still
 202             highly skewed with an upper limit <span class="bold"><strong>twice</strong></span>
 203             the median.
 204           </p>
 205 <p>
 206             When we have 1000 observations, the estimate of the standard deviation
 207             is starting to look convincing, with a range from 0.95 to 1.05 - now
 208             near symmetrical, but still about + or - 5%.
 209           </p>
 210 <p>
 211             Only when we have 10000 or more repeated observations can we start to
 212             be reasonably confident (provided we are sure that other factors like
 213             drift are not creeping in).
 214           </p>
 215 <p>
 216             For 10000 observations, the interval is 0.99 to 1.1 - finally a really
 217             convincing + or -1% confidence.
 218           </p>
 219 </div>
 220 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 221 <td align="left"></td>
 222 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 223       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 224       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 225       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 226         Distributed under the Boost Software License, Version 1.0. (See accompanying
 227         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>)
 228       </p>
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