Significados de Referência da Estatística Qui-quadrado: um Olhar Histórico-epistemológico
Resumo
Este artigo apresenta um estudo histórico-epistemológico sobre a estatístico Qui-quadrado. Para tanto, são utilizadas algumas noções teórico-metodológicas da Abordagem Onto-Semiótica (EOS) do conhecimento e do ensino matemático, que nos permitiram identificar quatro problemas que têm sido fundamentais para a evolução da estatística Qui-quadrado: teste de adequação, teste de independência, teste de homogeneidade e distribuição. Além disso, nas práticas matemático-estatísticas realizadas para resolver cada um destes problemas, foram identificados vários significados da estatística Qui-quadrado, o que permitirá estabelecer critérios epistemológicos que permitem, por um lado, propor níveis progressivos (do informal ao formal) do raciocínio inferencial para a referida estatística; e por outro lado, desenhar tarefas que visem promover a compreensão dos diversos significados do Qui-quadrado.
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Referências
BARNARD, George Alfred. 1992. Introduction to Pearson (1900) On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. In: KOTZ, S.; JOHNSON, N.L. (Eds.), Breakthroughs in Statistics, v. 2. New York: Springer, 1992. p. 1-10.
BATANERO, Carmen. Del análisis de datos a la inferencia: Reflexiones sobre la formación del razonamiento estadístico. Cuadernos de Investigación y Formación en Educación Matemática, v. 8, n. 11, p. 227-291, dic. 2013.
BAKKER, Arthur.; GRAVEMEIJER, Koeno. Learning to reason about distribution. In: BEN-ZVI, Dani; GARFIELD, Joan. (Eds.), The challenge of developing statistical literacy, reasoning, and thinking. Dordrecht: Kluwer Academic Publishers, 2004. p. 147-168.
COHEN, Louis.; MANION, Lawrence.; MORRISON, Keith. Research methods in education. London and New York: Routledge, 2011.
DEPAOLO, Concetta A.; ROBINSON, David F.; JACOBS, Aimee. Café Data 2.0: New Data From a New and Improved Café. Journal of Statistics Education, v. 24, n. 2, p. 85-103, jun. 2016.
DEVORE, Jay L. Probability & Statistics for Engineering and the Sciences. 7. ed. Mexico: Cengage Learning, 2008.
DOERR, Helen M.; DELMAS, Robert; MAKAR, Katie. A modeling approach to the development of students’ informal inferential reasoning. Statistics Education Research Journal, Auckland, v. 16, n. 2, p. 86-115, nov. 2017.
FELLERS, Pamela S.; KUIPER, Shonda. Introducing Undergraduates to Concepts of Survey Data Analysis. Journal of Statistics Education, v. 28, n. 1, p. 18-24, feb. 2020.
FISHER, Ronald Aylmer. On the interpretation of χ^2 from contingency tables, and the calculation of P. Journal of the Royal Statistical Society. v. 85, n. 2, p. 87-94, jan. 1922.
FISHER, Ronald Aylmer. Statistical methods for research workers. 5. ed. Edinburgh: Oliver and Boyd, 1934.
GALTON, Francis. IV. Statistics by intercomparison, with remarks on the law of frequency of error. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, v. 49, n. 322, p. 33-46, 1875.
GALTON, Francis. The Application of a Graphic Method to Fallible Measures. Journal of the Statistical Society of London. p. 262-271, jun. 1885.
GIBBS, Alison L.; GOOSSENS, Emery T. The Evidence for Efficacy of HPV Vaccines: Investigations in Categorical Data Analysis. Journal of Statistics Education. v. 21, n. 3, 2013.
GODINO, Juan Díaz; BATANERO, Carmen. Significado institucional y personal de los objetos matemáticos. Recherches en Didactique des Mathématiques, Grenoble, v. 14, n. 3, p. 325-355. 1994.
GODINO, Juan Díaz; BATANERO, Carmen; FONT, Vicenç. The onto-semiotic approach to research in mathematics education. ZDM, Berlin, v. 39, n. 1-2, p. 127-135, mar. 2007.
GODINO, Juan Díaz; BATANERO, Carmen; FONT, Vicenç. The onto-semiotic approach: implications for the prescriptive character of didactics. For the Learning of Mathematics, Vancouver, v. 39, n. 1, p. 37-42. 2019.
GODINO, Juan Díaz; FONT, Vicenç; WILHELMI, Miguel R.; LURDUY, Orlando. Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects. Educational Studies in Mathematics. v.77, p. 247-265, jul. 2011.
GREENWOOD, Major; YULE, George Undy. The statistics of anti-typhoid and anti-cholera inoculations, and the interpretation of such statistics in general. Proceedings of the Royal Society of Medicine, v. 8, p. 113-194, jun. 1915.
HALD, Anders. A history of parametric statistical inference from Bernoulli to Fisher, 1713-1935. Springer Science & Business Media, 2007.
HELMERT, Friedrich Robert. Über die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen. Z. Math. und Physik, v. 21, p. 192-218, 1876.
HEYDE, Chris; SENETA, Eugene. I. J. Bienaymé: Statistical Theory Anticipated. New York: Springer, 1977.
HOEKSTRA, Rink. Risk as an Explanatory Factor for Researchers' Inferential Interpretations. The Mathematics Enthusiast, v. 12, n. 1, p. 103-112, 2015.
JACOB, Bridgette. L.; DOERR, Helen. M. Statistical Reasoning with the sampling distribution. In: MAKAR, Katie; DE SOUSA, B.; GOULD, R. (Eds.), Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics. Voorburg, The Netherlands: International Statistical Institute, 2014. p. 1-6.
LANCASTER, H.O. Forerunners of the Pearson χ^2. Australian Journal of Statistics, n. 8, n. 3, p. 117-26, nov. 1966.
LEIGH, M.; DOWLING Alix .D. Water Taste Test Data. Journal of Statistics Education, v. 18, n. 1, 2010.
LUGO-ARMENTA, Jesús Guadalupe; PINO-FAN, Luis Roberto. Niveles de Razonamiento Inferencial para el Estadístico t-Student. Bolema: Boletim de Educação Matemática, v. 35, n- 71, en prensa, 2021.
MAKAR, Katie; RUBIN, Andee. A framework for thinking about informal statistical inference. Statistics Education Research Journal, Auckland, v. 8, n. 1, p. 82–105, may. 2009.
MAKAR, Katie.; RUBIN, Andee. Learning about statistical inference. In: BEN-ZVI, D.; MAKAR, K.; GARFIELD, J. (Eds.), International handbook of research in statistics education. Switzerland: Springer International. 2018. p. 261-294.
MAGNELLO, M. Eileen. Karl Pearson, paper on the chi square goodness of fit test (1900). In Landmark Writings in Western Mathematics 1640-1940. Elsevier Science, p. 724-731, 2005.
PEARSON, Karl. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. v. 50, n. 5, p. 157-175, 1900.
PEARSON, Karl. On the theory of contingency and its relation to association and normal correlation. Dulau and Company,1904.
PEARSON, Karl. On the probability that two independent distributions of frequency are really samples from the same population. Biometrika, v. 50, n. ½, p. 250-254, jul. 1911.
PEARSON, Karl. Tables for Statisticians and Biomnetricians. Cambridge: University Press, 1914.
PFANNKUCH, Maxine.; ARNOLD, Pip; WILD, Chris. J. What I see is not quite the way it really is: Students’ emergent reasoning about sampling variability. Educational Studies in Mathematics, New York, v. 88, n. 3, p. 343-360, mar. 2015.
PINO-FAN, Luis Roberto; FONT, Vicenç; GORDILLO, Wilson; LARIOS, Victor; BREDA, Adriana. Analysis of the meanings of the antiderivative used by students of the first engineering courses. International Journal of Science and Mathematics Education, v. 16, n. 6, p. 1091-1113, 2018. DOI: https://doi.org/10.1007/s10763-017-9826-2.
PINO-FAN, Luis Roberto; GODINO, Juan Díaz; FONT, Vicenç. Epistemic Facet of the Didactic-Mathematics Knowledge About The Derivative. Educação Matemática Pesquisa, v. 13, n. 1, p. 141-178, 2011.
REABURN, Robyn. Introductory statistics course tertiary students' understanding of p-values. Statistics Education Research Journal, v. 13, n. 1, p. 53-65, may 2014.
RIEMER, Wolfgang; SEEBACH, Günter. Rolling pencils - inferential statistics in the pencil case. In Understanding more mathematics with GeoGebra. Heidelberg: Springer Spektrum, p. 91-105, 2014.
ROCHOWICZ, John A. Bootstrapping analysis, inferential statistics and EXCEL. Spreadsheets in Education (eJSiE), v. 4, n. 3, p. 1-23, 2010.
ROSSMAN, Allan J. Reasoning about Informal Statistical Inference: One Statistician’s View. Statistics Education Research Journal, Auckland, v. 7, n. 2, p. 5-19, nov. 2008.
SALDANHA, Luis A.; THOMPSON, Patrick W. Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, v. 51, n. 3, p. 257-270, nov. 2002.
SEIER, Edith. An Early Start on Inference. In I. GAL (ed.), 9th International Conference on Teaching Statistics. Conference held in Flagstaff, Arizona, United States of America, 2014.
SNEDECOR, G.; IRWIN, M.R. On the chi-square test for homogeneity. Iowa State Coll. J. Sci, v. 8, n. 1, p. 75-81, 1933.
TARLOW, Kevin R. Teaching principles of inference with ANOVA. Teaching Statistics, v. 38, n. 1, p. 16-21, 2016.
WIJAYATUNGA, Priyantha. Geometric view on Pearson's correlation coefficient and a generalization of it to non-linear dependencies, Ratio Mathematica, v. 30, p. 3-21, 2016.
WOODARD, Victoria; LEE, Hollylynne; WOODARD, Roger. Writing Assignments to Assess Statistical Thinking, Journal of Statistics Education, v. 28, n. 1, p. 32-44, 2020.
YATES, Frank. Contingency tables involving small numbers and the χ^2 test. Supplement to the Journal of the Royal Statistical Society, v. 1, n. 2, p. 217-35, 1934.
YULE, George Undy. On the association of attributes in statistics: with illustrations from the material of the childhood society, &c. Philosophical Transactions of the Royal Society of London. Series A, v. 194, p. 257-319, 1900.
ZIEFFLER, Andrew; GARFIELD, Joan; DELMAS, Robert; READING, Chris. A framework to support research on informal inferential reasoning. Statistics Education Research Journal, Auckland, v. 7, n. 2, p. 40–58, nov. (2008).
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