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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.stat_tut.weg.st_eg.tut_mean_intervals"></a><a class="link" href="tut_mean_intervals.html" title="Calculating confidence intervals on the mean with the Students-t distribution">Calculating
  28           confidence intervals on the mean with the Students-t distribution</a>
  29 </h5></div></div></div>
  30 <p>
  31             Let's say you have a sample mean, you may wish to know what confidence
  32             intervals you can place on that mean. Colloquially: "I want an interval
  33             that I can be P% sure contains the true mean". (On a technical point,
  34             note that the interval either contains the true mean or it does not:
  35             the meaning of the confidence level is subtly different from this colloquialism.
  36             More background information can be found on the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">NIST
  37             site</a>).
  38           </p>
  39 <p>
  40             The formula for the interval can be expressed as:
  41           </p>
  42 <p>
  43             <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial4.svg"></span>
  44           </p>
  45 <p>
  46             Where, <span class="emphasis"><em>Y<sub>s</sub></em></span> is the sample mean, <span class="emphasis"><em>s</em></span>
  47             is the sample standard deviation, <span class="emphasis"><em>N</em></span> is the sample
  48             size, /&#945;/ is the desired significance level and <span class="emphasis"><em>t<sub>(&#945;/2,N-1)</sub></em></span>
  49             is the upper critical value of the Students-t distribution with <span class="emphasis"><em>N-1</em></span>
  50             degrees of freedom.
  51           </p>
  52 <div class="note"><table border="0" summary="Note">
  53 <tr>
  54 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
  55 <th align="left">Note</th>
  56 </tr>
  57 <tr><td align="left" valign="top">
  58 <p>
  59               The quantity &#945; &#160; is the maximum acceptable risk of falsely rejecting the
  60               null-hypothesis. The smaller the value of &#945; the greater the strength
  61               of the test.
  62             </p>
  63 <p>
  64               The confidence level of the test is defined as 1 - &#945;, and often expressed
  65               as a percentage. So for example a significance level of 0.05, is equivalent
  66               to a 95% confidence level. Refer to <a href="http://www.itl.nist.gov/div898/handbook/prc/section1/prc14.htm" target="_top">"What
  67               are confidence intervals?"</a> in <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
  68               e-Handbook of Statistical Methods.</a> for more information.
  69             </p>
  70 </td></tr>
  71 </table></div>
  72 <div class="note"><table border="0" summary="Note">
  73 <tr>
  74 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
  75 <th align="left">Note</th>
  76 </tr>
  77 <tr><td align="left" valign="top"><p>
  78               The usual assumptions of <a href="http://en.wikipedia.org/wiki/Independent_and_identically-distributed_random_variables" target="_top">independent
  79               and identically distributed (i.i.d.)</a> variables and <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">normal
  80               distribution</a> of course apply here, as they do in other examples.
  81             </p></td></tr>
  82 </table></div>
  83 <p>
  84             From the formula, it should be clear that:
  85           </p>
  86 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  87 <li class="listitem">
  88                 The width of the confidence interval decreases as the sample size
  89                 increases.
  90               </li>
  91 <li class="listitem">
  92                 The width increases as the standard deviation increases.
  93               </li>
  94 <li class="listitem">
  95                 The width increases as the <span class="emphasis"><em>confidence level increases</em></span>
  96                 (0.5 towards 0.99999 - stronger).
  97               </li>
  98 <li class="listitem">
  99                 The width increases as the <span class="emphasis"><em>significance level decreases</em></span>
 100                 (0.5 towards 0.00000...01 - stronger).
 101               </li>
 102 </ul></div>
 103 <p>
 104             The following example code is taken from the example program <a href="../../../../../../example/students_t_single_sample.cpp" target="_top">students_t_single_sample.cpp</a>.
 105           </p>
 106 <p>
 107             We'll begin by defining a procedure to calculate intervals for various
 108             confidence levels; the procedure will print these out as a table:
 109           </p>
 110 <pre class="programlisting"><span class="comment">// Needed includes:</span>
 111 <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">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 112 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 113 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
 114 <span class="comment">// Bring everything into global namespace for ease of use:</span>
 115 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
 116 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
 117
 118 <span class="keyword">void</span> <span class="identifier">confidence_limits_on_mean</span><span class="special">(</span>
 119    <span class="keyword">double</span> <span class="identifier">Sm</span><span class="special">,</span>           <span class="comment">// Sm = Sample Mean.</span>
 120    <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span>           <span class="comment">// Sd = Sample Standard Deviation.</span>
 121    <span class="keyword">unsigned</span> <span class="identifier">Sn</span><span class="special">)</span>         <span class="comment">// Sn = Sample Size.</span>
 122 <span class="special">{</span>
 123    <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
 124    <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
 125
 126    <span class="comment">// Print out general info:</span>
 127    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
 128       <span class="string">"__________________________________\n"</span>
 129       <span class="string">"2-Sided Confidence Limits For Mean\n"</span>
 130       <span class="string">"__________________________________\n\n"</span><span class="special">;</span>
 131    <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>
 132    <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">Sn</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 133    <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">"Mean"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sm</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
 134    <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>
 135 </pre>
 136 <p>
 137             We'll define a table of significance/risk levels for which we'll compute
 138             intervals:
 139           </p>
 140 <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>
 141 </pre>
 142 <p>
 143             Note that these are the complements of the confidence/probability levels:
 144             0.5, 0.75, 0.9 .. 0.99999).
 145           </p>
 146 <p>
 147             Next we'll declare the distribution object we'll need, note that the
 148             <span class="emphasis"><em>degrees of freedom</em></span> parameter is the sample size
 149             less one:
 150           </p>
 151 <pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">Sn</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
 152 </pre>
 153 <p>
 154             Most of what follows in the program is pretty printing, so let's focus
 155             on the calculation of the interval. First we need the t-statistic, computed
 156             using the <span class="emphasis"><em>quantile</em></span> function and our significance
 157             level. Note that since the significance levels are the complement of
 158             the probability, we have to wrap the arguments in a call to <span class="emphasis"><em>complement(...)</em></span>:
 159           </p>
 160 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">T</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>
 161 </pre>
 162 <p>
 163             Note that alpha was divided by two, since we'll be calculating both the
 164             upper and lower bounds: had we been interested in a single sided interval
 165             then we would have omitted this step.
 166           </p>
 167 <p>
 168             Now to complete the picture, we'll get the (one-sided) width of the interval
 169             from the t-statistic by multiplying by the standard deviation, and dividing
 170             by the square root of the sample size:
 171           </p>
 172 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">w</span> <span class="special">=</span> <span class="identifier">T</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">double</span><span class="special">(</span><span class="identifier">Sn</span><span class="special">));</span>
 173 </pre>
 174 <p>
 175             The two-sided interval is then the sample mean plus and minus this width.
 176           </p>
 177 <p>
 178             And apart from some more pretty-printing that completes the procedure.
 179           </p>
 180 <p>
 181             Let's take a look at some sample output, first using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm" target="_top">Heat
 182             flow data</a> from the NIST site. The data set was collected by Bob
 183             Zarr of NIST in January, 1990 from a heat flow meter calibration and
 184             stability analysis. The corresponding dataplot output for this test can
 185             be found in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">section
 186             3.5.2</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
 187             e-Handbook of Statistical Methods.</a>.
 188           </p>
 189 <pre class="programlisting">   __________________________________
 190    2-Sided Confidence Limits For Mean
 191    __________________________________
 192
 193    Number of Observations                  =  195
 194    Mean                                    =  9.26146
 195    Standard Deviation                      =  0.02278881
 196
 197
 198    ___________________________________________________________________
 199    Confidence       T           Interval          Lower          Upper
 200     Value (%)     Value          Width            Limit          Limit
 201    ___________________________________________________________________
 202        50.000     0.676       1.103e-003        9.26036        9.26256
 203        75.000     1.154       1.883e-003        9.25958        9.26334
 204        90.000     1.653       2.697e-003        9.25876        9.26416
 205        95.000     1.972       3.219e-003        9.25824        9.26468
 206        99.000     2.601       4.245e-003        9.25721        9.26571
 207        99.900     3.341       5.453e-003        9.25601        9.26691
 208        99.990     3.973       6.484e-003        9.25498        9.26794
 209        99.999     4.537       7.404e-003        9.25406        9.26886
 210 </pre>
 211 <p>
 212             As you can see the large sample size (195) and small standard deviation
 213             (0.023) have combined to give very small intervals, indeed we can be
 214             very confident that the true mean is 9.2.
 215           </p>
 216 <p>
 217             For comparison the next example data output is taken from <span class="emphasis"><em>P.K.Hou,
 218             O. W. Lau &amp; M.C. Wong, Analyst (1983) vol. 108, p 64. and from Statistics
 219             for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. Miller and J.
 220             N. Miller, Ellis Horwood ISBN 0 13 0309907.</em></span> The values result
 221             from the determination of mercury by cold-vapour atomic absorption.
 222           </p>
 223 <pre class="programlisting">   __________________________________
 224    2-Sided Confidence Limits For Mean
 225    __________________________________
 226
 227    Number of Observations                  =  3
 228    Mean                                    =  37.8000000
 229    Standard Deviation                      =  0.9643650
 230
 231
 232    ___________________________________________________________________
 233    Confidence       T           Interval          Lower          Upper
 234     Value (%)     Value          Width            Limit          Limit
 235    ___________________________________________________________________
 236        50.000     0.816            0.455       37.34539       38.25461
 237        75.000     1.604            0.893       36.90717       38.69283
 238        90.000     2.920            1.626       36.17422       39.42578
 239        95.000     4.303            2.396       35.40438       40.19562
 240        99.000     9.925            5.526       32.27408       43.32592
 241        99.900    31.599           17.594       20.20639       55.39361
 242        99.990    99.992           55.673      -17.87346       93.47346
 243        99.999   316.225          176.067     -138.26683      213.86683
 244 </pre>
 245 <p>
 246             This time the fact that there are only three measurements leads to much
 247             wider intervals, indeed such large intervals that it's hard to be very
 248             confident in the location of the mean.
 249           </p>
 250 </div>
 251 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 252 <td align="left"></td>
 253 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 254       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 255       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 256       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 257         Distributed under the Boost Software License, Version 1.0. (See accompanying
 258         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>)
 259       </p>
 260 </div></td>
 261 </tr></table>
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