History, and students' understanding of variance in statistics

Abstract

In this article we examine students' difficulties in understanding variance and standard deviation in introductory statistics, taking into consideration relevant elements from the historical evolution of these concepts. This approach (i) improved our understanding of students' difficulties concerning these concepts; (ii) underlined some important deficiencies in the usual introductory teaching of the subject; (iii) highlighted significant elements that could enrich introductory teaching.

Notes

¹ Here the teacher used a common argument: using SD instead of variance partially resolves the dimensional problem, but students argued that this was not satisfactory because the SD was obtained through the use of meaningless quantities.

² Students in the first group, who were weak in algebra, encountered more difficulties with this than those in the second.

³ The MLS was initially developed in the treatment of geodesic and astronomical problems at the beginning of the nineteenth century (initially by Legendre (1805), Gauss (1809, 1816, 1821, 1823), Laplace (1810–1812) and others (see Plackett 1972; Stigler 1986, chs 1, 3, 4; Stigler 1999, ch17; Sheynin 2005, ch 9; Hald 2007, chs 6, 7; Kourkoulos and Tzanakis 2008b). However, it was difficult to transfer statistical methods from geodesy and astronomy to biological and, even more, to social sciences. Pioneering work by Galton, Edgenworth, Pearson, Yule and others was necessary to elaborate a new conceptual framework adequate for regression and correlation in these sciences (Stigler 1986, chs 5, 8–10 and 1999 part I; Porter 1986, part 3; Magnello 1998). It is characteristic of the conceptual difficulties of treating such problems that only in 1897, relying on theoretical arguments, did Yule propose a generalised method of linear regression for problems in social sciences based on the MLS (Stigler 1986, chs 9, 10).

⁴ For further discussion on these (and some other) models see Tzanakis and Kourkoulos 2006; Kourkoulos et al. 2006; Kourkoulos and Tzanakis 2008b).

⁵ For a discussion on Legendre's work see Stigler 1986, 11–15, 55–61.

⁶ In 1893–1894, Pearson introduced the terminology ‘standard deviation’ and its definition (David 1995; Magnello 1996, 48).

⁷ Waterson's pioneering work passed essentially unnoticed to his contemporaries (Truesdell 1975; Brush 1976, ch 3)

⁸ It worth mentioning that much earlier Euler (1729, 1782) and Daniel Bernoulli (1738) proposed gas models in which temperatures were proportional to the squared speed of their microscopic constituents (for Euler whirling speed). In 1716, James Hermann proposed a model for gases close to Bernoulli's. Later Herapath (1816, 1821) also proposed a model close to that of Bernoulli, but he considered the ‘truth Temperature’ as proportional to scalar molecular momentum rather than to molecular energy (Truesdell 1975; Brush 1976, ch 2; Brush 1983, I 1.5.)

⁹ Earlier Waterson (1845) had derived a first, partial version of the theorem (but see note 7).

¹⁰ We chose to investigate these models first because, on the one hand, they have important interpretative elements related to variance and its properties, and on the other hand, they require only some elementary physics and so were appropriate for our students (prospective primary school teachers).

¹¹ Analysis of the data (individual interviews, classroom discussion, written assignments, and tests) indicated that, depending on the class, from 59% to 70% of the students understood both models and the interpretation of variance in each context; from 10% to 20% understood only one model; while from 20% to 26% failed to understand the models and/or coordinate their elements, hence, to interpret variance correctly. It is worth mentioning that all but one of the students in the third category were weak in elementary algebra, and most in elementary arithmetic also. About half of these students also had difficulty with one or more of the relevant physical concepts.

¹² The extension of these models to two or three dimensions can also be used fruitfully in teaching other important statistical concepts, like the MLS, residual sum of squares, Pearson's correlation coefficient, sum of random variables, and so on (see Farebrother and Schyns 2002; Kourkoulos and Tzanakis 2008a).