Cheating or Coincidence? Statistical Method Employing the Principle of Maximum Entropy for Judging Whether a Student Has Committed Plagiarism
Author(s)
M. P. Silverman
ABSTRACT
Elements of correspondence (“coincidences”) between a student’s solutions to an assigned set of quantitative problems and the solutions manual for the course textbook may suggest that the stu-dent copied the work from an illicit source. Plagiarism of this kind, which occurs primarily in fields such as the natural sciences, engineering, and mathematics, is often difficult to establish. This paper derives an expression for the probability that alleged coincidences in a student’s paper could be attributable to pure chance. The analysis employs the Principle of Maximum Entropy (PME), which, mathematically, is a variational procedure requiring maximization of the Shannon-Jaynes entropy function augmented by the completeness relation for probabilities and known information in the form of expectation values. The virtue of the PME as a general method of inferential reasoning is that it generates the most objective (i.e. least biased) probability distribution consistent with the given information. Numerical examination of test cases for a range of plausible conditions can yield outcomes that tend to exonerate a student who otherwise might be wrongfully judged guilty of cheating by adjudicators unfamiliar with the surprising properties of random processes.
Elements of correspondence (“coincidences”) between a student’s solutions to an assigned set of quantitative problems and the solutions manual for the course textbook may suggest that the stu-dent copied the work from an illicit source. Plagiarism of this kind, which occurs primarily in fields such as the natural sciences, engineering, and mathematics, is often difficult to establish. This paper derives an expression for the probability that alleged coincidences in a student’s paper could be attributable to pure chance. The analysis employs the Principle of Maximum Entropy (PME), which, mathematically, is a variational procedure requiring maximization of the Shannon-Jaynes entropy function augmented by the completeness relation for probabilities and known information in the form of expectation values. The virtue of the PME as a general method of inferential reasoning is that it generates the most objective (i.e. least biased) probability distribution consistent with the given information. Numerical examination of test cases for a range of plausible conditions can yield outcomes that tend to exonerate a student who otherwise might be wrongfully judged guilty of cheating by adjudicators unfamiliar with the surprising properties of random processes.
Cite this paper
Silverman, M. (2015) Cheating or Coincidence? Statistical Method Employing the Principle of Maximum Entropy for Judging Whether a Student Has Committed Plagiarism. Open Journal of Statistics, 5, 143-157. doi: 10.4236/ojs.2015.52018.
Silverman, M. (2015) Cheating or Coincidence? Statistical Method Employing the Principle of Maximum Entropy for Judging Whether a Student Has Committed Plagiarism. Open Journal of Statistics, 5, 143-157. doi: 10.4236/ojs.2015.52018.
References
[1] Kelly, T. (2011) College Plagiarism Reaches All Time High: Pew Study. Huffington Post (1 September 2011).
http://www.huffingtonpost.com/2011/09/01/college-plagiarism-all-ti_n_944252.html [2] Blum, S.D. (2009) Academic Integrity and Student Plagiarism: A Question of Education, Not Ethics. The Chronicle of Higher Education (20 February 2009).
http://chronicle.com/article/Academic-IntegrityStud/32323/ [3] Odom, T.W. (2015) Cheating in Schools in Rampant. But There’s an Easy Fix. Washington Post (13 March 2015).
http://www.washingtonpost.com/posteverything/wp/2015/03/13/cheating-in-schools-is-rampant-but-theres-an-easy-fix/ [4] Parker, K., Lenhart, A. and Moore, K. (2011) The Digital Revolution and Higher Education. Pew Research Center: Internet, Science & Tech (28 August 2011).
http://www.pewinternet.org/2011/08/28/the-digital-revolution-and-higher-education/ [5] Olafson, L., Schraw, G. and Kehrwald, N. (2014) Academic Dishonesty: Behaviors, Sanctions, and Retention of Adjudicated College Students. Journal of College Student Development, 55, 661-674.
http://dx.doi.org/10.1353/csd.2014.0066 [6] Vandehey, M., Diekhoff, G. and LaBeff, E. (2007) College Cheating: A Twenty-Year Follow-Up and the Addition of an Honor Code. Journal of College Student Development, 48, 468-480.
http://dx.doi.org/10.1353/csd.2007.0043 [7] McCabe, D.L. (2005) Cheating among College and University Students: A North American Perspective. International Journal for Educational Integrity, No. 1.
http://www.ojs.unisa.edu.au/index.php/IJEI/article/view/14 [8] Scanlon, P.M. and Neumann, D.R. (2002) Internet Plagiarism among College Students. Journal of College Student Development, 43,374-385. [9] Blum, S.D. (2009) My Word!: Plagiarism and College Culture. Cornell University Press, Ithaca. [10] Grant, B. (2015) HIV Scientist Pleads Guilty to Fraud. The Scientist (26 February 2015).
http://www.the-scientist.com/?articles.view/articleNo/42285/title/HIV-Scientist-Pleads-Guilty-to-Fraud/ [11] Mood, A.M., Graybill, F.A. and Boes, D.C. (1974) Introduction to the Theory of Statistics. 3rd Edition, McGraw-Hill, New York, 405-406. [12] Fenton, N. (2011) Improve Statistics in Court. Nature, 479, 36-37.
http://dx.doi.org/10.1038/479036a [13] Kendall, M.G. and Stuart, A. (1963) The Advanced Theory of Statistics, Volume 1: Distribution Theory. 2nd Edition, Hafner, New York, 198-201. [14] Hill, R. (2005) Reflections on the Cot Death Cases. Significance, 2, 13-16.
http://dx.doi.org/10.1111/j.1740-9713.2005.00077.x [15] Gigerenzer, G. (2003) Reckoning with Risk: Learning to Live with Uncertainty. Chapter 8, Penguin E-Book. [16] Gardner-Medwin, T. (2005) What Probability Should a Jury Address? Significance, 2, 9-12.
http://dx.doi.org/10.1111/j.1740-9713.2005.00076.x [17] Diaconis, P. and Mosteller, F. (1989) Methods for Studying Coincidences. Journal of the American Statistical Association, 84, 853-861.
http://dx.doi.org/10.1080/01621459.1989.10478847 [18] Fisher, R.A. (1924) A Method of Scoring Coincidences in Tests with Playing Cards. Proceedings of the Society for Psychical Research, 34, 181-185. [19] Parzen, E. (1960) Modern Probability Theory and Its Applications. Wiley, Hoboken, 46-47. [20] Fisher, R.A. (1929) Tests of Significance in Harmonic Analysis. Proceedings of the Royal Society A, 125, 54-59.
http://dx.doi.org/10.1098/rspa.1929.0151 [21] Silverman, M.P. and Strange, W. (2009) Search for Correlated Fluctuations in the β+ Decay of Na-22. Europhysics Letters, 87, Article ID: 32001.
http://dx.doi.org/10.1209/0295-5075/87/32001 [22] Silverman, M.P. (2014) A Certain Uncertainty: Nature’s Random Ways. Cambridge University Press, Cambridge, 157-160, 565-567. [23] Uspensky, J.V. (1937) Introduction to Mathematical Probability. McGraw-Hill, New York, 19-20. [24] Shannon, C.E. (1948) A Mathematical Theory of Communication. Bell System Technical Journal, 27, 379-423.
http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x [25] Greiner, W., Neise, L. and Stocker, H. (1987) Thermodynamics and Statistical Mechanics. Springer, Berlin, 149-152. [26] Jaynes, E.T. (1957) Information Theory and Statistical Mechanics. Physical Review, 106, 620-630.
http://dx.doi.org/10.1103/PhysRev.106.620 Jaynes, E.T. (1957) Information Theory and Statistical Mechanics, II. Physical Review, 108, 171-190.
http://dx.doi.org/10.1103/PhysRev.108.171 [27] Jaynes, E.T. (1968) Prior Probabilities. IEEE Transactions on Systems Science and Cybernetics, SSC-4, 227-241. Reprinted in Rosenkrantz, R.D. and Jaynes, E.T., Eds. (1989) Papers on Probability, Statistics, and Statistical Physics. Kluwer, 116-130. [28] Fisher, R.A. (1925) Theory of Statistical Estimation. Proceedings of the Cambridge Philosophical Society, 22, 700- 725. [29] Keynes, J.M. (1962) The Principle of Indifference. A Treatise on Probability. Chapter IV, Harper Torchbook, New York, 41-64. [30] Silverman, M.P., Strange, S., Silverman, C.R. and Lipscombe, T.C. (1999) On the Run: Unexpected Outcomes of Random Events. The Physics Teacher, 37, 218-225.
htttp://dx.doi.org/10.1119/1.880232
[1] Kelly, T. (2011) College Plagiarism Reaches All Time High: Pew Study. Huffington Post (1 September 2011).
http://www.huffingtonpost.com/2011/09/01/college-plagiarism-all-ti_n_944252.html [2] Blum, S.D. (2009) Academic Integrity and Student Plagiarism: A Question of Education, Not Ethics. The Chronicle of Higher Education (20 February 2009).
http://chronicle.com/article/Academic-IntegrityStud/32323/ [3] Odom, T.W. (2015) Cheating in Schools in Rampant. But There’s an Easy Fix. Washington Post (13 March 2015).
http://www.washingtonpost.com/posteverything/wp/2015/03/13/cheating-in-schools-is-rampant-but-theres-an-easy-fix/ [4] Parker, K., Lenhart, A. and Moore, K. (2011) The Digital Revolution and Higher Education. Pew Research Center: Internet, Science & Tech (28 August 2011).
http://www.pewinternet.org/2011/08/28/the-digital-revolution-and-higher-education/ [5] Olafson, L., Schraw, G. and Kehrwald, N. (2014) Academic Dishonesty: Behaviors, Sanctions, and Retention of Adjudicated College Students. Journal of College Student Development, 55, 661-674.
http://dx.doi.org/10.1353/csd.2014.0066 [6] Vandehey, M., Diekhoff, G. and LaBeff, E. (2007) College Cheating: A Twenty-Year Follow-Up and the Addition of an Honor Code. Journal of College Student Development, 48, 468-480.
http://dx.doi.org/10.1353/csd.2007.0043 [7] McCabe, D.L. (2005) Cheating among College and University Students: A North American Perspective. International Journal for Educational Integrity, No. 1.
http://www.ojs.unisa.edu.au/index.php/IJEI/article/view/14 [8] Scanlon, P.M. and Neumann, D.R. (2002) Internet Plagiarism among College Students. Journal of College Student Development, 43,374-385. [9] Blum, S.D. (2009) My Word!: Plagiarism and College Culture. Cornell University Press, Ithaca. [10] Grant, B. (2015) HIV Scientist Pleads Guilty to Fraud. The Scientist (26 February 2015).
http://www.the-scientist.com/?articles.view/articleNo/42285/title/HIV-Scientist-Pleads-Guilty-to-Fraud/ [11] Mood, A.M., Graybill, F.A. and Boes, D.C. (1974) Introduction to the Theory of Statistics. 3rd Edition, McGraw-Hill, New York, 405-406. [12] Fenton, N. (2011) Improve Statistics in Court. Nature, 479, 36-37.
http://dx.doi.org/10.1038/479036a [13] Kendall, M.G. and Stuart, A. (1963) The Advanced Theory of Statistics, Volume 1: Distribution Theory. 2nd Edition, Hafner, New York, 198-201. [14] Hill, R. (2005) Reflections on the Cot Death Cases. Significance, 2, 13-16.
http://dx.doi.org/10.1111/j.1740-9713.2005.00077.x [15] Gigerenzer, G. (2003) Reckoning with Risk: Learning to Live with Uncertainty. Chapter 8, Penguin E-Book. [16] Gardner-Medwin, T. (2005) What Probability Should a Jury Address? Significance, 2, 9-12.
http://dx.doi.org/10.1111/j.1740-9713.2005.00076.x [17] Diaconis, P. and Mosteller, F. (1989) Methods for Studying Coincidences. Journal of the American Statistical Association, 84, 853-861.
http://dx.doi.org/10.1080/01621459.1989.10478847 [18] Fisher, R.A. (1924) A Method of Scoring Coincidences in Tests with Playing Cards. Proceedings of the Society for Psychical Research, 34, 181-185. [19] Parzen, E. (1960) Modern Probability Theory and Its Applications. Wiley, Hoboken, 46-47. [20] Fisher, R.A. (1929) Tests of Significance in Harmonic Analysis. Proceedings of the Royal Society A, 125, 54-59.
http://dx.doi.org/10.1098/rspa.1929.0151 [21] Silverman, M.P. and Strange, W. (2009) Search for Correlated Fluctuations in the β+ Decay of Na-22. Europhysics Letters, 87, Article ID: 32001.
http://dx.doi.org/10.1209/0295-5075/87/32001 [22] Silverman, M.P. (2014) A Certain Uncertainty: Nature’s Random Ways. Cambridge University Press, Cambridge, 157-160, 565-567. [23] Uspensky, J.V. (1937) Introduction to Mathematical Probability. McGraw-Hill, New York, 19-20. [24] Shannon, C.E. (1948) A Mathematical Theory of Communication. Bell System Technical Journal, 27, 379-423.
http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x [25] Greiner, W., Neise, L. and Stocker, H. (1987) Thermodynamics and Statistical Mechanics. Springer, Berlin, 149-152. [26] Jaynes, E.T. (1957) Information Theory and Statistical Mechanics. Physical Review, 106, 620-630.
http://dx.doi.org/10.1103/PhysRev.106.620 Jaynes, E.T. (1957) Information Theory and Statistical Mechanics, II. Physical Review, 108, 171-190.
http://dx.doi.org/10.1103/PhysRev.108.171 [27] Jaynes, E.T. (1968) Prior Probabilities. IEEE Transactions on Systems Science and Cybernetics, SSC-4, 227-241. Reprinted in Rosenkrantz, R.D. and Jaynes, E.T., Eds. (1989) Papers on Probability, Statistics, and Statistical Physics. Kluwer, 116-130. [28] Fisher, R.A. (1925) Theory of Statistical Estimation. Proceedings of the Cambridge Philosophical Society, 22, 700- 725. [29] Keynes, J.M. (1962) The Principle of Indifference. A Treatise on Probability. Chapter IV, Harper Torchbook, New York, 41-64. [30] Silverman, M.P., Strange, S., Silverman, C.R. and Lipscombe, T.C. (1999) On the Run: Unexpected Outcomes of Random Events. The Physics Teacher, 37, 218-225.
htttp://dx.doi.org/10.1119/1.880232