NWD 2008
Wortels van statistische kernbegrippen:
Arthur Bakker
Freudenthal Instituut
Universiteit Utrecht
Uitgangspunt Freudenthal
The young learner recapitulates the learning process of mankind, though in a modified way. He repeats history not as it actually happened but as it would have happened if people in the past would have known something like what we do know now. It is a revised and improved version of the historical learning process that young learners recapitulate.
‘Ought to recapitulate’ – we should say. In fact we have not understood the past well enough to give them this chance to recapitulate it. (1983b, p. 1696)
1. Blaadjes en vruchten aan de boom
In an ancient Indian story, which was finally written down in the fourth century AD,
the protagonist Rtuparna estimated the number of leaves and fruit on two great
branches of a spreading tree (Hacking, 1975). He estimated the number on the basis
of one single twig, which he multiplied by the estimated number of twigs on the
branches. He estimated 2095, which after a night of counting turned out to be very
close to the real number.
2. Hoe groot is de vloot en hoeveel soldaten zitten er op?
Homer gives the number of ships as 1,200 and says that the crew of each Boetian ship
numbered 120, and the crews of Philoctetes were fifty men for each ship. By this, I
imagine, he means to express the maximum and minimum of the various ships’ companies...
If, therefore, we reckon the number by taking an average of the biggest and
smallest ships... (Rubin, 1971, p. 53)
3. Hoe hoog is de muur van Platea?
(The problem was for the Athenians)... to force their way over the enemy’s surrounding
wall... Their method was as follows: they constructed ladders to reach the top of
the enemy’s wall, and they did this by calculating the height of the wall from the number
of layers of bricks at a point which was facing in their direction and had not been
plastered. The layers were counted by a lot of people at the same time, and though
some were likely to get the figure wrong, the majority would get it right, especially as
they counted the layers frequently and were not so far away from the wall that they
could not see it well enough for their purpose. Thus, guessing what the thickness of a
single brick was, they calculated how long their ladders would have to be... (Rubin,
1971, p. 53)
4. Gemiddelde bij Grieken
By the mean of a thing I denote a point equally distant from either extreme, which is
one and the same for everybody; by the mean relative to us, that amount which is neither
too much nor too little, and this is not one and the same for everybody. For example,
let 10 be many and 2 few; then one takes the mean with respect to the thing if
one takes 6; since 106 = 62, and this is the mean according to arithmetical proportion
[progression]. But we cannot arrive by this method at the mean relative to us. Suppose
that 10 lb. of food is a large ration for anybody and 2 lb. a small one: it does not follow
that a trainer will prescribe 6 lb., for perhaps even this will be a large portion, or a
small one, for the particular athlete who is to receive it; it is a small portion for Milo,
but a large one for a man just beginning to go in for athletics. (Aristotle, Nichomachean Ethics, book II, chapter vi, 5; italics added)
If therefore, for instance, two persons each had merchandise valued at 20,000 sesterces
and one lost 10,000 due to water damage, the one with the saved merchandise
should contribute according to his 20,000, but the other on the basis of the 10,000.
(Spruit, 1996; translation from Latin and Dutch)
Figuur 2: Deel van Een tracaet van averien, ghemaeckt by Quintyn Weytsen (1641)
Figuur 3: Köbel (1535) over hoe een voet (lengtemaat) bepaald kan worden. Vraag 16 mannen om hun voeten achter elkaar te plaatsen. é
6. Rekenen bij epidemieën (tweede eeuw)
A town bringing forth five hundred footsoldiers like Kfar Amiqo, and three died there
in three consecutive days  it is a plague... A town bringing forth one thousand five
hundred footsoldiers like Kfar Akko, and nine died there in three consecutive days 
it is a plague; in one day or in four days  it is not a plague. (Rabinovitch, 1973, p. 86)
7. Bepalen van werkelijke waarde door Wright (1599)
Merk op dat een v als u gelezen moet worden, en de meeste u’s als v of w.
Exact trueth amongst the vnconstant waues of the sea is to bee looked for, though
good instruments bee neuer so well applied. Yet with heedfull diligence we come so
neare the trueth as the nature of the sea, our sight and instruments will suffer vs. Neither
if there be disagreement betwixt obseruations, are they all by & by to be reiected;
but as when many arrows are shot at a marke, and the marke afterwards away, hee may
bee thought to worke according to reason, who to find out the place where the marke
stood, shall seeke out the middle place amongst all the arrowes: so amongst many different obseruations, the middlemost is likest to come nearest the truth.
(Eisenhart, 1974, p. 52)
8. Vermoedens over foutenverdelingen
In 1756, Simpson made this shift to looking at the law of errors in astronomy when he used simple probability functions to argue that the mean of several observations was a better estimate of a true value than a single observation (Steinbring, 1980). The first distribution of errors he proposed was a discrete uniform distribution, that is with equal probabilities for all error values v, v+1,...1, 0, 1,..., v (Figure below a). Next, he assumed a discrete isosceles triangle distribution with probabilities proportional to 1, 2,..., v1, v, v+1, v,..., 2, 1 (Figure b), from which he obtained a continuous isosceles triangle distribution one year later (Figure c).
a. discrete uniform (1756)
b discrete isosceles triangle (1756)
c. continuous isosceles triangle (1757)
Soon after Simpson had launched his idea of probability distributions, other scientists proposed alternative laws of error (Figure below). Among them were Lagrange, Lambert, Daniel Bernoulli, Laplace, and Gauss.
a Lambert (1765): flattened semicircle
b Lagrange (1776): continuous uniform
c Lagrange (1776): continuous parabolic
d Lagrange (1781): cosine function
e Laplace (1781): log function
f Laplace (1774): double exponential
g Gauss (1809), Laplace (1810):
normal distribution
Galton lyrisch over de normale verdeling
It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect from Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed ones, starts potentially into existence. (Natural Inheritance, Galton, 1889, p. 62)
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error.” The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete selfeffacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along. (ibid., p. 66)
Freudenthal over generaal Knapp
In de 19^{de} eeuw, liet Generaal Knapp zijn recruten op een rij staan in de veronderstelling dat hij een normale verdeling zou zien.
Toen hij [generaal Knapp) tot zijn schrik bemerkte, dat de kruinen niet naar links èn rechts afliepen, dus niet zoiets als een normale kromme aftekenden, liet hij ze heel eigenaardige exercities uitvoeren, waarvan de finesses mij niet duidelijk zijn geworden.
Enkele bronnen
Bakker, A. (2001). De mediaan in het middelpunt. Nieuwe Wiskrant, 21(1), 2831.
Bakker, A. (2003). The early history of statistics and implications for education. Journal of Statistics Education, 11(1). www.amstat.org/publications/jse
Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools. Utrecht, the Netherlands: CD Beta Press.
Bakker, A., & Gravemeijer, K. P. E. (2006). An historical phenomenology of mean and median. Educational Studies in Mathematics, 62(2), 149168.
Cournot, A. A. (1843/1984). Exposition de la théorie des chances et des probabilités [Treatise of the theory of chances and probabilities]. Paris: Librairie philosophique J. Vrin.
David, H. A. (1995). First (?) occurences of common terms in mathematical statistics. The American Statistician, 49, 121133.
David, H. A. (1998). First (?) occurences of common terms in probability and statistics A second list, with corrections. The American Statistician, 52, 3640.
Eisenhart, C. (1974). The development of the concept of the best mean of a set of measurements from antiquity to the present day. 1971 ASA Presidential Address. Unpublished manuscript.
ESS (1981). Encyclopedia of statistical sciences (Eds. Kotz, S. & Johnson, N.L.). New York: Wiley & Sons.
Freudenthal, H. (1983b). The implicit philosophy of mathematics: History and education. In Proceedings of the International Congress of Mathematicians (pp. 16951709). Warsaw.
Galton, F. (1889). Natural inheritance. London: Macmillan.
Hacking, I. (1990). The taming of chance. Cambridge: Cambridge University Press.
Porter, T. M. (1986). The rise of statistical thinking, 18201900. Princeton, NJ: Princeton University Press.
Rabinovitch, N. L. (1973). Probability and statistical inference in ancient and medieval Jewish literature. Toronto: University of Toronto Press.
Stigler, S. M. (1977). Eight centuries of sampling inspection: The trial of the Pyx. Journal of the American Statistical Association, 72, 439500.
Stigler, S. M. (1986). The history of statistics. The measurement of uncertainty before 1900. Cambridge, MA: Harvard University Press.
Stigler, S. M. (1999). Statistics on the table. The history of statistical concepts and methods. Cambridge, MA: Harvard University Press.
Boek van Bakker is verkrijgbaar bij w.hofman@fi.uu.nl (E 17,50) maar als pdf ook op internet: http://igiturarchive.library.uu.nl/dissertations/20040513153943/UUindex.html
