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“Flatten the Curve!” But Which Curve?

A Historical Inquiry
« Aplatir la courbe ! » Oui, mais quelle courbe ? Une enquête historique
Éric Brian
p. 233-246

Abstracts

This article considers a phrase commonly found in the COVID-19 literature: “flatten the curve”. It examines the historical evolution of two centuries of statistical reasoning on the dispersion of uncertainty. Earlier extrapolations of Gaussian distributions are discussed, and statistical modelling of this type, based on misleading a priori assumptions, is analysed. Some rigorous alternatives are proposed for computation and recommendation purposes.

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1“Flatten the curve”, was a common battle cry in March and April 2020 when the first wave of the COVID-19 pandemic was at its height. But this phrase, repeated as a so-called scientific argument, relies on assumptions about mathematical extrapolations, conceptualized in Kantian philosophical language as an a priori underlying the scientific analysis. We will analyse here the underlying notion associated with the expression “flatten the curve” and how it might lead to incorrect reasoning on the matter in hand. We will examine the history of statistics over slightly more than two centuries, and along the way we will encounter earlier errors caused by the same a priori assumption on statistical distributions. The bad news is that while this error was diagnosed a whole century ago, it has more recently produced some tragic consequences in non-medical domains. The good news is that finding a public and convincing discourse on the COVID-19 pandemic is possible without jeopardizing the social authority of scientists in their pseudo-scientific public interventions.

1. “What curve?”

  • 1 W. Preiser, G. van Zyl & A. Dramowski, 2020; R. M. Anderson, H. Heesterbeek, D. Klinkenberg & T. D (...)
  • 2 E. Wolfson & C. Wilson, 2020.
  • 3 S. Roberts, 2020.
  • 4 K. Sepkowitz, 2020.

2Around the second half March 2020, various articles appeared in medical journals giving projected numbers of infections and the hypothetical impacts of preventive measures.1 The media then circulated this information more widely,2 for instance presenting this comparative pair of curves produced by the US Centers for Disease Control (see Figure 1, reproduced in The New York Times3 and by CNN4).

Figure 1. Intended impact of social distancing measures (including explicit captions)

Figure 1. Intended impact of social distancing measures (including explicit captions)

Source. Centers for Disease Control and Prevention.

  • 5 J. E. Harris, 2020.
  • 6 J. Courtney, 2020.

3Meanwhile econometricians published papers displaying various trends and impacts based on similar frequency curves,5 and statisticians published technical advice for drawing the graph in a simple manner.6 This is the well-known “bell curve”, whose name evokes its characteristic shape. Its equation is:
Image

  • 7 P.-S. Laplace, 1786, pp. 423-467.
  • 8 É. Brian, 2001.
  • 9 É. Brian, 1990, pp. 305-314.
  • 10 C. Kramp, 1799, pp. 195-202.
  • 11 E.-É. Duvillard de Durand, 2010 [1813].
  • 12 A. Quetelet, 1846; id., 1848; id., 1869.
  • 13 É. Brian, 1989; id., 1991.

where x is the horizontal coordinate, f(x) the vertical coordinate corresponding to frequencies, µ the coordinate of the central peak where frequency is highest, and σ a parameter of dispersion around this centre (on Figure 1, higher for the lightest curve than for the darkest one), understood today as a measure of the standard deviation around this centre. This formula was proposed by Pierre-Simon Laplace (1749-1827)7 and later by Carl Friedrich Gauss (1777-1855) and the bell curve is usually associated with these two names. The scientific and broader contexts of this research are well known8 and its scientific uses too.9 The issue at that time was the dispersion of observation errors in astronomy, and Laplace, after several other attempts, considered that this shape was a good fit, and above all easy to compute – by hand at that time. He encouraged scholars to build tables like those familiar to later statisticians before the digital era. Kramp,10 an astronomer working in Strasbourg, published such a table, as did Duvillard, an economist working in insurance.11 Both men worked under the influence of the research promoted by the Paris Academy in the late eighteenth century. A young Belgian astronomer, Adolphe Quetelet (1796-1874), visited this Academy in the mid-1820s where he learnt both the calculation techniques and the wider ideas promoted by the mathematician Joseph Fourier (1768-1830) who considered that statistical compilations should be the empirical foundation for the moral and economic sciences. Quetelet collected a vast amount of anthropometric data and observed that for a homogeneous population, the frequency distribution of measurements (heights or sizes of various parts of the body, or various demographic criteria) obeyed Laplace’s second law of errors. From this point he extrapolated an analogy between variations in physical and moral phenomena and dispersion of errors in the physical sciences. This was his “social physics” where the key concept was the “average man”; every human individual was considered as one variation, his difference with respect to the average being taken as analogous to a measurement error.12 From 1853 up until his death, Quetelet organized an international congress of statistics where thousands of statisticians – mathematicians together with senior administrative officers – learned his doctrine and shared their experience and know how. This international activism and the methods it forged in the nineteenth century fostered the spread of statistical knowledge in many countries.13

  • 14 B. Mandelbrot, 1983.
  • 15 N. Akrour, 2015.
  • 16 O. Le Courtois & C. Walter, 2014.
  • 17 É. Brian, 2009; id., 2015.

4Let us discuss now why the bell curve is not fit for purpose and how these wonderful bumps are obtained. Homogeneous data are collected on a daily basis over a given period. These are past numerical observations. Anything else deals with the future and is therefore conjectural. These first data points are then fitted to the beginning of a Gaussian distribution of frequencies. All comments are therefore consequences of this choice about the shape of randomness. The mathematical domain concerned is called stochastics, and throughout the twentieth century, various alternative hypotheses were explored. A good analogy was proposed by Benoît Mandelbrot (1924-2010): a few raindrops fall…14 Will this give us a British rain shower? A thunderstorm or a flood? Or nothing more than these premises? Counting the drops will not tell us. But depending upon the choice of stochastic modelling hypotheses, the forecasted future may or may not fit these different scenarios. Strictly speaking, no calculus can demonstrate what the outcome will be. It will only shape uncontrolled extrapolations (uncontrolled, in that the hypotheses are not rationally controlled). This issue is well known today in meteorology and finance. A recent meteorological PhD thesis illustrated the vast range of results that can be obtained on the basis of different initial hypotheses: all possible combinations of bumps are shown.15 In finance, various authors have explored models under non-Gaussian distributions of randomness16 and I myself discussed the empirical and theoretical limits of Gaussian modelling in a book published some years back.17

5So, in short, the predilection for Gaussian distributions of randomness is arbitrary, fostered by outdated statistical reasoning and readily accessible computer power. Let me comment upon some of the false ideas conveyed by this arbitrariness. First, it presupposes a regular increase towards a peak. Second, after the summit, the decrease is assumed to be symmetrical, with frequencies progressively tending towards zero. But should the frequencies stay significantly above zero, or should some regular fluctuations occur, no bell curve can be designed to fit the observations. Public discourse in the media and among decision-makers is nonetheless shaped by this conceptual arbitrariness of hypotheses, to the point where the very framework has collapsed. Suddenly the camel that had one hump now has two, with the emergence of additional “peaks” announcing possible dangerous resurgences of the virus (even if no single bell curve displays two or more bumps). The bell curve has outgrown its initial function and, to ward off fears, has become a double peak.

6At the end of the nineteenth century a similar Gaussian tropism operated like the vision of an oasis in the desert, causing perplexity even among statisticians. It is a pity to see that the memory of this episode has been lost by today’s specialists. Before concluding I will recall it briefly.

  • 18 L. Schweber, 2006.

7Dr. Louis-Adolphe Bertillon (1821-1883) was the head of the Paris city statistical office and as such a proactive member of Quetelet’s international statistical congress and a promoter of his methods and doctrines. He was also one of the founders of the anthropological society of Paris, the father of Dr. Jacques Bertillon (1851-1922), a promoter of demography, who himself later held the same position in the office, and of Alphonse Bertillon (1853-1914), a French pioneer of forensic techniques such as anthropometry and fingerprinting.18

  • 19 L.-A. Bertillon, 1876a, pp. 296-324 [table p. 303 ; graph. p. 306 & fold extra p.].
  • 20 L.-A. Bertillon, 1876b, pp. 265-271 & 286-308.
  • 21 C. Blanckaert, 1981.

8In 1876, Bertillon senior’s most outstanding year, he published two reference articles on statistical methodology based on the same dataset. The first, “Moyenne” (Mean), appeared in the Dictionnaire encyclopédique des sciences médicales, a dictionary for medical professionals (he himself was a qualified doctor),19 and the second, “La Théorie des moyennes en statistiques” (Theory of statistical means) was published in the Journal de la Société statistique de Paris.20 Both were illustrated with a table of the heights of conscripts and an accompanying graph. The puzzling point was the case of conscripts recorded for the Doubs department (today in Bourgogne Franche-Comté). Unlike the general population of French conscripts, the curve of height frequencies had two bumps and Louis-Adolphe Bertillon saw them as two typical distributions corresponding to two different races – Celts and Burgundians – assumed to be present in the region. Quetelet’s and Bertillon’s uses of the bell curve must be seen in the context of the intellectual debate between monogenists and polygenists.21 This controversy was intense among naturalists and anthropologists, playing out simultaneously in both scientific and theological arenas: the case for monogenism was based on the argument that all humans are descendants of a single couple. But an empirical curve with two peaks illustrating two racial types argued in favour of polygenism. Contemporary historians and archaeologists added considerations on how the region had been peopled (by Celts in the Bronze Age, and Burgundians around the fifth century, without consideration for the subsequent circulation and mixing of populations). Let us consider this double-peak curve on Figure 2.

Figure 2. Heights of conscripts in feet and inches

Figure 2. Heights of conscripts in feet and inches

Note. Horizontally: heights of conscripts in feet and inches, grouped by one-inch intervals. Vertically: frequencies of conscripts of each height. Continuous curve with one bump: frequencies [inappropriately called “probability”] of each height for France as a whole. Dotted curve with two bumps, equivalent frequencies for the Doubs department).

Sources. L.-A. Bertillon, 1876a, pp. 296-324 [table p. 303; graph. p. 306 & fold extra p.]; id., 1876b, pp. 17, 265-271 & 286-308.

  • 22 J. Bertillon, 1886.
  • 23 J. Bertillon, 1895.
  • 24 Abusive extrapolations in demography have been analysed by Lucien Febvre in his foreword to the En (...)

9Amid the monogenism vs. polygenism controversy, the double peak aroused curiosity and perplexity. Three years after his father’s death, Jacques Bertillon published a revised graph in a later issue of the same journal: “The height of men in France”.22 The data are similar, but neither corrected on the basis of various numerical hypotheses, as previously, nor displayed as smoothed curves. They simply give raw numbers and broken lines forming two successive triangles. The two peaks remained visible, but not the ideal of an average man or its division into two races. In 1895, the same Jacques Bertillon published an administrative statistics textbook to be used, among other things, for the training of government statistical officers.23 Here too, he used broken lines and abandoned his father’s tropism. Today, it is clear to everyone that the bell curve obsession drove some statisticians towards an empirical justification of racism.24 While these anthropological extrapolations later fell out of favour, statisticians’ predilection for Gaussian distributions remained: it was the most comfortable hypothesis for modelling random dispersion. Laplace’s initial argument on simplicity was still a forceful one.

2. Interpretation, conclusion, and recommendations

  • 25 B. Bru, M.-F. Bru & O. Bienaymé, 1997. This reformulation is the Bienaymé-Chebyshev inequality whi (...)
  • 26 É. Durkheim & M. Mauss, 2017 [1903].
  • 27 M. Halbwachs, 2010 [1912].
  • 28 M. Fréchet & M. Halbwachs, 2019 [1924].

10At the end of the nineteenth century, mathematicians – strictly speaking – were already criticizing statisticians – namely administrative officers. First and foremost was Irénée-Jules Bienaymé (1796- 1878), the man who contributed to the reformulation of the so-called “law of large numbers”.25 Meanwhile, Quetelet’s method achieved widespread renown among German statisticians in a field known as “moral statistics”. This field inspired certain empirical aspects of Émile Durkheim’s early conception of sociology. But he and some of his more committed followers integrated this domain into the broader perspective of post-Kantian theory of knowledge.26 One among them, Maurice Halbwachs, devoted one of his dissertations to a philosophical critique of Quetelet’s average man.27 These two lines of reasoning, the mathematical one and the philosophical one, came together in a course taught at the University of Strasbourg immediately after the first World War.28

11This course was intended for to arithmeticians – administrative statisticians, businessmen, accountants, actuaries – working in the newly established French regions. The course was designed to be rigorous from the dual perspectives of both mathematics and philosophy of sciences, requiring students to possess basic prior mathematical knowledge, and based on realistic and simple examples. The bell curve was questioned (p. 210) and Maurice Fréchet, the mathematician among the two authors, wrote a letter to the Academy of Sciences explaining their intention, and specifically highlighting their critique of the bell curve.

  • 29 A. De Foville, 1907.

12The mathematician and the philosopher were making a timely point: in the first decades of the twentieth century, the bell curve was a powerful statistical totem. In an article published in 1907 in the Journal de la société statistique de Paris, Alfred de Foville (1842-1913), who had been its president in 1886, expressed his strong attachment to the bell curve.29

Figure 3. Scheme of the bell curve according to de Foville

Figure 3. Scheme of the bell curve according to de Foville

Note. Horizontally: Axis of heights. Vertically: number of men of each height. “M” indicates the mean. (Note that the curve stops before reaching the extremities.)

Sources. A. De Foville, 1907, p. 323.

13Introducing Figure 3 reproduced above, de Foville wrote “What for Quetelet was a revelation and what for his disciples became a tradition, is the trend displayed by many variables […] We must salute this symbolic curve which appears to have acquired the Creator’s sympathy”. “A minimum, a maximum and a mean in the middle; this is the structure of many phenomena”. And a few pages further on, introducing Figure 4, with reference to other statisticians, “Should we consider that human mentalities fit the curve?”

Figure 4. Moral applications of the bell curve according to de Foville

Figure 4. Moral applications of the bell curve according to de Foville

Note. Captions from top to bottom: Talent, Mediocrity, Weakness, Lower bound of social utility [sic].

Sources. A. de Foville, 1907, p. 328-329.

14De Foville commented: “Here is an image that you will not forget once you have seen it. […] It suggests that the lottery of intelligence is organized no differently than the lottery of weights or sizes, a triumph for Quetelet”. Statisticians did not forget the bell curve and it has remained a key component of their traditional toolbox, even though a century of progress in stochastics has revealed its arbitrary nature.

  • 30 É. Brian & C. Walter, 2007.

15A few years ago, alongside an actuary familiar with recent developments in stochastics, we asked economists from various theoretical schools to write a paper defining the concept of “intrinsic value” according to their own school of thought. In one way or another, all of them saw this fundamental value as a central balancing point between neighbouring plus and minus values, with negligible extreme dispersions. All the authors shared a Quetelesian or Gaussian representation of financial phenomena. But the problem here is that when a systemic crisis occurs, if your computers, your colleagues, and you are expecting this type of equilibrium around an average, then everyone has a distorted view of reality.30 The above mentioned publications in meteorology and finance explore alternative stochastic models.

  • 31 E. Panofsky, 1927.
  • 32 E. Panofsky, 1951.
  • 33 E. Cassirer, 1923-1929.

16Are there earlier historical examples of similar a priori reasoning among scholars? The art historian Erwin Panofsky (1892-1968) studied two of them. The first is perspective as an understanding of space.31 Through a detailed study of artworks, Panofsky was able to detect a shift in the perception and representation of space, characterized by a mental construction that we call perspective. He gave a second example in a wonderful synthesis on architecture and scholasticism.32 Both fields were governed by a symbolic structure based on tripartition: scholastic summae, like pillars in cathedrals of the same era, were conceived as ternary decompositions. The philosophical background for Panofsky’s work was Ernst Cassirer’s philosophy of symbolic forms,33 a theoretical line that can be compared with early Durkheimism (but this is not the place for a lengthy discussion on this point). Panofsky concluded that scholastic thought could be characterized by a habitus, a set of mental acquisitions producing a habit of seeing the matter of a scholastic summa as needing a ternary organization, or the architectural elements combined to construct a cathedral to be assembled in a similar way. Today – this is my analytical conclusion – even after more than a century of mathematical and epistemological revision, the bell curve operates as a statistical habitus: a pandemic is suspected, so statisticians project upon their sparse data their usual arbitrary simplistic stochastic hypothesis. Finally, the bell curve being an element of common knowledge, a media rush amplified the dubious early conclusions that opportunism had encouraged.

17What can be said about the spread of the pandemic? Whoever you are, be it an epidemiologist, an economist, a statistician, a media commentator, or someone with any kind of responsibility:
– Do not extrapolate upon arbitrary projections. For instance, do not give a date for a so-called peak, do not give a trend, do not comment on a so-called next wave. All these comments will most probably be contradicted by future events and you will lose credibility, all the more so at a time of crisis.
– Stick to observed data (past, even the most recent past), they have plenty to tell.
– Explain transmission in simple terms (avoiding contradictory injunctions from one day to the next).
– Promote simple approaches for preventing transmission.
Not being a medical physician, but a doctor of mathematics and history, I can comment no further on this topic.

18A last word to explain that this paper is based on extensive research in these two disciplines. But I have heard so many people who ask “What’s the point?”, that I am pleased to answer “You see, history of science and epistemology can help to challenge the routinized application of scientific techniques, even in times of emergency”.

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Bibliography

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Notes

1 W. Preiser, G. van Zyl & A. Dramowski, 2020; R. M. Anderson, H. Heesterbeek, D. Klinkenberg & T. D. Hollingsworth, 2020.

2 E. Wolfson & C. Wilson, 2020.

3 S. Roberts, 2020.

4 K. Sepkowitz, 2020.

5 J. E. Harris, 2020.

6 J. Courtney, 2020.

7 P.-S. Laplace, 1786, pp. 423-467.

8 É. Brian, 2001.

9 É. Brian, 1990, pp. 305-314.

10 C. Kramp, 1799, pp. 195-202.

11 E.-É. Duvillard de Durand, 2010 [1813].

12 A. Quetelet, 1846; id., 1848; id., 1869.

13 É. Brian, 1989; id., 1991.

14 B. Mandelbrot, 1983.

15 N. Akrour, 2015.

16 O. Le Courtois & C. Walter, 2014.

17 É. Brian, 2009; id., 2015.

18 L. Schweber, 2006.

19 L.-A. Bertillon, 1876a, pp. 296-324 [table p. 303 ; graph. p. 306 & fold extra p.].

20 L.-A. Bertillon, 1876b, pp. 265-271 & 286-308.

21 C. Blanckaert, 1981.

22 J. Bertillon, 1886.

23 J. Bertillon, 1895.

24 Abusive extrapolations in demography have been analysed by Lucien Febvre in his foreword to the Encyclopédie française, vol. 7 (1936).

25 B. Bru, M.-F. Bru & O. Bienaymé, 1997. This reformulation is the Bienaymé-Chebyshev inequality which bounds the probability of a difference between a random variable and its mathematical expectation.

26 É. Durkheim & M. Mauss, 2017 [1903].

27 M. Halbwachs, 2010 [1912].

28 M. Fréchet & M. Halbwachs, 2019 [1924].

29 A. De Foville, 1907.

30 É. Brian & C. Walter, 2007.

31 E. Panofsky, 1927.

32 E. Panofsky, 1951.

33 E. Cassirer, 1923-1929.

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List of illustrations

Title Figure 1. Intended impact of social distancing measures (including explicit captions)
Credits Source. Centers for Disease Control and Prevention.
URL http://journals.openedition.org/histoiremesure/docannexe/image/13544/img-1.jpg
File image/jpeg, 161k
Title Figure 2. Heights of conscripts in feet and inches
Caption Note. Horizontally: heights of conscripts in feet and inches, grouped by one-inch intervals. Vertically: frequencies of conscripts of each height. Continuous curve with one bump: frequencies [inappropriately called “probability”] of each height for France as a whole. Dotted curve with two bumps, equivalent frequencies for the Doubs department).
Credits Sources. L.-A. Bertillon, 1876a, pp. 296-324 [table p. 303; graph. p. 306 & fold extra p.]; id., 1876b, pp. 17, 265-271 & 286-308.
URL http://journals.openedition.org/histoiremesure/docannexe/image/13544/img-3.jpeg
File image/jpeg, 511k
Title Figure 3. Scheme of the bell curve according to de Foville
Caption Note. Horizontally: Axis of heights. Vertically: number of men of each height. “M” indicates the mean. (Note that the curve stops before reaching the extremities.)
Credits Sources. A. De Foville, 1907, p. 323.
URL http://journals.openedition.org/histoiremesure/docannexe/image/13544/img-4.jpg
File image/jpeg, 227k
Title Figure 4. Moral applications of the bell curve according to de Foville
Caption Note. Captions from top to bottom: Talent, Mediocrity, Weakness, Lower bound of social utility [sic].
Credits Sources. A. de Foville, 1907, p. 328-329.
URL http://journals.openedition.org/histoiremesure/docannexe/image/13544/img-5.jpeg
File image/jpeg, 1.0M
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References

Bibliographical reference

Éric Brian, ““Flatten the Curve!” But Which Curve?”Histoire & mesure, XXXV-2 | 2020, 233-246.

Electronic reference

Éric Brian, ““Flatten the Curve!” But Which Curve?”Histoire & mesure [Online], XXXV-2 | 2020, Online since 01 January 2024, connection on 16 April 2024. URL: http://journals.openedition.org/histoiremesure/13544; DOI: https://doi.org/10.4000/histoiremesure.13544

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About the author

Éric Brian

Historian of sciences and social scientist, EHESS, Paris; editor of Revue de synthèse (Brill); Centre Maurice-Halbwachs (CMH, EHESS, ENS, CNRS). E-mail: eric.brian@ehess.fr

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Copyright

The text and other elements (illustrations, imported files) are “All rights reserved”, unless otherwise stated.

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