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Volume 58 2003/3

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Wilhelm Lexis: The Normal Length of Life as an Expression of the “Nature of Things”

Jacques Véron []* Jacques Véron, Institut National d’Études Démographiques, 133 bd Davout, 75980 Paris Cedex 20, Tel: 33 0(1) 56 06 21 76, Fax: 33 0(1) 56 06 21 99 Jean-Marc Rohrbasser []*

Résumé de l'article

At the first International Congress of Demography held in Paris in 1878, the German statistician Wilhelm Lexis argued for the notion of a normal length of human life governed by a law of nature. His conception was based on Quetelet’s “average man” and the law of normal distribution of errors formulated by Laplace and Gauss.
The normal length of life, which differs from the average length of life, is, according to Lexis, a “true value” characteristic of the mortality of the human species.
Lexis classifies ages at death into three groups, of which the most important is “the normal group”. He seeks to define its boundaries by distinguishing normal deaths from “premature” ones. Normal mortality is then described by three values: normal age at death, the proportion of deaths included in the normal group, and probable error.
The discussions generated by Lexis’s paper, most notably Bertillon’s remarks on infant mortality, reveal distinct perceptions of the mortality process. A few years later, specialists such as Bodio, Perozzo, Levasseur, and Pareto took up the theory and method of Lexis, acknowledging the originality of his mathematical and statistical analysis of mortality, though not endorsing his hypothesis of a law of nature. Lors du premier Congrès international de démographie qui se tient à Paris en 1878, le statisticien allemand Wilhelm Lexis défend l’idée d’une durée normale de la vie humaine qui relèverait d’une loi de la nature. Il se fonde sur les travaux concernant l’homme moyen de Quetelet et utilise la loi de distribution normale des erreurs, formulée par Laplace et Gauss.
La durée normale de la vie, qui diffère de la durée moyenne de la vie, est, selon Lexis, une « vraie valeur », caractéristique de la mortalité de l’espèce humaine.
Lexis distingue trois groupes d’âges au décès, dont un est privilégié, « le groupe normal » ; il s’emploie à en définir les frontières en distinguant les décès normaux de ceux qui sont « prématurés ». La mortalité normale est alors décrite par trois valeurs : l’âge normal au décès, la proportion de décès appartenant au groupe normal et l’erreur probable.
Les discussions que suscite la communication de Lexis, et tout particulièrement les commentaires de Bertillon sur la mortalité des enfants, sont révélatrices de perceptions distinctes du processus de mortalité. Quelques années plus tard, des savants tels que Bodio, Perozzo, Levasseur et Pareto font écho à la théorie et à la méthode de Lexis, lui concédant un traitement mathématique et statistique original de la mortalité, sans toutefois reprendre à leur compte l’hypothèse d’une loi de la nature. Durante el primer congreso internacional de demografía, que tuvo lugar en París en 1878, el estadístico alemán Wilhelm Lexis defendió la teoría de que existe una duración normal de la vida humana, ley de la naturaleza. Tal perspectiva sintetiza la concepción del “hombre medio” de Quetelet y la ley de distribución normal de los errores, formulada por Laplace y Gauss.
La duración normal de vida, diferente de la duración media de vida, es, según Lexis, un “valor real”, característico de la mortalidad de la especie humana.
Lexis distingue tres grupos de edad, uno de los cuales, el “grupo normal”, es privilegiado, y define la frontera entre las defunciones normales y las “prematuras”. La mortalidad normal se define a partir de tres valores: la edad normal de defunción, la proporción de defunciones en el grupo normal y el error probable.
Las discusiones que suscitó la comunicación de Lexis, y en concreto los comentarios de Bertillon relativas a la mortalidad infantil, revelan percepciones distintas del proceso de mortalidad. Unos años más tarde, investigadores tales como Bodio, Perozzo, Levasseur y Pareto se hicieron eco de la teoría de Lexis y le reconocieron un tratamiento matemático y estadístico original de la mortalidad, aunque no retuvieron la hipótesis de la existencia de una ley natural.

Plan de l'article

• Mortality: mere statistical regularity or law of nature?

• A single model of man

— Quetelet: harmony and stability of laws

— The status of the average

— Observation errors and accidental variations

• The normal lifetime

— A distribution of ages at death

— Normal age at death

• Reception of Lexis’s hypothesis in the late nineteenth century

— The 1878 Congress of Paris: Bertillon’s commentary

— An application of Lexis’s theory to Italian data: Luigi Perozzo

— The 1887 Congress of Rome: Bodio and “the normal measure of life”

— Levasseur’s La population française and Pareto’s Cours d’économie politique

Life expectancy at birth — one of the key measures for the description of mortality — does not give a satisfactory account of recent trends in mortality such as the extension of oldest-old longevity and, more generally, the phenomenon of delayed aging. Väinö Kannisto (Population: An English Selection, 13(1), 2001) has noted that other measures such as the modal length of life are better suited to the observation of survival to old age and of longevity, and were used in the past by Lexis, in particular to identify the natural length of life. In this article, Jacques Véron and Jean-Marc Rohrbasser present Lexis’s approach and situate it in the intellectual context of the late nineteenth century. The main concern at that time was the search for the “laws of the human species” that transcended individual singularities — or, in more statistical terms, for an average that was independent of accidental variations. Other authors later took up and developed Lexis’s method, though without preserving its more philosophical content.

At the ﬁrst International Congress of Demography held in Paris in 1878, the German statistician Wilhelm Lexis [1] gave a paper in French entitled “On the normal length of human life and on the theory of the stability of statistical ratios” [2]. In the words of Jacques Bertillon, Lexis’s treatment of the subject displayed “a consummate mathematical science [that] sheds light [on] demographic studies”. In this paper, Lexis claims that “the concept of normal lifetime [has] its signiﬁcance in the nature of things”.

For Lexis, everyone should live the same length of time — the normal length of life — but some of us are prevented from doing so by particular circumstances. It is therefore possible to distinguish “normal” deaths, which occur at the normal age of death or are randomly distributed around that age, from premature deaths of adults and, a fortiori, deaths of children.

Lexis’s concept of the length of life can be understood as a synthesis of two elements: ﬁrst, Quetelet’s contributions to sociology via the notion of “l’homme moyen” or “average man”; second, the law of normal distribution of errors formulated by Laplace and Gauss [3].

I. Mortality: mere statistical regularity or law of nature?

By computing a series of probabilities of dying, a pattern of mortality can be identiﬁed. But the question arises — and has been the subject of a recurrent debate in the history of demographic thought — of whether this pattern is a construct of the human intellect or, on the contrary, is part of the very nature of things. In his writings on the length of human life, Lexis favoured a statistical approach capable of bringing to light the orderly progression of mortality, but he saw that order as pertaining to the laws of nature.

While the conception held by Lexis had been the most common in the eighteenth century, it was not so in the nineteenth century. For example, in the 1854 volume of Statistique de la France on population trends, the “law of mortality” was characterized only by a series of numbers without any higher order being mentioned:

“Looking at the general table […], one observes […] that one-sixth of all children die in their ﬁrst year; one-ﬁfth do not reach the age of 2, one-fourth the age of 3, and one-third the age of 12. One-half remain at age 38, one-third at age 59, one-fourth at age 65, one-ﬁfth at age 69, and one-sixth at age 72.

This survival expresses the law of mortality […].” [4]

The emphasis in this example is only on numerical regularities, yet these are not treated as self-evident. Their identiﬁcation requires subjecting the data to a speciﬁc treatment. Mortality, measured by the proportion of those who do not reach the ages of 1, 2, and 3, is respectively

. But to get the “next” proportion of

, as is suggested by the progression in the risk with age, one must look not at the next highest age, i.e. 4 years, but at 12 years. The same is true for the proportions of survivors at adult ages —

etc. — since they correspond to seemingly unrelated ages (38 years, 59 years, 65 years, etc.). The law of mortality is therefore a construction of the observer.

In somewhat similar fashion, in an article published in 1875 [5], the actuary Georges de Serbonnes attempts to give a purely arithmetical account of the variation in the expected length of life (Table 1):

“[…] the expectation of life between 20 and 60 years of age, the active period of human existence, seems to be governed by a law whose key is the cabalistic number seven.”

Serbonnes expresses this “law” by a decreasing arithmetic progression of a ratio of

, and concludes “that we [use up] seven-tenths of a year per year” [6]. He does not attempt to characterize the phenomenon of mortality otherwise than by this arithmetic progression.

Table 1
Expectation of life by age, based on the “so-called ‘new experience’ English table”

IMGIMGAge (in years) Expectation of life (...IMGIMF

Lexis proceeds from an entirely different perspective. His research on mortality is conducted using the concept of the average, but an average that is not merely the summary of a set of data values. For him, it is the expression of a natural law of mortality.

II. A single model of man

Lexis lays claim quite explicitly to the intellectual heritage of Adolphe Quetelet [7]. For him, as for the Belgian astronomer, the social body exhibits an essential unity, guaranteed by the laws of nature. The investigations of mortality are an application of the conceptual model developed by Quetelet.

1. Quetelet: harmony and stability of laws

In his Anthropométrie (1871), Adolphe Quetelet sets out his vision of social phenomena very clearly:

“Individual man has generally been studied with care, but little thought has been given to examining the social body to which he belongs, and whose different properties are not only of great importance, but should elicit the keenest attention by virtue of the remarkable laws presiding over their unity.” [8]

Quetelet sees the unity of the individual and the social body as a sign of the existence of unchanging and consistent laws of nature:

“The more one studies the works of creation, the more one has to admire the laws that ensure their harmony and stability. These laws, which regulate the functioning of worlds and assign to each its movement and rank, are no less wondrous in regard to the tiniest specks of dust scattered over their surface.” [9]

As early as his 1835 work on “social physics”, Quetelet argued that it was necessary to go beyond the observation of singularities, since they were obstacles to perceiving “the laws of the human species”:

“Above all, we need to lose sight of man taken in isolation, and view him as merely a fraction of the species. By stripping him of his individuality, we will eliminate all that is merely accidental; and the individual particularities that have little or no effect on the mass will disappear by themselves, enabling us to apprehend the general results.” [10]

To exclude what is accidental it is important to ﬁnd the right observation distance [11]. When we examine things at too close a range, notes Quetelet, we see only inﬁnite diversity, and observation limited to individual cases does not allow us to identify the “admirable laws”.

The correct distance from the object can correspond to a certain number of observations. For the image of a circle to become visible, writes Quetelet, we must be able to observe a sufﬁcient number of points, and we cannot see a rainbow by examining only “drops of water in isolation”. Likewise, man cannot be understood unless we have a sufﬁcient number of members of the human species:

“[…] the more individuals we observe, the more the individual particularities, whether physical or moral, are effaced, giving sharper relief to the series of general facts by virtue of which society exists and endures.” [12]

Calculating an average is tantamount to placing ourselves at an observation distance large enough for the unity of a phenomenon to become apparent.

2. The status of the average

In his quest for typical values [13], Quetelet actually endows the average — an expression of the deep nature of things — with an ontological status:

“By bringing together individuals of the same age and sex, and taking the average of their individual constants, we obtain constants that I assign to a ﬁctitious being whom I call the average man of that population.” [14]

The purpose of Quetelet’s average man is not to summarize a set of different values by use of an arithmetic average, but to identify a central value:

“The man I am considering here […] is the average around which the social elements oscillate. This, if you like, will be a ﬁctitious being for whom all things will occur in conformity with the average results obtained for a society.” [15]

Similarly, for Lexis the normal length of human life is not equated with the mean length of life. Because calculation of the latter, for a given birth cohort, requires no prior theorization of the mortality process, Lexis questions its scientiﬁc relevance:

“Everyone knows, indeed, that what is called the mean length of life of a generation is a purely arithmetical term, and in no way a true average as understood by Qu[e]telet.”

The “true” average is inseparable from the characterization of a normal type.

Lexis believes that observation of a large number of persons is the basis on which to form an idea of the single model of man, since the cases are concentrated around a particular value. This distribution, he maintains, is governed by “the well-known law of accidental errors”.

Like the Belgian astronomer, therefore, Lexis effectively endows the average with an ontological status, since it can be used, for example, to approximate a “mortality in itself”. He does not stop at a simple analysis of mortality in terms of dispersion around a central value; he suggests that this dispersion is governed by a law.

When applied to mortality, the Quetelet-Lexis model leads to postulating the existence of a value for the length of life determined by the essence of phenomena, a value around which individual lengths of life are distributed in accordance with the “binomial law”.

Lexis set himself the aim of proving the existence of a law of mortality that is both a statistical reality and a regulator of the natural order. For Lexis, as for Quetelet, chance is compatible with the existence of “ﬁxed principles”.

3. Observation errors and accidental variations

Astronomy played an important part in the development of error theory, since repeated observation of the same celestial body yields non-identical measures. The question thus arose of how to identify a “true value” on the basis of disparate data.

Jean III Bernoulli, in the supplement to the Encyclopédie, suggested choosing the “middle” of the obtained values in order to minimize the measurement error [16]. As an astronomer, Quetelet equated the multiplicity of observations obtained when examining a homogeneous human population with the multiplicity of observations of a single celestial body. As Michel Armatte has pointed out, Quetelet established an epistemologically fundamental analogy between average man and the centre of gravity of physical bodies:

“[…] this same average is the locus of a true, abstract magnitude, a single referent for these different individuals, a theoretical man created outright by the mean value. Theoretical man ﬁlls the void of the referent of these measures, a referent that existed in the heaven of Astronomy but not in human society. Theoretical man is the subject of a social mechanics that can be patterned on celestial mechanics […].” [17]

Lexis was clearly inspired by Quetelet and his analysis of observation errors in astronomy when he discussed accidental errors in relation to the mortality process. The example he gives of the game of bowls (boules) can be interpreted as an analogy between the diversity of measurements of a single phenomenon and the multiplicity of observed cases. A player throws a large number of bowls and tries to reach a target located at a given distance; the player’s skill is assumed to remain the same. Most of the bowls will scatter, in approximately two halves, on the near and far sides of the target:

“The density of the fallen projectiles will be greatest in the immediate vicinity of the targeted distance; the deviations on either side will decrease in number with their size, and the overall distribution of the bowls will be broadly consistent with the theoretical formula whose speciﬁc value is determined solely by the precision of the attempts, i.e. by the player’s skill, which is assumed to be constant. The projectiles distributed in this fashion correspond to the group of deaths that I have designated as normal.” [18]

For Lexis, the normal lifetime is the “true value” characteristic of the mortality of the human species.

III. The normal lifetime

In Lexis’s conception of the mortality process, each species is characterized physiologically by a capital of years to live: the normal length of life. For the human species, this length is between 70 and 80 years. To calculate the value of the normal length of life, the German statistician classiﬁes deaths by age into three groups, of which the most important is the “normal group”.

1. A distribution of ages at death

Deaths are distributed across all ages, but some — those Lexis calls “normal” — are clustered around a central value: the normal type (Figures 1 and 2). If the population studied is large enough, the deviations from the central value will be, Lexis asserts, “in accord with the formula Qu[e]telet calls the binomial law”.

Figure 1

End points of individual lives

Note: Point m, the “death point,” represents the death of a person whose length of life is am. The points shown by ^* in the area bounded by line B denote premature deaths of children. The + signs between lines B and P are the premature deaths of adults. The area between straight lines P and N contains a mixture of premature deaths of adults and normal deaths, indicated by dots. N is the normal age at death and the interval [N–ε, N+ε] contains one-half of the normal deaths. Beyond line –ε, all deaths are normal.

The method adopted by Wilhelm Lexis involves distributing all ages at death into three groups:

the “normal type” group;
infant deaths;
“other individuals having succumbed to a premature death virtually unrelated to the degree of age”.

In the normal group, the distribution of ages at death exclusively obeys the law of accidental errors [19]. By contrast, the ages at death in the group of infant deaths are totally unrelated to the normal length of human life since this group “has not even begun to compete to reach the natural limits of life”. The third group, that of adult deaths, is also characterized by premature deaths; the deaths of persons in this group are due to circumstances unrelated to their age.

Since infant and youth mortality are purely accidental in Lexis’s model [20], any computation that includes child deaths produces bias in the estimate of the length of human life. Taking infant and youth mortality into account when determining a summary indicator of length of life for a population would be equivalent to computing the height of a population by combining measurements taken on children and on adults.

In the example referred to earlier of the game of boules, the player is presumed to throw “a certain number of bowls that are unﬁt for serious attempts and […] not even aimed at the target”. These bowls travel only a small fraction of the distance separating them from the target, and they may be compared to “children who die in their infancy”. Premature adult deaths resemble bowls “at risk of being stopped by unforeseen obstacles at any point on their paths up to a certain limit”.

To identify the normal group of deaths, it is necessary to deﬁne the boundaries with precision, and therefore to characterize what distinguishes normal deaths from premature adult deaths. A death at 55, for example, may be either a normal death or a premature adult death. There is, writes Lexis, “a transitional region where the extreme elements of the normal group and of the premature-deaths group overlap […] but, in the tables I have examined, this hidden part of the normal group forms less than a quarter of the total [of normal deaths]” [21].

The “hidden part” of the normal group of deaths (Figure 2) corresponds to the deaths counted as premature adult deaths when in fact they belong to normal mortality, i.e. to the dispersion around the central value.

Figure 2

Mortality curve by age

Note: The three groups of deaths are shown: premature child deaths (descending section of curve between birth and age B), premature adult deaths (area containing deaths shown by + sign between age B and that corresponding to – ε) and normal death (all those bounded by the bell-shaped curve).

Source: Lexis, 1878, p. 449.

In Zur Theorie der Massenerscheinungen in der menschlichen Gesellschaft (On the Theory of Mass Phenomena in Human Society), Lexis sets out in very clear terms the problem posed by the distinction between premature deaths and normal deaths:

“One might be confused by the overlapping of premature deaths and normal deaths. What needs to be realized, however, is that the causes of a death regarded as premature when it affects a 30-year-old, may, when a person in an age group over 50 is involved, occasion a death legitimately classiﬁed among the ‘normal’ cases. In older age groups, there are also genuinely accidental cases that could be regarded as abnormal deaths, but their number is proportionally so small that one can ignore them.” [22]

Classifying deaths into normal and premature requires measuring the dispersion around the central value. This leads Lexis to characterize normal mortality by three values: normal age at death, the proportion of deaths belonging to the normal group, and probable error.

2. Normal age at death

In Zur Theorie der Massenerscheinungen… [23], Lexis presents in detail the calculation of the probable error, of which he provided only the results in his 1878 paper. The French mortality data compiled by Jacques Bertillon showed the largest number of male deaths occurring in the 70-75 age group (Table 2).

Table 2
Extract from Jacques Bertillon’s life table for men, reproduced by Wilhelm Lexis (unspecified year)

IMGIMGMen Dying at age according to the ta...IMGIMF

Lexis observes that from a total of 500 births [24], 40 deaths occurred between the ages of 70 and 75. He accordingly designates the central value of that age group, i.e. 72.5 years, as the normal age at death. Since Lexis works from the hypothesis that all deaths over that age are “normal deaths”, the evolution of mortality after age 72.5 enables him to compute the probable error, i.e. the age interval in which a person has a one in two chance of dying.

Given that the mortality curve obeys a law of accidental errors and that all deaths of men over 72.5 years are normal deaths, Lexis uses the curve’s symmetrical property to estimate the number of normal deaths occurring before the normal age at death (Figure 3). Of the 100 deaths occurring after the normal age at death [25] (72.5 years), 58 take place between the ages of 72.5 and 80 (Table 2). Consequently, 116 deaths of men aged 65-80 [26] belong to the normal group: these deaths therefore represent 116/200ths or 58/100ths of all normal deaths.

Figure 3

Distribution of male deaths in France in normal age group

IMGIMGDistribution of male deaths in France in normal ag...IMGIMF

Note: Table 2 includes six deaths over age 90, which, by virtue of the curve’s symmetry, can be classiﬁed into four deaths at ages 90-95 and two deaths at over 95.

Source: after Lexis.

Since reality is assumed to conform to theory, the probability that an observation error lies between – u and u is given by the following integral [27]:

From the numerical table giving for each u the value F(u), for F(u) = 0.58 we obtain a value of 0.57 for u.

Knowing the probable error x and the value u of the function F(u), Lexis calculates the precision parameter h:

Since u = 0.57 and x = 7.5 years (difference between 80 years and 72.5 years), h is equal to 0.076.

Lexis is then able to determine the probable error [28]. This is used to deﬁne the interval — centred on the normal age — in which there is a one in two chance of dying. In this case, the value of F(u) is 0.5, since there must be a 50% chance of being in the interval; by interpolation, the table gives a value of 0.4769 for u, and a precision parameter h of 0.076; the probable error x given by

is 6.275 years. Consequently, half of all men in the normal group die between the ages of 72.5 – 6.275 and 72.5 + 6.275, i.e. between the ages of 66.225 and 78.775.

In addition, of the total 500 deaths [29], 100 occur after age 72.5 and are all assumed to be normal. Given the symmetry of the normal-deaths curve, 100 occur before age 72.5 and are also normal. Consequently, there are 200 normal deaths, which therefore represent 40% of all deaths in the generation studied.

Wilhelm Lexis characterizes normal mortality by three values, which are, in the case of France and for males: 72.5 years for the normal age at death; 40% of male deaths in the normal group of deaths; 6.275 years for the probable deviation around the normal age.

In Zur Theorie der Massenerscheinungen… [30], Lexis reports the characteristic values of normal mortality for other European regions and countries in the second half of the nineteenth century (Table 3). From this it appears that, except in France, Switzerland, and Bavaria, the normal age at death for women exceeds that of men, although the differences are generally small. The only signiﬁcant exceptions are Sweden and Belgium, with differences of 3 and 5.5 years respectively to the advantage of women.

Table 3
Estimate of normal age at death, normal group as share of total, and probable error for selected European regions and countries in second half of nineteenth century

IMGIMGMales Females Normal age at death (y...IMGIMF

For each sex, the differences in normal length of life can be large, even within Europe. The normal length of life of men in Norway is seven years longer than the value calculated for Belgium. For women, the largest gap is six years.

The concentration of deaths in the normal group is also variable. Nearly one-half of Norwegian men die at ages situated in the normal group of deaths; in Bavaria, the proportion is only 31%. In Norway, the concentration of deaths in the normal group is even slightly more marked for women (54%).

The probable error is nearly always in the range of 6-7 years. Belgium is an exception, with a probable error for men of almost 9 years.

IV. Reception of Lexis’s hypothesis in the late nineteenth century

The discussions generated by Wilhelm Lexis’s paper at the 1878 Congress — primarily the commentary of Bertillon — reveal contrasting perceptions of the mortality process. A few years later, such prominent ﬁgures in the scientiﬁc world of the time as Luigi Bodio, Luigi Perozzo, Émile Levasseur, and Vilfredo Pareto drew on the German statistician’s theory in their own work.

1. The 1878 Congress of Paris: Bertillon’s commentary

In his response to the paper delivered by Lexis, Jacques Bertillon [31]appears to endorse the author’s theory, since according to him it allows a “division of human life into several categories”. He also plays down the signiﬁcance of expectation of life at birth in the case of a high dispersion of ages at death:

“[…] our results lose much of their precision when we combine probabilities so very different as we do when calculating the duration of the mean length of life. In France, it is about forty years, even though this is exactly one of the ages at which death most rarely occurs. Hence, in the assertion of this average value there is something strangely at odds with what we all know.” [32]

Bertillon points out that an expectation of life of forty years does not indicate “the true probability of death” since the highest risks occur “in the ﬁrst years of life, or well after the ages of 65, 70, and 75” [33]; hence it does not explain the distribution of deaths by age.

In his paper to the Paris Congress, Lexis divided total deaths into different categories, but the only one he considered to be really important was the “normal group” [34]. Jacques Bertillon was favourable to the proposed classiﬁcation, but differed from Lexis in drawing a parallel between infant and youth mortality, and old-age mortality: “it would therefore be useful for us to examine these two age groups separately: those who have had an enduring calling for life, and those who have made but a ﬂeeting appearance on the world’s stage” [35].

Bertillon was in fact seeking to establish a “break” in the distribution of deaths by age, based on the observation that mortality is minimal in adolescence. This break, he maintained, could be used to delimit an initial homogeneous set of deaths: “[…] the chances of dying are lowest at ages 10-15; I would be quite inclined to treat separately the lives of those who die after their 15th year” [36].

While repeatedly emphasizing the interest of Lexis’s paper, Jacques Bertillon fails to note the originality of this vision of a normal mortality as an expression of the “nature of things” and which characterizes “the general mortality conditions in a generation” for each country and in different periods. The only concern of Bertillon seems to be the statistical analysis of mortality data.

2. An application of Lexis’s theory to Italian data: Luigi Perozzo

In an article published in 1879, Luigi Perozzo [37] applies Lexis’s theory to Italian data for the period 1872-76 [38]. For the Italian statistician, the theory, which he describes as a law (la legge Lexis), may express merely a numerical series or, on the contrary, an authentic natural phenomenon [39].

Perozzo uses the Lexis method directly to estimate the three characteristic values of normal mortality for Italy. He bases his calculations on data extracted from bills of mortality.

The 1879 article quotes two letters from Lexis [40], written in French, in which he comments on Perozzo’s work. This correspondence gives Lexis the opportunity to specify the ﬁeld of application of his theory, which he also terms “hypothesis” [41]. The calculations must be performed on data relating to a single generation (“a single group of births”), which is not the approach followed by Perozzo, who instead uses data compiled by calendar year. For the use of the bills of mortality to be valid, the number of births must not vary too greatly over time:

“One must conclude from this […] that the three elements of the theoretical curve, namely, normal age, normal deviation and normal group (per cent), have remained roughly stable for the 1766-1816 generation, since otherwise the age groups of deaths for a single chronological period could never be substituted for those of a life table.” [42]

Wilhelm Lexis notes that the normal group of deaths is very small in Italy by comparison with the other European countries. The reason is Italy’s high “infant” mortality [43]. In his letter to Perozzo, Lexis also introduces marital status as a source of “divergence” between computed and observed values: “the groups of living persons who supply the deaths undergo changes from two causes: death and entry into another civil status”. The normal age of single persons may thus be lower since, in surviving, they tend to marry. In consequence, their deaths are included in the estimation of the normal age of death for married people.

Luigi Bodio himself returned to the Lexis hypothesis a few years later in the course of a congress held in Italy.

3. The 1887 Congress of Rome: Bodio and “the normal measure of life”

At the 14 April session of the Congress of the International Statistical Institute (ISI) held in Rome in 1887, Luigi Bodio [44] reviewed the conditions of hygiene and health in Italy. He ended his presentation with a discussion of mortality curves by age [45].

Like Bertillon, Bodio focused primarily on the division of deaths into three age groups. He seemed to distance himself somewhat from Lexis’s method yet, at the same time, to take it up for his own purposes:

“Examining the trajectory of the [mortality] curve, Mr. Lexis believed he could infer from it the normal measure of life. The increase in mortality around the 72nd year and its symmetrical decrease on either side of this peak indicate the existence of a type to which men conform. Just as nature gives men of a speciﬁc nationality a certain stature and a certain weight for each age group, it also endows them, depending on certain climate conditions, etc., with a speciﬁc vitality, thanks to which they ordinarily reach the age of 72, rather than 68 or 75.” [46]

The conclusion of the paper is, however, unequivocally in favour of the Lexis method since for Bodio it yields a “nearly exact coincidence of the theoretical curve, which could be plotted from two or several ordinate values, with the actual curve, based on experimental data”.

Two important works, published by economists in the 1890s, contained further references to Lexis — proof of the genuine interest his research aroused in the scientiﬁc community of the day.

4. Levasseur’s La population française and Pareto’s Cours d’économie politique

In 1891, Émile Levasseur [47] published the second volume of La Population française [48] work in which he mentioned Lexis in connection with the “two periods of high mortality”.

Levasseur returns to the notion of a division of ages at death:

“There are two age groups, therefore, that supply death with an abundant harvest: infancy, because there are many infants, and many of them die; and old age between the years of 60 and 80, because it experiences high mortality and its ranks are still sufﬁciently full to provide a large contingent. Between ages 5 and 60, death harvests little, because the living are endowed with powerful resistance; after age 80, it reaps nearly all, but the ears of wheat have become scarce.” [49]

The economist seems to adopt Lexis’s view of the mortality process when he employs, in a highly detailed form, the comparison between human life and the throwing of an object at a target [50]:

“A player has beside him a stack of disks that he takes one after the other without choosing them [51]; he casts them with all his strength in the same direction towards his target. Some of these disks, slipping from his hands, fall ﬂat at his feet without rolling; others, being ill-formed, do not roll, or they break and pile up a few steps from the player, forming a heap that is the stack of missed shots.

The others roll, initially at high speed, and reach roughly the same distance, because they are cast by the same hand and with an almost constant force. Towards the end of their journey, they slow down, falter, and eventually fall; they too pile up in large numbers around the target. This second pile is so arranged as to form a peak and two slopes that descend smoothly along a binomial curve.

A small number of disks have stopped on the way because they were cast less skillfully by the player or met an obstacle.

Beyond the target, also scattered here and there on the ground, are disks to which an extraordinary spin had been imparted.

The playing ﬁeld is the course of life, and the disks stand for men. The vital force that inhabits men is comparable to the force that the player imparts to the disks. The constitutions that are not viable stop at the outset, like the ill-formed disks. The vigorous bodies go approximately up to the mean term: this term (which differs from expectation of life, which we discuss later) lies between ages 70 and 75; few go beyond it. An adolescent of good constitution can therefore reasonably hope to reach the age of 70-75, though it must be remembered that some disks are left along the way.” [52]

Levasseur does not use the expression “normal life,” although that is the logical conclusion of his remarks. He speaks of the “mean term” (terme moyen) of life, which he ﬁnds useful to distinguish from expectation of life and clearly identiﬁes as the age interval containing the maximum number of deaths. The “vital force” referred to by Levasseur is comparable to the concept of force used in physical science. In determining the length of life of each individual, the mortality process — for both Levasseur and Lexis — is analogous to a throw propelled by a “roughly constant force”. In the absence of what can be assimilated to accidental errors, each person must, as a result of nature, live a speciﬁed number of years: the “mean term” of life in Levasseur’s terminology, the “normal length” of life for Lexis.

Another inﬂuential author who referred to Lexis’s theory was Vilfredo Pareto [53]. In volume 1 of his Cours d’économie politique (1896) [54], Pareto devotes a chapter to “personal capital” (capitaux personnels), in which he addresses population issues — in particular the analysis of the relationship between demographic growth and economic growth, the critique of Malthusian theory, and the estimation of the value of man [55].

Pareto begins his discussion of “personal capital” with a study of the “survival law” of a population, in which he refers to the “[t]heory of Mr. Lexis” [56]. Pareto does not mention the German statistician’s paper at the Congress of Paris but quotes his Zur Theorie der Massenerscheinungen…, published the previous year. However, Pareto reproduces the curve of deaths by age as published in the 1878 Congress proceedings (Figure 2), not the version included in the German work of 1877 [57].

In fact, the Italian economist conﬁnes himself to a simple presentation of the curve, as constructed and interpreted by Lexis. He refers to the “so-called ‘normal’ deaths” and notes that, according to Lexis, “the maximum ordinate [of the normal-death curve] is the normal age at death,” and he repeats — without examining it in detail — the comparison between mortality and “aiming at a target.” Pareto also refers to the studies by Perozzo and Bodio on normal mortality, but nowhere alludes to the notion of a natural order of mortality.

* * *

At the International Congress of Demography of 1878, Lexis presented his theory of normal life, already elaborated in 1877 in Zur Theorie der Massenerscheinungen in der menschlichen Gesellschaft. The theory is based on a distribution of deaths into three age groups. One of these encompasses the length of life that each individual could hope for in the absence of premature mortality. In the same way that projectiles scatter at random around the centre of a target, so individual lengths of life are distributed around a central value — “normal life” — according to the law of accidental errors.

Lexis undertakes an original mathematical and statistical analysis of mortality. His approach is both empirical and theoretical, as was noted a few years later by Bodio at the Congress of the International Statistical Institute in Rome. For Lexis, mathematical probability “ﬁnds itself, so to speak, at the heart of the collective phenomenon”, and mass phenomena are “empirical expressions” of that probability. In Zur Theorie…, Lexis had already drawn a distinction between the empirical approach that permits measurement of the phenomenon, and the theoretical approach that explains it. The “theoretical formula […], claims the statistician, represents the course of things in accord with the abstract law of probability, and it gives the only possible rational explanation of the symmetry of deaths”.

Wilhelm Lexis attributes the stability of certain values, of which the normal length of human life was an example, to “the intervention of controlling forces, calculated compensations, or coercive laws”. Over and above the variations that are governed by the law of accidental errors, the values — though stable — may vary over the long-term. Lexis calls these “physical” variations; they are secular variations “of a historical nature”, related to the “changes in the essential constitution of the collective life of a population”. The length of human life and its change over time are conceived by Lexis as an expression of “the nature of things”.

Acknowledgments

The authors would like to express their warm thanks to Martine Deville, without whom some basic source material would have been unavailable. They also thank the referees for their highly constructive comments.

NOTES

[*]

Institut National d’Études Démographiques, Paris.This article is an expanded version of a paper given at the “Lexis in Context” seminar, Max Planck Institut für demograﬁsche Forschung (Rostock, 28-29 August 2000).Translated by Jonathan Mandelbaum.

[1]

Lexis (1837-1914) studied law, mathematics, and natural science. After teaching mathematics at the Bonn lyceum (high school), he served as Extraordinary Professor of Political Economy at the University of Strasbourg, and Professor of Geography, Ethnography, and Statistics at Dorpat (now Tartu, Estonia). In 1878, he held the Chair of Political Economy at the University of Freiburg im Breisgau (Grand Duchy of Baden). In 1889, he became Vice-President of the International Statistical Institute and, in 1895, the head of the ﬁrst Department of Actuarial Science at the University of Göttingen.

[2]

W. Lexis, “Sur la durée normale de la vie humaine et sur la théorie de la stabilité des rapports statistiques”, Annales de démographie internationale, 1878, II, Paris, pp. 447-62.

[3]

Väinö Kannisto has applied Lexis’s method to recent data: “Mode and dispersion of the length of life”, Population: An English Selection, 13-1, 2001, pp. 159-72.

[4]

Statistique de la France, second series, vol. IV, part 1, “Mouvement de la population pendant l’année 1854,” Strasbourg, Berger-Levrault, 1857, p. xliii.

[5]

Georges de Serbonnes, “La vie moyenne de la Table H^MF”, Le Moniteur des Assurances, 84, 15 September 1875, pp. 346-348.

[6]

Hence

[7]

“Qu[e]telet’s remarkable research has taught us the interesting fact that individuals belonging to a given nationality are more or less exact copies of a model of given proportions […]” (Lexis, 1878, p. 447).

[8]

Adolphe Quetelet, Anthropométrie ou mesure des différentes facultés de l’homme, Brussels, Muquardt, 1871, p. 5.

[9]

Quetelet, 1871, p. 10.

[10]

Adolphe Quetelet, Sur l’homme et le développement de ses facultés ou essai de physique sociale, Paris, Bachelier, 1835; reprint Fayard, Paris, 1991, p. 31.

[11]

As Quetelet noted in the Bulletin of the Central Statistical Commission of the Kingdom of Belgium — a commission of which he was currently chairman — this distance makes it possible to develop a science of collective phenomena:“[…] by losing sight of individuals, one can unravel, through the social phenomena that dominate the masses, a set of laws that one determines with extreme precision. The initial obstacle was the belief in man’s free will; one knew that his will is an elusive cause, located outside the scope of any law; from this, it was concluded that the effects of that cause would thus be impossible to determine; but one overlooked the fact that man’s will ceases to act beyond certain limits where science begins, and that the effects, which appeared to be so powerful — such as those that people have always thought to be present at the birth of things — could be judged practically non-existent if they are examined from a collective viewpoint. Experience indeed soon proved to the most lucid observers that individual wills neutralize one another amid general wills.”Bulletin de la Commission centrale de statistique, Kingdom of Belgium, Ministry of the Interior, vol. VIII, Brussels, 1860, pp. 433-34.

[12]

Quetelet, 1835, p. 37 (author’s italics).

[13]

Adolphe Quetelet, Du système social et des lois qui le régissent, Paris, Guillaumin, 1848, book 1, section 1, chapter 2, pp. 13-14.

[14]

In his Anthropométrie…, Quetelet (1871, p. 22) reasserts the value of multiplying the observations of a given type: “If I had a hundred different persons measure the Apollo Belvedere, and if I then carefully computed the average of the hundred numbers denoting the size of the head, or of the eyes, or of the mouth, etc., I could with all these averages reconstruct the primitive type. This would be impossible only in the absence of a general type, whose existence everything points to here”.

[15]

Quetelet, 1835, p. 44.

[16]

Michel Armatte, “Théorie des erreurs, moyenne, et loi ‘normale’”, in Jacqueline Feldman, Gérard Lagneau, and Benjamin Matalon (eds), Moyenne, milieu, centre. Histoires et usages, Paris, EHESS, 1991, pp. 67 and 68.

[17]

Michel Armatte, “La moyenne à travers les traités de statistique du XIX^e siècle”, in Jacqueline Feldman et al., op. cit., p. 93.

[18]

Lexis, 1878, p. 450.

[19]

In Zur Theorie der Massenerscheinungen in der menschlichen Gesellschaft, published in Freiburg im Breisgau in 1877, Lexis had already used the metaphor of the game of boules, specifying a distance to the target “of about 70 feet”, to state that the bowls were distributed on the near side and far side of the target in accordance with mathematical error theory, the dispersion being determined by the player’s skill. Note that this distance of 70 feet recalls a life span of 70 years.

[20]

The term “accidental” expresses, in the case of young children, the occurrence of an event that brings life to a premature end; in the case of the normal group of deaths, the same term describes a random dispersion around a central value.

[21]

Lexis, 1878, p. 451.

[22]

Lexis, 1877, p. 45. (Translator’s note: quotation translated into English from the authors’ French rendering of the original German.)

[23]

Lexis, 1877, pp. 48-9.

[24]

“The ﬁgures are reduced to a cohort of 500 births […]” (Lexis, 1878, p. 451).

[25]

Implying that the total number of normal deaths is 200.

[26]

72.5 ± 7.5.

[27]

This is Laplace’s integral. For a description of the statistical analysis of accidental errors, see, for example, Émile Borel, Traité du calcul des probabilités et de ses applications, vol. 1, fascicle II, R. Deltheil, “Erreurs et moindres carrés”, Paris, 1930.

[28]

Or probable deviation.

[29]

Cf. note 24.

[30]

Lexis, 1877, p. 63.

[31]

The Bertillon family distinguished itself in France in the second half of the nineteenth century. Louis-Adolphe Bertillon, a physician, took a special interest in hygiene and the statistics of the causes of death; his son Jacques, also a physician, was a founder of the International Statistical Institute. His ﬁelds of interest were population decline and alcoholism (cf. Jacques and Michel Dupâquier, Histoire de la démographie, Paris, Perrin, 1985, pp. 401-406). Louis-Adolphe was the son-in-law of Achille Guillard, who introduced the term “demography” in his Éléments de statistique humaine or démographie comparée…, Paris, Guillaumin, 1855.

[32]

Bertillon in Annales de démographie internationale, 1878, p. 461.

[33]

This argument had been expounded by Christiaan Huygens in 1669, in his correspondence with his brother Lodewijk. The Huygens brothers also discuss this comparison between averages and probability.

[34]

“[…] the physiological conditions of our species naturally include a certain normal length of life […]” (Lexis, 1878, p. 450).

[35]

Bertillon in Annales de démographie internationale, 1878, p. 461.

[36]

Bertillon in Annales de démographie internationale, 1878, p. 461.

[37]

Luigi Perozzo was at that time an engineer at the Italian Statistical Ofﬁce. He is known for the stereogram, a spatial representation of the dynamics of a population.

[38]

Luigi Perozzo, “Distribuzione dei morti per età”, Annali di Statistica, 2nd series, vol. 5, 1879, pp. 75-93.

[39]

“[…] possono dubitare se la veriﬁcazione della legge sia dovuta all’artiﬁciale produzione delle cifre a cui si applica od esprime un vero fatto naturale”, (Perozzo, 1879, p. 77).

[40]

The ﬁrst letter is addressed to “Louis” Bodio, with a request that he forward it to Luigi Perozzo; the second is addressed directly to Perozzo.

[41]

“I thank you [‘Louis’ Perozzo] for your positive reaction, in your interesting study, to my theory or rather my hypothesis.” (Lexis in Annali di Statistica, 1879, p. 81).

[42]

Lexis in Annali di Statistica, 1879, p. 81.

[43]

“[…] since 55% of deaths belong to classes 0-15, there remain only 45% to be divided between premature [deaths] and the normal group. In Norway, for example, three-quarters of children born alive survive beyond age ﬁfteen, and that is why the normal group is far larger there than in Italy.” (Lexis in Annali di Statistica, 1879, p. 82).

[44]

Luigi Bodio, at the time, was Director General of Statistics of the Kingdom of Italy. He had attended the session of 9 July 1878, in which Lexis had presented his concept of the normal lifetime.

[45]

Lexis attended this session of 14 April.

[46]

Luigi Bodio, “Quelques renseignements sur les conditions hygiéniques et sanitaires de l’Italie”, in Bulletin de l’Institut international de statistique, vol. II, part 1, 1887, pp. 264-84.

[47]

A liberal economist, Pierre Émile Levasseur (1828-1911) was Professor at the Collège de France and Administrator of the institution, a member of the Institut de France, and President of the Société de Statistique de Paris. His works include a multi-volume Histoire des classes ouvrières en France (History of the Working Classes in France) (1859-67).

[48]

The ﬁrst volume was published in 1889.

[49]

Levasseur, La Population française, 1891, book II, chapter 13, p. 178.

[50]

Levasseur speaks of a “disk” (disque) rather than a “bowl” (boule) as Lexis does.

[51]

Hence chance plays a role.

[52]

Levasseur, 1891, book II, chapter 13, pp. 178-9.

[53]

Vilfredo Pareto (1848-1923) succeeded Léon Walras to the Chair of Political Economy at the University of Lausanne. He contributed to the “neo-classical revolution” in economics.

[54]

Vilfredo Pareto, Cours d’économie politique, vol 1, 1896, and vol 2, 1897, new edition by G. H. Bousquet and G. Busino, Geneva, Droz, 1964.

[55]

Pareto makes several references to Levasseur’s work.

[56]

“Mr. Lexis has produced an important theory, of which we shall say a few words” (Pareto, 1896, § 158, p. 77).

[57]

Pareto’s name does not appear in the list of those attending the 1878 Congress: cf. Annales de démographie internationale, 1878, pp. 303-6.

Volume 58 2003/3

Wilhelm Lexis: The Normal Length of Life as an Expression of the “Nature of Things”

Jacques Véron [*] Jacques Véron, Institut National d’Études Démographiques, 133 bd Davout, 75980 Paris Cedex 20, Tel: 33 0(1) 56 06 21 76, Fax: 33 0(1) 56 06 21 99 Jean-Marc Rohrbasser [*]

I. Mortality: mere statistical regularity or law of nature?

II. A single model of man

III. The normal lifetime

IV. Reception of Lexis’s hypothesis in the late nineteenth century

Acknowledgments

NOTES

Jacques Véron []* Jacques Véron, Institut National d’Études Démographiques, 133 bd Davout, 75980 Paris Cedex 20, Tel: 33 0(1) 56 06 21 76, Fax: 33 0(1) 56 06 21 99 Jean-Marc Rohrbasser []*