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- // 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.
- */
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