Journal of Human Sport and Exercise

The power law model applied to the marathon world record

Andrés Bernardo Fernández-Revelles, Eduardo García Mármol



In September 2013 the world record in the marathon men's race was broken. The aim of this study is to apply to the 2013 Berlin Marathon a mathematical model based on the power law that analyses the marks distribution and checks its connection. The results show that the correlations obtained in all the different categories have been very significant, with a result of (r ≥ 0.978; p < 0.000) and a linear determination coefficient of (R2 ≥ 0.969). As a conclusion it could be said that the power law application to the 2013 Berlin Marathon Men's race has been an useful and feasible study, and the connection between the data and the mathematical model has been so accurate.


Ranking; Power law; Marathon; World record; Space-Time distribution


Alvarez-Martinez, R., Martinez-Mekler, G., y Cocho, G. (2011). Order-disorder transition in conflicting dynamics leading to rank-frequency generalized beta distributions. Physica a-Statistical Mechanics and Its Applications, 390(1), 120-130.

Amaral, L. A. N., Scala, A., Barthelemy, M., y Stanley, H. E. (2000).Classes of small-world networks.Proceedings of the National Academy of Sciences of the United States of America, 97(21), 11149-11152.

Campanario, J. M. (2010a). Distribution of changes in impact factors over time.Scientometrics, 84(1), 35-42.

Campanario, J. M. (2010b). Distribution of ranks of {beta}-decay half-lives [physics.gen-ph].arXiv:1011.5390v1.

Campanario, J. M. (2010c). Distribution of Ranks of Articles and Citations in Journals.Journal of the American Society for Information Science and Technology, 61(2), 419-423.

Campanario, J. M. (2010d). Self-Citations That Contribute to the Journal Impact Factor: An Investment-Benefit-Yield Analysis. Journal of the American Society for Information Science and Technology, 61(12), 2575-2580.

Campanario, J. M. (2011a). Empirical study of journal impact factors obtained using the classical two-year citation window versus a five-year citation window. Scientometrics, 87(1), 189-204.

Campanario, J. M. (2011b). Large Increases and Decreases in Journal Impact Factors in Only One Year: The Effect of Journal Self-Citations. Journal of the American Society for Information Science and Technology, 62(2), 230-235.

Carbone, V., y Savaglio, S. (2001). Scaling laws and forecasting in athletic world records. Journal of Sports Sciences, 19(7), 477-484.

Coile, R. C. (1977). Lotkafrequency-distribution of scientific productivity. Journal of the American Society for Information Science, 28(6), 366-370.

del Rio, M. B., Cocho, G., y Naumis, G. G. (2008). Universality in the tail of musical note rank distribution.Physica a-Statistical Mechanics and Its Applications, 387(22), 5552-5560.

Edwards, R., y Collins, L. (2011). Lexical Frequency Profiles and Zipf's Law. Language Learning, 61(1), 1-30.

Egghe, L. (2009a). Mathematical derivation of the impact factor distribution.Journal of Informetrics, 3(4), 290-295.

Egghe, L. (2009b). A Rationale for the Hirsch-Index Rank-Order Distribution and a Comparison With the Impact Factor Rank-Order Distribution.Journal of the American Society for Information Science and Technology, 60(10), 2142-2144.

Egghe, L. (2010a). The distribution of the uncitedness factor and its functional relation with the impact factor.Scientometrics, 83(3), 689-695.

Egghe, L. (2010b). A New Short Proof of Naranan's Theorem, Explaining Lotka's Law and Zipf's Law.Journal of the American Society for Information Science and Technology, 61(12), 2581-2583.

Egghe, L. (2011a). The impact factor rank-order distribution revisited. Scientometrics, 87(3), 683-685.

Egghe, L. (2011b). Mathematical relations of the h-index with other impact measures in a Lotkaian framework. Mathematical and Computer Modelling, 53(5-6), 610-616.

Egghe, L. (2012). Study of rank- and size-frequency functions and their relations in a generalized Naranan framework.Mathematical and Computer Modelling, 55(7-8), 1898-1903.

Egghe, L. (2013). Study of the rank- and size-frequency functions in the case of power law growth of sources and items and proof of Heaps' law. Information Processing y Management, 49(1), 99-107.

Fernández-Revelles, A. B. (2013).Modelomatemático de ley de potenciasaplicado al maratón.RevistaHabilidadMotriz, 41.

Garfield, E. (1980).Bradford law and related statistical patterns.Current Contents(19), 5-12.

Hong, H. S., Ha, M., y Park, H. (2007). Finite-size scaling in complex networks. Physical Review Letters, 98(25).

Joyner, M. J., Ruiz, J. R., y Lucia, A. (2011). Last Word on Viewpoint: The two-hour marathon: Who and when? [Letter]. Journal of Applied Physiology, 110(1), 294-294.

Katz, J. S., y Katz, L. (1999). Power laws and athletic performance.Journal of Sports Sciences, 17(6), 467-476.

Laherrere, J., y Sornette, D. (1998). Stretched exponential distributions in nature and economy: "fat tails" with characteristic scales. European Physical Journal B, 2(4), 525-539.

Lavalette, D. (1996). Facteurd'impact: impartialitéou impuissance? Report, INSERM U350.Orsay, France: Institut Curie-Recherche, Bât, 112, Centre Universitaire, 91405.

Mansilla, R., Koppen, E., Cocho, G., y Miramontes, P. (2007).On the behavior of journal impact factor rank-order distribution.Journal of Informetrics, 1(2), 155-160.

Naumis, G. G., y Cocho, G. (2007). The tails of rank-size distributions due to multiplicative processes: from power laws to stretched exponentials and beta-like functions. New Journal of Physics, 9.

Naumis, G. G., y Cocho, G. (2008). Tail universalities in rank distributions as an algebraic problem: The beta-like function. Physica a-Statistical Mechanics and Its Applications, 387(1), 84-96.

Newman, M. E. J. (2005).Power laws, Pareto distributions and Zipf's law. [Review]. Contemporary Physics, 46(5), 323-351.

Phillips, J. R. (2010, 19-01-2011). Online Curve Fitting and Surface Fitting Web Site Retrieved 30-09-2013, 2013, from

Popescu, I. (2003). On a Zipfs Law extension to impact factors. Glottometrics, 6(83-93).

Savaglio, S., y Carbone, V. (2000). Human performance - Scaling in athletic world records. Nature, 404(6775), 244-244.

SCC-Events. (2013, 30-09-2013). BMW 40 Berlin Marathon Retrieved 30 Septiembre, 2013, from

Waltman, L., y van Eck, N. J. (2009). Some comments on Egghe's derivation of the impact factor distribution. Journal of Informetrics, 3(4), 363-366.


Copyright (c) 2019 Journal of Human Sport and Exercise

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.