Fair and square

por: Angela Cardoso
probabilidades metricas

The printed number and the rise of the statistical program 

This is an essay I wrote for the module History of Science in January  2021.


Engaging philosophically is something I consider a social endeavour though it doesn’t  exclude periods of individual solitude and a fair amount of despair. In my case, students and  teachers from another country lived something that triggered me. Teachers were deprived of  one of their traditional responsibilities, that of assessing students, and students, who once  had a person to address themselves regarding their grades, now had a black box, which, at  best, would “reply” is just executing a set of rules.  

It’s May 2020, and due to COVID-19 pandemic, it was announced students in England  wouldn’t sit for GCSEs, AS and A levels exams (see note 1). As the exams were cancelled, the question  of how to evaluate and grade students popped up. The Office of Qualifications and  Examinations Regulation (Ofqual), whose mission is to “maintain standards and confidence  in qualifications in England”, proposed to address the situation through the use of an  algorithm.  

Seeing pictures of students protesting and holding cards “Teachers know my potential,  algorithm does not” or “my grades, not my postcode” in BBC (A-levels and GCSEs, 2020)  made me wonder why choosing an algorithm to decide student’s grades was better suited  than teachers’ judgement. And what was this statistical model made of anyway? According  to Ofqual, teachers were asked to provide data for each student and for every subject . It  included an estimated grade, taking into consideration previous data, such as mock-tests  and class assignments, and a ranking comparing every student at the school within the  same estimated grade. One of the assumptions was grades attributed by the algorithm  would be consistent with schools and students’ previous performance, even without exams  taking place. The organisation affirmed the statistical model aimed at maintaining the  standards both between centres and over time in order to prevent grade inflation, but  alerted to the fact there could be issues for students whose grades fall outside the pattern of  results in that centre in recent years. Another organisation, the Royal Statistical Society  (RSS), whose motto is Data | Evidence | Decisions, alerted “Ofqual to the statistical issues  relating to exam grading and assessment in 2020” (Witherspoon, email to Ofqual, 2020,  April 9) and proposed to help by bringing in an advisory expert group to the discussion. The  grade results provided by the algorithm provoked protests all over the country and escalated up to the point of calling the Prime Minister to intervene (Adams, Stewart, 2020,  August 16).  

This situation illustrates well our condition of statistical subjects, where many parts of our  lives are under the influence of a probabilistic and statistical umbrella, even if these  inferences and judgements may be wrong (Wissenschaftskolleg zu Berlin, 2011, January 6).  This essay seemed an adequate opportunity to expand reflection on the subject, therefore I  will be covering what were some of the contingencies in the origin and development of  probabilities and how they became part of the way we think, hence of who we are.  Untangling this narrative is like analysing a plot: what were the conditions in place that  created a nutritious ground to these ideas? Who were the characters who fed these ideas  and how were they related? Which moments, either of eruption or apparent stagnation,  were important to it becoming a thing? It’s not my aspiration to expose in a small essay  what others have done so brilliantly in a journey of a lifetime but I will adventure on referring  to some of the most relevant moments which allow us to understand the students’  probabilistic and statistical situation.  

This essay has three parts. First, I will start by explaining the origins of classical probability  in a deterministic conception and its rise as a new mathematical program and new  rationality. Second, I will move towards the enumeration of people and their habits and how  it changed the idea of determinism, eroding it. Here I will focus on the law of large numbers,  normal distribution and the birth of the average man. Finally I will conclude by summarising  the main points and suggesting further research to address our probabilistic and statistical  thinking.  

The Mechanics of the heavens and order on Earth  

In the first half of the seventeenth century, natural philosophers such as Galileo thought  nature had causes which were determined by God and that they could be expressed  through the language of mathematics. Then, in order to be able to describe nature  mathematically, it was first necessary to understand its causes ( Gigerenzer, Swijtink,  Porter, Daston, Beaty, & Kruger, 1989, p. 11). It’s in this rational context that “Philosophiæ  Naturalis Principia Mathematica” by Sir Isaac Newton thrives as the first unified program of  modern physics, bringing together the disciplines of astronomy, physics and mathematics.  Newton helped to create the idea of an universe where there was a single force, universal  gravitation, determining the motion of objects: from planets to falling objects everything was  predetermined. Universal gravitation enabled retrodictions and predictions of the positions  of the planets, considering its present configuration.  

The devising of Newton’s theory had an intimate connection on how probabilities started to  be understood. Traditionally seen as a qualitative measure of an opinion backed by  authority, probability slowly became a quantitative measure of certainty. At the core of this  change is the intersection of a new rationality which was leaving behind the traditional idea  of certainty with the desire of applying mathematics to new areas of experience.  ( Gigerenzer, Swijtink, Porter, Daston, Beaty, & Kruger, 1989, p. 1). Without the  entanglement between both, none of them alone would have been enough to advance  probabilities as we know them today (Hacking,1990).  

In the classical period of probabilities, from circa 1660 to 1840, there were three  undiscerning ways to which mathematicians referred to as probabilities.  

The first was about equal possibilities based on physical symmetry which was applied to  gambling scenarios. For instance, the work of Blaise Pascal in understanding the  relationship between risks and uncertain pay-off is an example of this new rationality where  it was “not enough to consider how good or bad an outcome is in itself, but also the  likelihood that it will come to pass (Arnauld and Nicole, [1662] 1965, pp. 352-3 in  Gigerenzer et al., 1989 p. 5).  

The second way referred to probabilities as observed frequencies of events, through analysing  different kinds of data collected since the sixteenth century. The permeation of the  probabilities outside the realm of gambling can be seen by the hand of the Dutch Christiaan  Huygens who continued Pascal’s work and applied probabilities to the study of equitable  contracts. Chance and risk as random events were at the heart of the dealings in the  Amsterdam Stock Exchange where financing marine ventures required the need to to  determine what needed to be paid in exchange contracts, in order to guarantee equity and  justice. Another of its applications was in calculating the price of annuities, which evolved to  become life insurance (Gigerenzer et al., 1989 p. 6).  

Finally, the third concept of probabilities is connected to the notion of sufficient knowledge and  it arose from analysing procedures in courtrooms and miracles. If order in nature was  universal, invariable and eternal and a sign of divine providence, then the question on how  to interpret miracles was yet to be answered. One of the works which contributed to the  development of this idea was David Hume’s essay on miracles. He defended that a miracle  was less probable than a naturalistic explanation, raising the notion of intrinsic evidence;  and rejected the evidence arising from testimony, considered as extrinsic evidence on past events as an inference to predict future ones. It is the connection of intrinsic and extrinsic  evidence that characterises probabilities as models of causation and degrees of subjective  certainty or belief (Gigerenzer et al., 1989 p. 9, p.28).  

These three interpretations regarding probabilities led to a distinction a posteriori between  objective and subjective probabilities. Still, this distinction shouldn’t be seen as a binary  opposition though but rather as a complex web weaved by classical probabilists who also  used both meanings randomly. For example, Pierre-Simon Laplace was an astronomer who  used both objective and subjective elements in his theories. Moreover, Laplace was also  one of the founders of the classical theory of probability and an avid supporter of  determinism to allow the control and domination of nature which would be possible under  the condition we have sufficient knowledge. This conception of a predetermined universe  carved the way for probability to rise as a new mathematical program and a new rationality  where probability was then to be understood a state of mind relative to human knowledge  rather than a state of the physical world.  

Circa 1840 A.A. Cournot wrote “the double sense of probability, which at once refers to a  certain measure of our knowledge”, this is subjective probabilities, “and also to a measure  of the possibility of things independently of the knowledge we have of them», this is the  equal possibilities based on physician symmetry and frequencies. (Daston, 2007). Hacking  distinguishes probabilities as modelling and as inference , when referring to objective and  subjective probabilities respectively (Hacking, 1990 p. 98).  

Though it seemed mathematics could then be applied to other domains of experience, there  was a catch in the program: the variability found in data represented a problem. Classical  probabilists faced huge criticisms and the program was discredited (Wissenschaftskolleg zu  Berlin, 2011, January 6). What initially survived was the objective conception of probability  applied in gambling and actuarial problems. To successfully push the program forward and  connect both the objective and subjective senses of probability two more elements were  needed: the first element required probability as a measure of knowledge or state of mind  had to be shown in practice, meaning it needed to attract people who were motivated to find  in society the equivalent of Newton’s gravitational universal law; and the second element  was evidence needed to be quantified for which the availability of printed numbers was key.  

The printed number and the rise of statistics  

The idea of gathering numbers about populations can be traced back to the eighteenth  century when government officials such as Leibniz, the “philosophical godfather of Prussian  official statistics» (Hacking, 1990 p. 18), who saw the potential of a central office estimating  population and their behaviour to serve different administrations such as taxation and  military service. Nevertheless, a universal statistical law of human nature didn’t emerge in  Prussia but in England and France in government offices.  

This vision was shaped in the beginning of the nineteenth century. By this time, there was a  solid printing network in place and an increase of the capacity to print propelled by the  Industrial Revolution. This made the appearance of an avalanche of printed numbers  technically possible, developed and implemented by the rising modern states and their new  bureaux, which aimed at creating stability and avoiding unbalances to peace after the  turbulence of the Napoleonic era.  

The new bureaucratic state institutions were instrumental to gather and diffuse the  avalanche of numbers and it engendered the need of creating categories where individuals  could fall into. The idea was to show that in practice society would follow into the same  universal umbrella of predictability and control. In order to do so, everything was  enumerated, from birth, deaths, average date of marriage, voting predisposition and, in  particular, wrongdoings, such as suicides (Hacking, 1990, p. 64). It is in the wrongdoings  that statistics finds its social mission, the one of improving society by applying reason. 

One of the most relevant statistical realisations is the law of large numbers. It is a part of a  theorem refined and presented by Siméon Denis Poisson in 1837 which expresses the  stability of mass phenomena. It acknowledged there were regularities in the societies and  this became a fact which “didn’t need” to be empirically checked because it was how things  were, the collected numbers showed a frequency in the many events such as the number of  deaths and births. Though there were many sceptics to Poisson’s theorem, the proposition  about the law of large numbers had the soil to flourish for some decades once the stability  of relative frequencies was associated with a sign of divinity. Statistical law was now  admitted in social affairs (Hacking, 1990, p. 104).  

Nevertheless, the regularities in wrongdoings such as crime and suicide were difficult to  understand in a deterministic universe, so the law of large numbers was later discarded as  

explanation. The question of what could be the equivalent of the laws of regularity which  govern the universe to the ones that govern individuals remained opened.  

Maxwell, a scientist in the field of mathematics physics and Quetelet, an astronomer and  statistician, had a different vision regarding what determinism was and defended that many  of the irregularities found by bureaucrats were also found in physics such as in the  statistical laws of thermodynamics. Quetelet thought it wasn’t necessary to know everything  about all the properties in absolute detail to address the puzzle of societal wrongdoings and  favoured society and not the individual as the basis of social statistics. He introduced the  curve of normal distribution at a societal level, which was already used in physics. The  curve of normal distribution, described observational errors in astronomy and is composed  by the mean and measure of dispersion. The dispersion is very relevant because it  determines if an average is or not reliable. When it is, it is called a normal distribution and is  represented by what we know today as a Bell Curve. If the universe also had its own errors,  then it was only normal society had theirs too, determinism was not at stake.  

Underlying this notion is the concept of “normality”. Quetelet, by accepting deviations and  errors present in the universe and society, gave birth to the notion of the average man  (Gigerenzer et al., 1989 p. 53). The average man became an objective notion of “a” man  against whom all others can be compared to and judge if they are average or a deviation.  The Bell curve or normal distribution is still used in social matters. For instance, the  intelligence quotient (IQ) remains one of the most controversial products of statistical  reasoning expressed through a Bell curve. Used to measure intelligence, it conceives it in a  universal and supposedly neutral way.  

Moreover, the law of large numbers and normal distributions are present in the students’  exams case portrayed in the introduction. Ofqual’s algorithm was applied because it was  considered there was sufficient data which provided evidence about the stability and  regularity in centres’ performance in the past years and it also had a way to compare  students to each other through distribution of their grades. Ofqual also expected some  students whose grades would fall out of the patterns or the normal distribution curve, but  considered it wouldn’t compromise the large scale average and predictability.  

On that account, the variations found in astronomers’ observations had consequences: if  some variability was found in the universe, then some chance and randomness needed to  be readmitted both in physics and society. Though variability didn’t put an end to  determinism it led to an erosion, what Ian Hacking defines as taming chance. Nevertheless,  the laws of nature differ from those of society: while elements of the universe don’t have an  ambition of being considered normal or constant, humans do look forward to being seen as such, meaning we will try to adapt to what the rules are, which will consequently affect what  is considered to be “the average man”. Despite the initial failure of the probabilistic program,  we became statistical subjects thriving to be considered normal people. This situation was  only possible due to the avalanche of numbers which helped create the statistical laws of  societies. Yet, without the belief that there was an equivalent of a newtonian-laplacian  universe in society, the statistical laws might have never been inferred in those numbers  (Hacking, 1990).  


We have seen how probabilities and statistics are connected with determinism and how  they changed through the course of the classical period of probabilities.  

The classical probabilities set in motion a new mathematical program and a new conception  of how to think about certainty, trying to exclude chance. The regularity of the universe was  desired on earth, to allow insights, predictions and control in all spheres of human life.  Though the program had many sceptics, probabilities and statistics carved its way into the  core of our rationality and are engraved in social sciences and in our rationality.  Probabilities were a huge success at many levels. For instance, they are an epistemological  success because when we think about evidence, data, experiments and credibility we do it  in probabilistic terms; it is a success in logic due to inference theory. As the history of  probabilities and statistics is an ongoing endeavour, further considerations may be added  for future research. I suggest, for example, if a plausible comparison can be made between  the relevance of printed number in the nineteenth century and the relevance of data points  in contemporary society for the purpose of determining our behaviour; and understanding if  algorithms are the most recent techno-deterministic flavour of our society, governing all  spheres of our life from whom we may vote, whom we may be attracted to or what are we  going to buy. These questions are valid because they seem supported by the fact that  despite the evidence of some level of indeterminism in the universe brought in by other  theories in physics, such as quantum theory, determinism remained mostly unshaken.  

Note 1:

GSE, AL are the UK equivalent to school-leaving or university entrance examinations in different  subjects. GSE stands for General Certificate of Secondary Education, AS for Advanced Subsidiary level  and A is a combination of advanced levels.  


Adams, R., Stewart, H. (2020, August 16). The Guardian. Retrieved from https:// www.theguardian.com/education/2020/aug/16/boris-johnson-urged-to-intervene-as-exam results-crisis-grows 

A-levels and GCSEs: How did the exam algorithm work? (2020, August 20). BBC.  Retrieved from https://www.bbc.com/news/explainers-53807730 

A-levels and GCSEs: U-turn as teacher estimates to be used for exam results. (2020, August  17). BBC. Retrieved from https://www.bbc.com/news/uk-53810655 

Coughlan, S. (2020, August 14). Why did the A-level algorithm say no? BBC. https:// www.bbc.com/news/education-53787203  

Daston, L. (2007). The History of Emergences. Isis, 98(4), 801-808. Retrieved from https:// doi.org/10.1086/529273 

Gigerenzer, G., Swijtink, Z., Porter, T., Daston, L., Beatty, J., & Kruger, L. (1989). The Empire of  Chance: How Probability Changed Science and Everyday Life (Ideas in Context). Cambridge:  Cambridge University Press. doi:10.1017/CBO9780511720482  

Hacking, I. (1990). The Taming of Chance (Ideas in Context). Cambridge: Cambridge University  Press. doi:10.1017/CBO9780511819766  

Kline, R. R. (2001). Technological determinism. In N. J. Smelser, & P. B. Baltes (Eds.),  International encyclopedia of the social & behavioral sciences (pp. 15495-15498). Oxford:  Pergamon. doi:https://doi.org/10.1016/B0-08-043076-7/03167-3 Retrieved from https:// www.sciencedirect.com/science/article/pii/B0080430767031673  

Lemov, R. (2017). Archives-of-Self: The Vicissitudes of Time and Self in a Technologically  Determinist Future. In L. Daston (Ed.), Science in the Archives: Pasts, presents, futures (pp.  247-270). The Chicago University Press: Chicago and London.  

Ofqual about us. (n.d.) Retrieved from https://www.gov.uk/government/organisations/ofqual/ about 

Social Science Research Council. (2019, February 12). 2019 SSRC Fellow Lecture: Lorraine  Daston [Video]. YouTube. https://www.youtube.com/watch?v=6xErFnyjMAA 

Witherspoon, S. (2020, April 9). [Email to Ofqual]. RSS alerts Ofqual to the statistical issues  relating to exam grading and assessment in 2020. Royal Statistical Society. Retrieved from  https://rss.org.uk/RSS/media/File-library/News/2020/RSS_Ofqual_30042020_SFW_final.pdf 

Wissenschaftskolleg zu Berlin. (2011, January 6). Lorraine Daston – The Rule of Rules, or How  Reason Became Rationality [Video]. Vimeo. https://vimeo.com/18497646. 


Árboles para nuestro futuro

por: Angela Cardoso

Un futuro más verde 

Decidimos juntarmos a Reforesta y ofrecer un árbol a cada organización con la cual tuvimos la posibilidad de colaborar este año. No podemos decir que juntos ya plantámos un bosque, pero 27 árboles ya nos permite sonreir al pensar que en unas décadas alguíen podrá caminar bajo su sombra. Que colaboremos contigo un día y plantemos una en tu nombre como agradecimiento por estar con nosotros en un camino hacia la sostenibilidad de nuestras organizaciones y de nuestro planeta. 


La toma de decisiones con Bentoism

por: Angela Cardoso

Este mes de Octubre participé en dos conferencias ágiles la Agile Tour Toulouse y la Agile Tour Vilnius y hablé de toma de decisiones con Bentoism.   

¿Porque tomar decisiones es difícil?

Es difícil porque no estamos en un dominio de algo que se pueda medir cuantitativamente como mayor, menor o igual que otra opción, si no en un dominio normativo, en que lo que cuenta son nuestras preferencias, o interés propio. Desde Adam Smith, y con su punto culminante en el artículo publicado en 1970 “La responsabilidad social de los empresarios es incrementar sus ganancias” de Milton Friedman, el interés proprio suele estar reducido a uno mismo. Otras teorías como los Commons de Elinor Ostrom desafían esta visión del interés propio. 

Además del interés propio en la toma de decisiones, el otro factor importante es el tiempo. El tiempo tiene una capacidad de transformar nuestros valores pero a la hora de decidir nos enfrentamos a la dificultad de imaginar el futuro, más fácil es recordar el pasado.    

Entonces, ¿cómo podemos mirar la toma de decisiones más allá del corto plazo y de una visión reducida del interés proprio? 

Escuché hablar del Bentoism con Yancey Strickler, autor de This could be our Future y cofundador de Kickstarter hace un año en el Web Summit. Desde ahí que viene formando una comunidad de practica internacional que intenta explorar los conceptos de yo y nosotros, ahora y en el futuro, basándose en la filosofía de las cajas de comida japonesas: Bento.

La caja Bento, muchas veces preparada por um miembro de la familia para otro, está asociada a una conveniencia, a algo que es práctico. La caja tiene diferentes compartimentos para los diferentes alimentos, con el objetivo de tener una alimentación rica y variada. Su tamaño nos satisface al 80%, dejándonos con ganas de comer más tarde.

La metáfora de la caja Bento con la toma de decisiones nos permite reflexionar porqué hay personas que, además de sus ocupaciones profesionales, se involucran para organizar conferencias sin ninguna compensación durante muchos años seguidos. ¿No habrá entonces algo más que una visión reducida interés proprio? ¿Como se puede practicar el Bentoism de manera individual y en equipo?

¿Quieres saber más y praticar? Lo que necesitas es sencillo: papel, algo con que escribir y apuntarte al meet-up aquí.