text: // students_t_example3.cpp
  // Copyright Paul A. Bristow 2006, 2007.
  
  // Use, modification and distribution are subject to the
  // Boost Software License, Version 1.0.
  // (See accompanying file LICENSE_1_0.txt
  // or copy at http://www.boost.org/LICENSE_1_0.txt)
  
  // Example 3 of using Student's t.
  
  // A general guide to Student's t is at
  // http://en.wikipedia.org/wiki/Student's_t-test
  // (and many other elementary and advanced statistics texts).
  // It says:
  // The t statistic was invented by William Sealy Gosset
  // for cheaply monitoring the quality of beer brews.
  // "Student" was his pen name.
  // Gosset was statistician for Guinness brewery in Dublin, Ireland,
  // hired due to Claude Guinness's innovative policy of recruiting the
  // best graduates from Oxford and Cambridge for applying biochemistry
  // and statistics to Guinness's industrial processes.
  // Gosset published the t test in Biometrika in 1908,
  // but was forced to use a pen name by his employer who regarded the fact
  // that they were using statistics as a trade secret.
  // In fact, Gosset's identity was unknown not only to fellow statisticians
  // but to his employer - the company insisted on the pseudonym
  // so that it could turn a blind eye to the breach of its rules.
  
  // The Students't distribution function is described at
  // http://en.wikipedia.org/wiki/Student%27s_t_distribution
  
  #include <boost/math/distributions/students_t.hpp>
     using boost::math::students_t;  // Probability of students_t(df, t).
  
  #include <iostream>
     using std::cout;    using std::endl;
  #include <iomanip>
     using std::setprecision;    using std::setw;
  #include <cmath>
     using std::sqrt;
  
  // This example of a two-sided test is from:
  // B. M. Smith & M. B. Griffiths, Analyst, 1982, 107, 253,
  // from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 58-59
  // J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907
  
  // Concentrations of lead (ug/l) determined by two different methods
  // for each of four test portions,
  // the concentration of each portion is significantly different,
  // the values may NOT be pooled.
  // (Called a 'paired test' by Miller and Miller
  // because each portion analysed has a different concentration.)
  
  // Portion  Wet oxidation Direct Extraction
  //   1           71            76
  //   2           61            68
  //   3           50            48
  //   4           60            57
  
  const int portions = 4;
  const int methods = 2;
  float data [portions][methods] = {{71, 76}, {61,68}, {50, 48}, {60, 57}};
  float diffs[portions];
  
  int main()
  {
     cout << "Example3 using Student's t function. " << endl;
     float mean_diff = 0.f;
     cout << "\n""Portion  wet_oxidation  Direct_extraction  difference" << endl;
     for (int portion = 0; portion < portions; portion++)
     { // Echo data and differences.
        diffs[portion] = data[portion][0] - data[portion][1];
        mean_diff += diffs[portion];
        cout << setw(4) << portion << ' ' << setw(14) << data[portion][0] << ' ' << setw(18)<< data[portion][1] << ' ' << setw(9) << diffs[portion] << endl;
     }
     mean_diff /= portions;
     cout << "Mean difference = " << mean_diff << endl; // -1.75
  
     float sd_diffs = 0.f;
     for (int portion = 0; portion < portions; portion++)
     { // Calculate standard deviation of differences.
        sd_diffs +=(diffs[portion] - mean_diff) * (diffs[portion] - mean_diff);
     }
     int degrees_of_freedom = portions-1; // Use the n-1 formula.
     sd_diffs /= degrees_of_freedom;
     sd_diffs = sqrt(sd_diffs);
     cout << "Standard deviation of differences = " << sd_diffs << endl; // 4.99166
     // Standard deviation of differences = 4.99166
     double t = mean_diff * sqrt(static_cast<double>(portions))/ sd_diffs; // -0.70117
     cout << "Student's t = " << t << ", if " << degrees_of_freedom << " degrees of freedom." << endl;
     // Student's t = -0.70117, if 3 degrees of freedom.
     cout << "Probability of the means being different is "
      << 2.F * cdf(students_t(degrees_of_freedom), t) << "."<< endl; // 0.266846 * 2 = 0.533692
     // Double the probability because using a 'two-sided test' because
     // mean for 'Wet oxidation' could be either
     // greater OR LESS THAN for 'Direct extraction'.
  
     return 0;
  }  // int main()
  
  /*
  
  Output is:
  
  Example3 using Student's t function.
  Portion  wet_oxidation  Direct_extraction  difference
     0             71                 76        -5
     1             61                 68        -7
     2             50                 48         2
     3             60                 57         3
  Mean difference = -1.75
  Standard deviation of differences = 4.99166
  Student's t = -0.70117, if 3 degrees of freedom.
  Probability of the means being different is 0.533692.
  
  */
  