Comparing population distributions from
bin-aggregated sample data: An application to
historical height data from France
Jean-Yves Duclos∗, Josée Leblanc†, David Sahn‡
20th April 2009
This paper develops a methodology to estimate the entire population dis-
tributions from bin-aggregated sample data. We do this through the estima-
tion of the parameters of mixtures of distributions that allow for maximal
parametric flexibility. The statistical approach we develop enables compar-
isons of the full distributions of height data from potential army conscripts
across France’s 88 departments for most of the nineteenth century. These
comparisons are made by testing for differences-of-means stochastic dom-
inance. Corrections for possible measurement errors are also devised by
taking advantage of the richness of the data sets. Our methodology is of
interest to researchers working on historical as well as contemporary bin-
aggregated or histogram-type data, something that is still widely done since
much of the information that is publicly available is in that form, often due
to restrictions due to political sensitivity and/or confidentiality concerns.
Key words: Health, health inequality, aggregate data, 19th- century France,
JEL Classification: C14, C81, D3, D63, I1, I3, N3.
∗Institut d’Anàlisi Econòmica (CSIC), Barcelona, Spain, and Département d’économique and
CIRPÉE, Université Laval, Canada; email: email@example.com
†Department of Finance, Ottawa, Canada; email: Leblanc.Josee@fin.gc.ca
‡Cornell University, Cornell, USA; email: firstname.lastname@example.org
There are many reasons to consider dimensions of well-being other than in-
come or expenditure, both normative and practical. Following Sen (1985) and
others, for example, one may wish to consider well-being as multidimensional,
comprising characteristics such as , good health, nutrition, literacy, and freedom
of association. Income may be instrumentally important to achieve these ends, but
it is the capabilities themselves that are intrinsically important and merit recogni-
tion and measurement in their own right. Poverty can thus be defined as depriva-
tion of basic capabilities or the failure of certain basic functionings, not just low
levels of income. Deprivation of capabilities can also in turn contribute to low
material standards of living.
This paper focuses on health, which is certainly important in a multidimen-
sional understanding of well-being. In fact, even in a purely unidimensional wel-
farist framework, it can be argued that health contributes to welfare at least as
much as income. Income may not even be a sound directional indicator of overall
welfare in some environments. As Floud (1984) writes, “[t]here is little point in
an improvement in real wages which is bought at the expense of a miserable life
and an early death”. The example of the United States, for instance, shows that
although health and economic growth have generally converged during the twenti-
eth century, they diverged during the nineteenth century (“the antebellum period”)
(Costa and Steckel 1997); strong economic growth during the nineteenth century
coincided inter alia with a decrease in body-heights. Whether income is more
indicative of welfare than health in such circumstances is then open to debate.1
Beyond the conceptual arguments, there are many practical reasons to mea-
sure well-being in non-income dimensions. First, measurement difficulties may
be less of a problem for some non-income variables. Collecting income (or expen-
diture) data is a complex procedure that contrasts, for instance, with the relatively
straightforward procedure of collecting anthropometric data — data that also suf-
fer typically less from misreporting. Measurement errors can of course still affect
such data, but anthropometric data (unlike other self-reported and subjective mea-
sures of health) are more likely to be uncorrelated with important variables of
interest — such as the welfare variable itself. Note also that health can be more
1More generally, measures of health are often not highly correlated with incomes, either within
a given country or across countries (Haddad and Ahmed 2003, Behrman and Deolalikar 1988,
Behrman and Deolalikar 1990, Appleton and Song 1999), suggesting among other things that
health variables may provide significant information on welfare that is not captured by income
easily measured at the individual rather than at the household level, thus largely
avoiding the need to make difficult assumptions on the well-being of individuals
which requires an assessment of both the needs of individuals and how resources
are allocated among household members relative to needs.
Perhaps most importantly, the choice of a welfare indicator obviously depends
on the availability of data. Research on the distribution of well-being routinely
takes advantage of the availability of large-scale micro-data sets that provide de-
tailed information on money-metric and non-monetary indicators2. The choice of
welfare indicators is much more constrained for studies covering earlier historical
periodswhere the available data is more limited and often of variable quality. It
may for instance be the case that only part of the initially gathered welfare infor-
mation was preserved, or that it comes from sources whose primary goal was not
to capture data representative of the population3. Furthermore, historical house-
hold level data on incomes and expenditures are particularly difficult to collect
since economies were less monetized, transactions were often in-kind, consump-
tion was largely from home consumption rather than market purchases, and tax
authorities did not have good records of gathering and verifying information on
Fortunately, quality historical data are widely available from many countries
on one of the clearest manifestations of health and nutritional status, stature. It
is also now well established that one of the best global indicators of living condi-
tions is height, standardized for age and gender (de Onis, Frongillo, and Blossner
2000). More specifically,stature is the outcome of a combination of inputs that
affect nutrition and disease, such as the local health environment, access to clean
2Some of the non-monetary indicators found in the literature include body-height (Fogel, En-
german, Floud, Steckel, Trussell, Wachter, Sokoloff, Villaflor, Margo, and Friedman 1982, Steckel
and Floud 1997, Wagstaff 2002, Pradhan, Sahn, and Younger 2003, Sahn and Younger 2005),
body mass index (Costa and Steckel 1997), amount of abdominal fat (Costa and Steckel 1997),
birth weight (Costa 1999), life expectancy (Whitwell, Souza, and Nicholas 1997, Goesling and
Firebaugh 2004), the overall mortality rate or mortality before a certain age (Costa and Steckel
1997, Floud and Harris 1997, Weir 1997, Whitwell, Souza, and Nicholas 1997, Wagstaff 2002),
the prevalence of chronic or severe illness (Costa and Steckel 1997, Wagstaff 2002, Anson and
Sun 2004), the individual’s own assessment of his or her health (Deaton and Paxson 1998, Nolte
and McKee 2004), disabilities (Anson and Sun 2004), difficulties accomplishing tasks (Anson and
Sun 2004), and mental illness (Anson and Sun 2004).
3For example, Costa (1999)’s data on the birth weight of babies were obtained from hospital
archives. These archives are not necessarily complete, since hospitals will not have kept every
patient file since 1848; moreover, the hospital registries may represent a biased sample of the pop-
ulation, since wealthier women were over-represented among those who gave birth in hospitals.
water, nutrient intake, maternal health status, health technology, the organization
of work, and so forth. In short, stature captures “multiple dimensions of the indi-
vidual health and development and their socio-economic and environmental deter-
minants” (Beaton, Kelly, Kevany, Martorell, and Mason 1990)4. And in particular,
heights of young men entering adulthood is a cumulative indicator of their overall
health and nutritional status during their formative years, particularly the period
prior to the beginning of puberty.
Economic historians have thus expended considerable effort to examine
changes in anthropometric outcomes of various populations over time (Fogel, En-
german, Floud, Steckel, Trussell, Wachter, Sokoloff, Villaflor, Margo, and Fried-
man 1982, Steckel and Floud 1997, Weir 1997, Deaton and Paxson 1998 and
Goesling and Firebaugh 2004).5An important concern that arises in how histo-
rians use anthropometric indicators is whether the summary health statistics they
employ, particularly measures of central tendencies, can adequately capture the
distribution of health. Indeed, it is increasingly recognized that looking at en-
tire distributions of health, just as economists have long done with incomes, can
provide valuable information that would otherwise be hidden by summary health
statistics, such as means and the share of the population that falls before a norma-
tive standard, or cut-off point, that may define poor health6.
This paper gives prominence to the distributional analysis of health by exam-
ining both the evolution and the distribution of heights throughout France in the
nineteenth century. More specifically, the paper uses a particularly rich data set
collected on men who were called up for possible conscription into the French
army during this period. The screening of all men at the age of 20 for manda-
4This is also supported by the existence of a significant correlation between body-heights and
various indicators of health — see for instance Wagstaff (2002) — suggesting that the choice of
an indicator other than body-height would yield similar results. Moreover, adult stature is not only
a good indicator of prior episodes of, infection and chronic disease, but it has also been shown to
be an important determinant of risk of morbidity and mortality.
5A number of other studies have also looked at health inequalities, using statistics such as
covariances (such as Deaton and Paxson 1998 and Anson and Sun 2004), differences between
various percentiles (Costa 1999), distances from the mean for different social classes (Anson and
Sun 2004), or inequality indices (e.g., Wagstaff 2002, Pradhan, Sahn, and Younger 2003, Goesling
and Firebaugh 2004, Sahn and Younger 2005). Studies of health inequality have also attempted to
explain whether inequality in some factors, such as income, can transmit itself into health inequal-
ity — see for instance Weir (1997), Anson and Sun (2004), and Nolte and McKee (2004)). Other
studies have decomposed the evolution of health inequality across factors, such as Pradhan, Sahn,
and Younger (2003), Goesling and Firebaugh (2004), Sahn and Younger (2005), and Wagstaff
6For instance, see Pradhan, Sahn, and Younger (2003) and Deaton (2003).
tory military service involved a physical examination, including measuring their
height. Thus, we have a virtually complete census of heights for each year, dis-
aggregated by administrative department. Data in such abundance for a whole
century are arare find. Consequently, socio-economic improvements as well as
periods of adverse conditions in 19th-century France can be expected to have an
observable effect measured by the stature achieved at 20 years of age - which is
what we measure in our data.
While individual heights were recorded at the time of conscription, the data
we have available are limited to the number of men that fall into classes, or bins
of height intervals, for each year and department. This raises obvious challenges
given our objective to compare the entire distributions of heights. These difficul-
ties are not unique to our data, or the paper’s historical period of interest. Even
today, much of the information that is used and publicly available on distributions
of income come from bin-aggregated, or histogram-type, data — important exam-
ples of this are the popular World Income Inequality and POVCAL databases7. In
several countries, the unit data from early household surveys have not survived; in
other cases, access to distant and/or more recent microdata is restricted by politi-
cal sensitivity and confidentiality concerns. We therefore propose and implement
sample data through the estimation of the parameters of mixtures of distributions
that allow for maximal parametric flexibility8. While we only apply our method
to the historical height data from France, this approach will be of general interest
to both historians and other researchers working on contemporary bin level data
on household incomes, heights, and other indicators of well-being.
Our data are abundant, consisting of more than 6000 different distributions
of heights for the period 1819 to 1900 over 90-some departments. Using the
sampling distribution of the estimators of the means and of the cumulative dis-
tributions of heights, we test for differences in means and also implement tests
for robust comparisons across time and regions. We are also able to correct our
inference procedures for the possible presence of measurement errors. This is
rarely possible to do with the usual data used for comparing welfare; it is made
feasible here by the richness of the data. We correct for measurement errors by
measuring and taking into account the noise that is (possibly) introduced by mea-
surement errors in the many year-to-year comparisons of the distributions that are
7See www.wider.unu.edu/wii d/wiid.htm and www.worldbank.or g/LSMS/tools/povcal/
8See for instance Bandourian, McDonald, and Turvey (2002) for a review of some of the more
restrictive functional forms that have been proposed in the literature.
made possible by our data. This renders the broader comparisons in which we are
interested robust to both sampling and measurement errors. Again, this is done by
estimating the parameters of mixtures of distributions with the maximum degree
of parametric flexibility and by therefore exploiting all the statistical information
that is present in the available height data.
In short, this paper develops a methodology that allows us to estimate the en-
tire population distributions from the bin-aggregated sample data. We go on to
illustrate how this methodology can be applied to a rich data set from France of
20-year-old army conscripts, and thereafter employ the generated distributions to
address one of many possible questions on the evolution and the distribution of
health and welfare in 19th-century France. These data are introduced in Section
2. The methodology for deriving from bin data the entire distribution as well as
the average height of each department-year is described in Section 3.1 and extracts
the maximum possible amount of information from the data. Statistical tests of
differences in mean heights and in the entire distributions are performed using the
test statistics and sampling distributions presented in Section 3.2. Further details
in carrying out stochastic dominance tests appear in Section 3.3. The empirical
results are presented in Section 4, where two main questions are more particu-
larly considered to illustrate how our statistical procedures can be used. The first
question considers the evolution of body-heights in France from the beginning to
the end of the century; the second deals with the regional correlates of the distri-
bution of body-heights. Section 5 concludes the paper. The Appendix in Section
6 provides more technical details on the data, the estimation procedures, and the
adjustment for measurement errors.
The data we use are from the registries of potential conscripts into the French
army for the period 1819 to 1900, covering the 90-some departments of France9.
More precisely, they are from the "Comptes rendus statistiques et sommaires."
Depending on the year, they constitute either a "nearly" random sample, or a com-
plete census of young French men aged 20 years old for each year of the century
(this point is addressed in greater detail in the Appendix on page 22). They consist
of 6369 different datasets, each representing the distribution of body-heights for a
9We are very grateful to Gilles Postel-Vinay for his generous assistance in making available to
us, and helping us better understand the data. More details on the data set are also found in Sahn
and Postel-Vinay (forthcoming).
given department and a given year. In total, we have measurements on close to 15
million young French men over the course of the century.
A particularity of these data is that they are not available in a totally disag-
gregated form. Rather, they are grouped into "classes," each of which contains
individuals within a specific body-height range. The available data thus report the
number of individuals in each of these classes. Furthermore, neither the number
of classes, nor the boundaries between bins, were constant throughout the century.
The data from 1819 to 1829 are divided into 15 classes, those from 1830 to 1871
into 16 classes, and those from 1872 to 1900 into 9 classes. The class boundaries
for the various years are reproduced in Table 1. Note that the classes were based
on the old imperial measurements until 1867 (1 French inch = 27.07 millimeters)
so that 1.570 meters corresponds to 4 feet 10 inches, 1.597 meters to 4 feet 11
inches, etc. The metric measurements we employ are the public equivalent to the
imperial measurements used during these years.
These registries contain a considerable amount of information. Indeed, the
number of conscripts measured each year varies between one-quarter, and all, of
the male population aged 20. Between 1819 and 1830, approximately 80,000 men
were measured each year; from 1836 to 1885, approximately 150,000; and as of
1886, approximately 300,000. Only during this final period was the entire male
population aged 20 measured each year.
3Estimation and inference
3.1Estimation of complete distributions
That the data available for this study are grouped into classes raises difficulties
if the objective is to compare complete distributions of heights. To address this
challenge, we first need to estimate the continuous population height distributions
using the discontinuous sample histograms that are available. To do this, we solve
a system of (C − 1) equations in (C − 1) unknowns, where C is the number of
classes into which the heights are regrouped in the aggregated sample data. Each
of these equations captures the probability of belonging to one of the C classes of
height. Equation (1) defines such a probability for the class c of heights x, a class
whose lower bound is xcand upper bound is xc:
F(xc;ˆΘ) − F(xc;ˆΘ) =ˆHc,
parameters for this function, andˆHcis the proportion of heights between xcand
xcthat is observed in our data.
F is specified as a mixture of normal distributions, namely, as a weighted
sum of several normal distributions. We let the mixture use as many parameters
as is statistically possible given the grouped form of our data. This mixture of
normal distributions thus allows for the maximal possible amount of estimation
flexibility. Note that since the normal distribution is smooth, this property will
also be imposed on our estimated population height distributions.
Equation (2) provides an example of a “mixture” of three normal distributions
with a set of 9 parameters:
F(x;α1,α2,α3,µ1,µ2,µ3,σ1,σ2,σ3) = α1Φ
?x − µ1
?x − µ3
?x − µ2
+ (1 − α1− α2) Φ
where Φ is the distribution function of the standard normal distribution, and where
αd, µdand σdcorrespond respectively to the weight, the mean, and the standard
error of normal distribution d. Note that since α1+ α2+ α3 = 1, we can set
α3 = 1 − α1− α2. There are therefore 8 “free” parameters in (2). Thus, a
mixture of D ≥ 1 normal distributions contains 3D−1 free parameters. Similarly,
there are only (C − 1) “degrees of freedom” in data aggregated into C classes of
heights, since the probabilities of belonging to one class is one minus the sum
of the probabilities of belonging to the others. Hence, the problem is to solve a
system of equations such as (1), with c = 1,...,C − 1, using theˆHcobserved
probabilities of belonging to C classes and choosing D = C/3.
For some years, however, C/3 is not an integer. For those years we use instead
mixtures of (C + 1)/3 or (C + 2)/3 distributions, setting the last one (σC+1)
or two (µC+2and σC+2) parameters in the mixture to some pre-specified values.
These values are chosen as those that are estimated in the 1880 distribution of
individual heights, which is the only year for which we have access to the entire
set of individual-level data.
More technical details on the above estimation procedure can be found in the
Appendix, Section 6.2.
3.2 Test statistics
Once the parametersˆΘ in (1) are estimated for each year and each department,
we can proceed to assess the evolution of the distributions of health in nineteenth-
century France. We do this in two ways: first by comparing mean body-heights,
which is one of the most common procedures in the literature, and second by
comparing “health poverty rates”. Note that these poverty rates will be compared
across ranges of possible “health poverty lines”, which will amount to testing for
stochastic dominance of height distributions.
Mean height can be estimated as
The height poverty rate (“the poverty headcount”) is the proportion of individuals
below a height poverty line. Computing the poverty rate consists of evaluating the
distribution function at the poverty line z:
Once estimated, ˆ µ and F(z;ˆΘ) can be compared across departments and years.
For the comparisons of means, the null hypothesis is that the mean of distribution
B does not exceed the mean of distribution A, and the alternative hypothesis is
that it does. The test statistic that we use is then
ˆ µ =
ˆ µB− ˆ µA
var(ˆ µB) + ?
var(ˆ µ) =1
(x − ˆ µ)2dF(x;ˆΘ)
and n (which is always well above 500 in our data) stands for the number of sol-
diers over whom the aggregated bin data have been computed. Under the assump-
tion that population heights follow the flexible form given by (2) and that the two
means µBand µAare equal, the statistic (5) can be shown to follow asymptotically
a normal distribution with mean zero and unit variance. At the conventional 5%
level, the above null hypothesis will then be rejected if (5) is greater than 1.645.
For ordering poverty headcounts10, we use the test statistic
F(z;ˆΘA) − F(z;ˆΘB)
var(F(z;ˆΘB)) + ?
var(F(z;ˆΘi)) =F(z;ˆΘi) (1 − F(z;ˆΘi))
Under the null hypothesis of equality of the two distribution functions at z, and
that population heights follow the flexible form given by (2), the distribution
of F(z;ˆΘA) − F(z;ˆΘB) is asymptotically normal with mean 0 and variance
Note that the expressions ˆ µ, ?
are also as distribution-free as they can be. Note furthermore that the asymptotic
result for (7) is valid for the z located at the frontiers of the bins of the aggregated
data even when population heights do not follow exactly the flexible form given
by (2), since at such z we can estimate (7) and (8) directly from theˆHcin (1).
var(F(z;ˆΘB)) + ?
computed once the parametersˆΘ in system (1) are estimated. SinceˆΘ contain the
var(F(z;ˆΘA)) (since the samples from A and B are indepen-
var(ˆ µ), F(z;ˆΘA) and ?
var(F(z;ˆΘi)) are readily
A poverty comparison that uses (7) depends on the choice of the line z. It is
also evidently dependent on the choice of the distribution function as a “poverty
index”. To make the paper’s poverty comparisons more robust to such choices,
stochastic dominance tests can be performed by comparing poverty rates over
ranges of poverty lines. Pushing this approach farther, one can also compare cu-
mulative height distributions over the entire range of possible heights.
To see what this implies in terms of poverty rankings, note that the poverty
headcount F belongs to a general class of poverty indices, denoted as Π1(z+),
that can be defined with the help of two simple axioms and of a condition (see for
instance Duclos and Araar 2006, Part III). The first axiom, a monotonicity axiom,
says that an increase in the body-height of any one individual (provided that no
one else’s body-height decreases) should (weakly) reduce the value of a poverty
10See for instance Davidson and Duclos (2000, Theorem 1).
index. The second axiom, a symmetry axiom, says that interchanging the body-
height of any two individuals should not affect the poverty index. The condition
is that the poverty index should use a poverty line that is below z+.
We then say that distribution B poverty dominates11distribution A if and only
if the distribution function for B lies below that for A for all poverty lines in the
interval [0,z+]. Analytically, for generally-denoted poverty indices P(z) and a
distribution function F(x), we have
PA(z) ≥ PB(z)
⇔ FA(x) ≥ FB(x)
∀ P(z) ∈ Π1(z+)
∀ x ∈ [0,z+].
This results says that ordering poverty headcounts over all lines in [0,z+] also
ranks all poverty indices that meet the monotonicity and symmetry axioms, and
for whatever choice of poverty lines below z+.
For statistical and normative reasons (see Davidson and Duclos 2006), these
dominance tests are better implemented over ranges poverty lines ranging from of
z−to z+, rather than from 0 to z+(these tests are then denoted in the literature as
restricted stochastic dominance tests). Empirically, this interval will correspond
to [1.53,1.78], or from approximately the 3rd to the 97th percentile of the distri-
butions of heights observed in nineteenth century France. The null and alternative
hypotheses for the dominance tests that we conduct can then be written as:
F(zm;ˆΘA) ≤ F(zm;ˆΘB)
F(zm;ˆΘA) > F(zm;ˆΘB),
11In the first order, since we could also test for higher-order dominance comparisons — see
again Duclos and Araar (2006), Part III.
where the zi’s represent m points in the interval [z−,z+]. The null hypothesis is
an hypothesis of non-dominance of A by B. The alternative hypothesis is that B
The decision rule differs from that for simpler test hypotheses, since H0and
H1are sets of multiple hypotheses. The decision rule is to reject H0and conclude
that B dominates A if and only if each of the inequalities in H0can be rejected at
the 5% level. Since it involves testing separately over m hypothesis tests, this test
procedure is generally conservative: the 5% nominal level for each inequality test
in H0leads to a less than 5% probability of committing a Type I error of wrongly
rejecting the joint hypothesis of non-dominance of A by B when non-dominance
An illustration of a test procedure for stochastic dominance is provided by
Figure 1, which uses data from the Ain department. We test here whether distri-
bution B (1886) dominates distribution A (1819) (H0: 1886 does not dominate
1819). We see in the lower panel that the t test statistics (of equation (7)) exceed
the critical value (the dotted line) across the entire interval [1.53,1.78], so we can
reject H0and conclude that year 1886 dominates year 1819, that is, that year 1886
has less height for whatever choice of poverty measures in Π1(z+).
We now turn to the empirical results. Recall that the first step is to estimate an
entire population distribution function for each of our 6369 datasets. This is done
using the estimation techniques presented in Section 3. Once estimated, these
distributions generally fit the empirical distributions very well, as is illustrated
for instance in Figure 2. The histogram shows rectangles whose areaˆHc(see
(1)) comes from the aggregated data. The line is the estimated density function,
a mixture of normal density functions (whose cumulative distributive functions
appear in (2)).
Some estimated distributions are very close to the normal distribution, such
as the one of Figure 3. This is more often the case of the distributions with eight
parameters (a mixture of three normal distributions), e.g., for many of the dis-
tributions from 1872 to 1900. Other distributions are less smooth, as Figure 4
shows. It proved impossible to find a satisfactory solution to the system (1) for 3
out of the 6369 distributions of our database. This may be due to numerical limi-
tations in attempting to solve for systems of up to 15 equations in 15 unknowns,
in which each equation is a sum of as many as 6 normal distribution functions —
also computed numerically. It can also be because of difficulties inherent in fitting
empirical distributions that diverge (because, e.g., of sampling variability) widely
from the normal distribution. Representing less than 0.05% of our distributions,
these three distributions are dropped from our analysis; further details on them
can be found in Leblanc (2007).
4.1National distributions of heights
We first consider the overall distribution of heights of the French during the
19th century. Figure 5 illustrates mean body-heights in each department for every
year. Observe that there appears to be an upward trend in average body-heights.
This phenomenon is most clearly seen in Figure 6, which reproduces the national
mean of body-heights (i.e., all departments aggregated — more information can
be found in the Appendix) from 1819 to 1900. Note that the national mean seems
to have increased from around 163.5 to 165.5 centimeters between 1819 and 1900.
Figure 6 also suggests that this evolution of the mean was not constant. Finally,
observe in Figure 5 that there is considerable variation across departments.
Let us now turn to the year-to-year evolution of heights for each department.
For each year, Tables 2 and 3 show the percentage of departments within which
this year is better than (or dominates) the preceding year. Table 2 uses means and
Table 3 compares distribution functions at a fixed z.
Notice that there appears to be a great deal of variation in the means from
year to year. In Table 2, the percentages of departmental means dominating or
dominated by the preceding year hover around 15% to 35%. These percentages
therefore far exceed 5%, the level of the test (i.e. the proportion of times when
sampling error would have erroneously led us to conclude that there was domi-
nance when there was in fact none). The percentages of decline are also nearly as
high as those of increase.
Similar results in Table 3 are obtained for comparisons of the proportion of
individuals whose body-height was below 1.652 meters for the years 1819–1866
and 1.640 meters for 1867–1900.12The presumption that the variability in the
12These values were selected because they correspond to class boundaries (the proportion used
thus corresponds to the sum of the number of individuals in each class below this boundary) and
they differ because the class boundaries were changed between these two periods. They also
means actually springs from the raw data, and not from peculiarities in the esti-
mation methods, is thus supported by the fact that a similar dominance rate exists
in Table 3 as for means in Table 4.
There are two possible explanations to the relatively high rate of “acceptance”
of dominance. The first is that the distributions within a single department truly
do vary from one year to the next, and that the null hypothesis of non-dominance
of the distributions must naturally be rejected more often than the nominal 5%
level of the tests if we want our tests to have some power. The second, more cau-
tious, explanation starts with the presumption that there should be little difference
from one year to the next between the population distributions within a single de-
partment, and that we need to locate the source of the high rates of acceptance
of dominance in measurement errors. This would then admonish caution in our
interpretations of the dominance results.
The Appendix (Section 6.4) analyzes the effect of various possible sources of
measurement errors on the validity of our inference results. A relatively conserva-
tive approach that emerges from that analysis is to consider the year-year accep-
tance of dominance rankings to stem from department-year-specific measurement
errors that are identically, independently and normally distributed across the cen-
tury. This would suggest a standard deviation of those measurement errors of the
order of 1.7 times the size of the sampling error on the estimator of the mean —
or roughly 0.23 cm. Note that a standard deviation of 0.23 cm in the distribution
of (true) population average heights would also suffice to generate the high rates
of dominance acceptance that we observe in Table 2.
Thus, rather than conclude that one year dominates another on the usual ba-
sis of rejecting non-dominance at the nominal level of 5%, we will only draw
this conclusion if we can reject non-dominance of means at a nominal level of
20%, which is approximately the average rate of dominance rankings across de-
partments from one year to the next over the century. In the case of stochastic (or
distribution) dominance (Table 4), the corresponding average rate of dominance
rankings is about 1.5%. Even though this is smaller than the 5% nominal level
used in testing each of the null in the composite null H0in (10), recall from the
discussion on page 11 that the decision rule is to reject that composite null and
conclude that B dominates A only if all of the inequalities in H0can be rejected
at the nominal level. The test procedure is therefore inherently conservative, lead-
correspond to points outside of the tails of these distributions. In fact, approximately 60% of
individuals’ body-heights were less than 1.652 meters in 1819, and for 25% body-height was less
than 1.640 meters in 1886. Because of this, it is worth noting that only the information present in
the data (and not in the estimated distributions) were used for these tests.
ing to a probability of committing a Type I error that can be much lower than the
nominal level. Allowing for the presence of measurement errors in our context is
not enough to offset this. This explains why the 1.5% quoted above is below the
nominal 5% level.
4.3 Did the body-height of the French increase during the 19th
We then turn to the following question: “Did the body-height of the French
increase during the 19th century?” To answer this, we compare the distributions
of the first ten years of available data (1819–1828) with those of the last ten years
(1891–1900) on the basis of the elements discussed above — the means and the
entire distributions. Comparisons of department-years were performed depart-
ment by department, in order to account for “fixed effects” unique to each depart-
ment (slight variations in genetic heritage, different geophysical conditions, etc.).
If data are available for all years, this corresponds to a maximum of 100 com-
parisons per department. Overall, 7430 individual comparisons were performed.
In addition, aggregate analyses were also performed: for these same years, the
means and distribution for all of France were compared.
4.3.1Comparisons of means
Overall, 93.8% of the distributions at the end of the century had a statistically
greater mean than at the beginning. Thus, there was clearly a height progression
over the course of the century, even in light of the aforementioned conservative-
ness of our inference procedures.
One of the reasons why this proportion is not 100% can be found in devel-
opments that are specific to certain departments, as Figure 7 illustrates for the
Calvados department. We observe that, in this department, the evolution between
the beginning and the end of the century is not very pronounced, and that further-
more the mean was already high at the start of the century. Figure 8 provides,
however, a good illustration for the Ain department of the situation of a vast ma-
jority of the departments, namely a steady progression in body-height throughout
the century. In addition, for purposes of verification, we identified the percentage
of distributions at the beginning of the century that dominated the distributions at
the end of the century. This proportion was only 2%, which adds to the strong
statistical evidence of an increase in mean height over the course of the century.
Finally, the same test is performed for France as a whole, i.e. we tested for domi-
nance of the aggregate means for the years in question. These comparisons reveal
a 100% dominance rate of the end of the century over the beginning of the century
(and 0% for the beginning over the end). This, again, confirms the strong evidence
of a statistically significant increase in mean height.
4.3.2 Dominance tests
In order to test for a stronger normative ranking of the distributions, we test
for dominance of end-of-century distributions over start-of-century distributions
for values of z ranging from z−= 1.53 to z+= 1.78 at intervals of 0.0025 (or 101
comparisons per pair of distributions). Using this approach, we find that 35.6%
of distributions at the end of the century statistically and stochastically dominated
those at the beginning of the century. Even in light of the conservativeness of the
inference procedures outlined above, it therefore appears that a robust progression
in the distribution of body-heights can be observed during the century. This evo-
lution is confirmed by the percentage of dominance “in the other direction” — the
dominance of distributions from the beginning of the century over those at the end
of the century is near zero (more precisely, 0.02%). This is also confirmed by the
evolution of aggregate distributions for each year. Indeed, 100% of distributions
for France as a whole at the end of the century dominate those at the beginning of
the century (and 0% of those at the beginning dominate those at the end).
Conversely, we see that in the case of poverty rates, the percentage of domi-
nance of the end over the start of the century, by department, is substantially less
than in the case of the means. This finding that we are less likely to be able to
reject the null of non-dominance than the null of equality of means is expected
and frequently reported in the literature for anthropometric data (see, for exam-
ple, Sahn forthcoming and Sahn and Stifel 2002). The interpretation is that not
all percentiles of the distribution benefited equally from improvements in living
conditions. However, given that in the rejecting the null of no stochastic domi-
nance rests on a very strong criterion, i.e. that no point of distribution A can lie
beneath the corresponding point of B if this latter dominates the former, we limit
the “rigidity” in comparisons employing bounds z−and z+corresponding to 1.53
The results are illustrated in Figure 9 (and partially in Table 5). We observe
that, in general, the smaller the interval, the greater the proportion of dominance
(which makes sense because it reduces the number of inequalities in (11) that must
hold). For example, if we restrict the interval to [1.55,1.75] (approximately cor-
responding to percentiles 8.5 to 95) we obtain 53.1% dominance, or 17.5% more
than in the case initially examined. At the limit, if we evaluate this dominance
at a single point (the standard definition of a height poverty rate), we obtain a
very high percentage of dominance (in the Figure, this corresponds to all points
between the ordinate and the 45-degree line in the plane z−z+). For example, if
we set the poverty line at 1.57 m, then 98.2% of the time the poverty rate is sta-
tistically lower at the end of the century than at the beginning. We also see that if
we set the threshold at 1.62 m (as in ?), we obtain end-of-century dominance in
70.4% of cases.
In another vein, it is of some interest to examine Table 6, which shows the
differences in the results for stochastic dominance in distribution according to the
effectivenumberofpointscomparedin(11). Intheory, weshouldsetmtoinfinity,
but in practice this is of course impossible. Instead, we use m = 101. Notice that
the difference in the results obtained from between 6 and 101 points is small. Also
observe that there is little advantage to using more than 101 points. It is therefore
more the range, rather than the fineness of the intervals of thresholds, that matters
for testing for height dominance.
Table 5 summarizes the results of the comparison between the beginning and
the end of the century. Most decidedly, there is an increase in stature over the
course of the century, whether this is assessed using means or distributional dom-
inance. However, end-of-century dominance seems much greater when mean
body-height is used than when more robust stochastic dominance techniques are
4.4Are there regional differences in heights?
Let us now turn our attention to showing how our techniques can be employed
to address the spatial characteristic of the stature of conscripts. To illustrate, we
first examine the question of whether the heights of conscripts from Paris differ
from neighboring departments. Next we assess the extent of northeastern and
southwestern differences in stature. In both instances, we are interested in the
extent to which these differences are observed across the years.
4.4.1Paris and neighboring departments
In order to assess the height correlate of living in Paris, we conduct a year-by-
year comparison of department 7513(which includes Paris) with the neighboring
departments 60, 02, 51, 10, 89, 45, 28 and 27.14These latter are the departments
that immediately border the region Île-de-France (the geographical region includ-
ing Paris), and they can be located on Figure 10 (where they are underlined).15
Aggregating across the years, the department of Paris is globally dominated
by the bordering departments (Table 7). Indeed, 69.7% of the means of the neigh-
boring departments exceed the Paris means (and only 17.0% of the latter dominate
the former). Furthermore, 16.4% of the distributions of the adjacent departments
stochastically dominate those of Paris (versus 2.5% for the converse). This pro-
portion rises to 27.9% (versus 4.8%) if the dominance tests are restricted to the
interval [1.55,1.75]. This supports the results obtained previously by other re-
searchers regarding the poorer quality of life in Paris in the nineteenth century.
Figure 11 contrasts the evolution of the means in the department of Paris with
those of neighboring departments across the century. There appear to be three
distinct periods: one that lasted until 1830, another that ended in approximately
1886 (or 1890), and a final one until 1900. Between 1819-30, 56.6% (versus
21.7%) of means and 8.4% (versus 3.6%) of distributions for Paris dominate those
of the other departments. This last percentage increases to 16.9 (versus 6.0) if we
compare dominance curves on the interval [1.55,1.75]. This could be attributable
to the fact that, at the beginning of the century, the positive economic effects of
the population concentration dominated the negative effects of crowding, such as
increased infectious disease, as Paris had yet to become nearly as highly densely
populated as it would become later in the century.
Statistical tests for the years 1836-1886, like those for the entire century, in-
dicate that the bordering regions dominate Paris at an even more convincing rate:
82.1% (versus 5.6%) of the means of the other departments dominate those of
Paris and 20.7% (versus 0.0%) of the neighboring departments dominate Paris
in distribution. In the last part of the century (1890-1900), however, we obtain
more qualified results. Indeed, 58.0% (versus 28.4%) of the bordering depart-
ments dominate Paris in means. But as we have stressed throughout this paper,
14Oise, Aisne, Marne, Aube, Yonne, Loiret, Eure-et-Loir and Eure.
15Department 77 (Seine-et-Marne) was excluded from the comparison because its population
density, while less than that of Paris, far exceeds that of the other bordering departments owing to
its proximity to Paris (it is part of the Île-de-France region).
it is important to look beyond means and instead also focus on the changes in
the distribution. Employing the usual interval, [1.53,1.78], Paris stochastically
dominates the other departments (9.1% versus 5.7%), but in the [1.55,1.75] inter-
val the bordering departments mostly appear to dominate (17.0% versus 12.5%).
This suggests the interesting results that while mean height in Paris is lower, the
poorest fared better in Paris than elsewhere.16
4.4.2North-Eastern and South-Western regions
It has also been observed17that the population of the north-east part of France
are generally taller than those of the south-west. Some have even drawn a line
from Saint-Malo to Geneva to divide these two regions (cf. Figure 10). We use
this latter approach to determine whether this association is also suggested by the
data and the rigors of our methodology, which once again relies on much more
robust statistical testing of means and distributions than is found in the literature.
More precisely, we compare all departments that are entirely to the north with
those that are entirely to the south of this virtual demarcation (departments that
are either bisected by the line or very close to it were omitted).18
Our results corroborate earlier work that suggests that the population in the
northeast was, in fact, taller than that of the south-west. The means of the depart-
ments of the north dominate those of the southern departments in 86.4% of the
cases versus 5.9% in the opposite direction. Moreover, 32.8% (versus 0.4%) of the
distributions in the north stochastically dominate those of the south. These results
are even stronger — 51.1% versus 0.7% — if we employ the interval [1.55,1.75]
in testing for dominance. It is also of interest to observe what happens when
we examine north-south differences for two periods: 1819-1828 and 1891-1900.
Our results indicate that 87.5% of the means of the northern distribution dominate
those of the south, and the converse occurs in 4.2% of the cases at the beginning
of the century. The comparable numbers for the latter period are 85.5% and 7.6%,
16It may also be of interest to compare these results with those of ?), who regress the number
of individuals below a certain body-height (1.62 m) on several variables, including a dummy for
residing in Paris. They perform this regression on data from the first half of the century (1830–50)
as well as from the end of the century (1875–1900). They find a negative impact of living in Paris
for both periods, although of greater magnitude for the latter period.
17See ?) for instance.
18Departments wholly in the north: 02, 08, 10, 14, 21, 25, 27, 28, 39, 45, 50, 51, 52, 54, 55, 57,
59, 60, 61, 62, 67, 68, 70, 75, 76, 77, 78, 80, 88, 89, and 90; departments wholly in the south: 04,
05, 06, 07, 09, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 29, 30, 31, 32, 33, 34, 36, 38, 40, 42,
43, 44, 46, 47, 48, 49, 56, 63, 64, 65, 66, 69, 73, 79, 81, 82, 83, 84, 85, 86, and 87.
respectively. From the point of view of the mean, a few less distributions in the
north thus dominate those of the south and a few more from the south dominate
those from the north at the end of the century.
When we examine stochastic dominance, however, the results are not the
same. Indeed, while 25.8% versus 0.01% of the distributions of the north stochas-
tically dominate those of the south at the beginning of the century — 34.1% versus
0.4% dominate at the end of the century. Thus, there was not a decline, but rather
an increase in the number of distributions in the north that dominate those of the
south in poverty during the century. Residents of southern France thus appear to
have progressed slightly more rapidly on average, but certain percentiles of the
distribution have fared less well. Overall, and regardless of the point of com-
parison taken, inhabitants of the north-east of France broadly dominate those of
the south-west in stature. Table 8 summarizes the results of all the north-south
This study has focused on developing a methodology that enables us to gener-
ate complete population distributions from bin-aggregated or histogram-type sam-
ple data. By generating the entire population distributions from grouped data, we
are able to make robust statistical and normative comparisons of the distributions.
This is something that has previously not been done by economic historians that
have instead relied on primarily non-statistical comparisons of measures of central
tendency to examine changes in the stature of populations over time. Likewise,
many researchers using contemporary data face similar challenges as they are of-
ten limited to bin-aggregated data, due to political constraints or concerns over
confidentiality that prevent access to actual observations on households and indi-
We go on to illustrate our methodology to height data from France where we-
first generate height distributions, and then use them toanswer two main questions
regarding the stature of the French during the nineteenth century: How did it
evolve over the course of the century? And were there regional and geographi-
cal correlates with body-heights? The paper’s methodological procedures and the
richness of our data allows us to answer these questions using height distribu-
tions for the more than 6000 department-year distributions (representing around
15 millions young Frenchmen) and over close to a century.
The paper’s first important empirical conclusion is that body-height increased
significantly in France over the course of the century (from 1819 to 1900). Thus,
93.8% of the means and 35.6% of the distributions of the end of the century sta-
tistically dominate those of the beginning of the century (as opposed to 2% and
0.02% in the reverse direction). This provides statistically and normatively strong
evidence of a global improvement in living conditions over the course of this pe-
The second main empirical finding is that the body-height of the French var-
ied significantly with the geographical region in which they lived. First, Parisians
were generally shorter than those residing in neighboring departments. 69.7% of
means and 16.4% of distributions from bordering departments statistically dom-
inated those of Paris across the century, as opposed to 17% and 2.5% in the op-
posite direction. This may be attributable to overpopulation of the city of Paris,
which resulted in poor sanitation and drove up the cost of living, or to the mi-
gration of the poor (possibly shorter than the average) into the capital in search
for work. However, this observed first-order stochastic dominance was not con-
stant throughout the century. Indeed, Paris dominated its suburbs at the beginning
of the century (1819–30). The situation was then inverted to make room for a
strong dominance of the suburbs over Paris by the middle of the century — which
persisted, though to a lesser extent, toward the end of the century.
The second “geographical” effect found in this study is that the French of
the north-east were generally taller than those of the south-west. Thus, 86.4%
of means and 32.8% of distributions of the north dominated those of the south
throughout the century, as opposed to 5.9% and 0.4% in the opposite direction.
One possible explanation is that some parts of the south featured more difficult
geographic terrain (mountainous regions, etc.), and that industrialization and im-
provements in living conditions eveolved less rapidly.
In short, robust statistical tests, both differences-in-means and stochastic dom-
inance tests based on using data extracted from the conscription registries of the
French army, allow us to ascertain the existence of clear differences across time
and regions . The paper’s statistical methods are in particular designed to be more
robust than is usually the case, in that they allow for the presence of both sampling
and non-sampling errors while making maximal use of the available empirical in-
formation. The paper’s stochastic dominance tests allow ranking height distribu-
tions for almost any measure of poverty or social welfare based on body-height,
which stands in contrast to the usual comparisons based on summary statistics
such as median or mean heights.
Finally, the methodology developed and applied in this paper could be use-
fully applied to other types of grouped data, such as thoseon incomes and wealth,
in addition to answering other questions based on this paper’s own data. In the
case of the latter, for instance, future research could explore to what extent is the
increase in body-height during the century attributable to a rise in the mean, and
to what extent is it due to changes in the relative distribution. Likewise, it would
be possible to use the generated distributions to assess how inequality in heights
evolved over the course of the century as well as determine the contributions of
years and departments to health poverty and health inequality across the century.
6.1 Notes on the data
Though the data we use are rich, they also feature some drawbacks. First, be-
cause those called up to active service were drawn by lot, not all those selected
were measured (or had their measurements retained). Weir (1997) finds that prior
to 1872 approximately 19% of recruits were exempted because of physical de-
formities, 17% for legal reasons, 9% because of a weak constitution, and 7% for
being too short. Among those exempted for legal reasons, there is no reason to
presume that a bias is introduced into the data, since the exempted individuals then
do not in principle differ from typical Frenchmen. We might expect, however, that
intellectuals are richer, and thus taller, than the mean. This would imply that their
exclusion from the sample might introduce a bias. However, since they represent
a small proportion of the population, we can probably ignore this bias. Among
those exempted for health reasons,those who were too short were entered into a
category of their own (insufficient body-height), and thus are included in our data
as a category for which we know the upper bound. It is highly unlikely that those
rejected because of a weak constitution were too short, otherwise they would be
classified as such.
The only category that could pose real problems for our analysis is physi-
cal deformities. In this case, we can only assume that their distribution was the
same as that of the overall population in the contingent, i.e., whose body-height
exceeded a certain threshold.
Other concerns regarding the reliability of the data are that some individuals
may have been measured with their shoes on, the data may have been rounded to
the nearest unit, and that there was a switch from the imperial to the metric system
in the 1867 registries may have caused significant rounding errors (Weir 1997)19.
19In fact, it is likely that during the transition between the two systems, body-height was first
And finally, there is some ambiguity about the class boundaries after 1867. More
details on this and other data issues can be found in Leblanc (2007).
6.2Classification of heights
The most direct way to use the information contained in our aggregated data
is to apply an equation (of type (1)) to each available class. However, this will
not necessarily yield estimated distributions that are very close to the true distri-
butions, since it does not always impose sufficient constraints on the estimation.
In particular, since the first class (corresponding to individuals categorized as too
short) corresponds to (−∞,b1] (where b1is the minimum body-height for being
accepted into the army), direct application of this constraint would not impose
any value on the distribution function for statures shorter than b1, which could
lead to estimated distributions that are unrealistically far with any real distribution
of body-heights. Thus, we created a supplementary interval by dividing this inter-
val into two parts: (−∞,1.47]20and [1.47,b1]. Since the proportion of the male
population aged 20 whose body-height is below 1.47 meters should be negligible,
we set the proportion of the population in that class to zero and grouped all those
categorized as too short into the second class.
Also, for the years until 1871, we combined the two last classes provided by
the bin data, since the numbers in each one were virtually nil. This allowed us
to retain the same number of parameters as if we had not subdivided the first
interval, and to do so without loss of information (since the omitted class did not
contribute any additional information). Thus, the last class went from [1.923,∞)
to [1.896,∞) for the years prior to 1896, and from [1.96,∞) to [1.94,∞) for the
As to the years 1872 to 1900, we opted to treat the last class the same way
as the first class. Indeed, the last class for these years contained as many as 20%
of the sample, so using this class “as is” as a constraint on the estimation would
not necessarily yield a distribution with only a handful of individuals taller than
2 meters (which should be the case in reality). Consequently, we divided this
interval into two parts, i.e. [1.73,1.85] and [1.85,∞). We assumed that individuals
categorized in the class [1.73,∞) were, in fact, all in the first segment of this
interval, and that the proportion in the second segment was zero.
measured according to the imperial system, then the measure was rounded, and finally it was
transformed into a metric measure. Heyberger (2005) also underscores that the switch between
the two systems was performed in a transitional fashion prior to 1867.
20In practice, we used –1000 rather than −∞.
Since the number of constraints corresponds to the number of parameters, us-
ing the aforementioned constraints implies that the estimated distributions for the
years 1819–1829 feature 14 parameters; those for 1830–1871, 15; and those for
6.3 Estimation of national distributions of heights
In order to aggregate the distributions (and the means) of the departments for
purposes of generating the yearly distributions (or the means) for France as a
present in the censuses of 1851 (for 1819 to 1859), 1876 (for 1860 to 1885), and
1896 (for 1886 to 1900). For each department, we used as a weight the number of
men between 20 and 24 years old in the department divided by the total number
of men in that cohort in France that year.21
If we let pibe the number of men between 20 and 24 years old, and letˆΘirep-
resent the estimated parameters of the distribution of department i in a given year,
then the estimator and the variance of the estimator of the distribution function for
France are given by
Similarly, if we represent the estimated mean of department i by ˆ µi, the estimator
of the overall mean for France and the variance of that estimator are given by
var(ˆ µFRANCE) =
21There were some difficulties deriving the weights from 1860 to 1869 since departments 6, 73,
and 74 were added in 1860. Additionally, departments 57 and 67 were eliminated in 1870. To deal
with these problems we made a series of adjustments in weights of bordering departments.
6.4 Measurement errors
As mentioned in Section 4.2, there are several possible explanations to the
relatively high rate of “acceptance” of dominance in Tables 2 and 322. To examine
some of them, let xijk be the true height x of observation i for department j
and year k, and let an estimator of the true population mean µjkof height x for
department j and year k be given by ˆ µjk,
where ?ijkis an observation-specific measurement error that we assume to have
zero mean and to be uncorrelated with xijk; ηjkis a department-year-specific dis-
turbance term; and n denotes the sample size that is assumed for expositional
simplicity to be the same across all distributions.
Firstly, let us set ηjkin (16) to zero. Assuming the existence of the appropri-
ate population moments and using the central limit theorem, ˆ µjkcan be shown to
be asymptotically normally distributed with an asymptotic variance that is given
by n−1(var(xjk) + var(?jk)). The presence of the ?ijkmeasurement errors will
therefore increase the variance of the estimator relative to the case of the absence
of these errors, a variance that would equal n−1var(xjk). Confidence intervals
around ˆ µjkwill therefore also be wider than in the absence of measurement errors.
However, since var(xjk+ ?jk) can be estimated from the available sample infor-
mation, and since ˆ µjkis a consistent estimator of the true mean by the law of large
number, the coverage probability of the confidence interval can be maintained to
some desired level. Note also that the importance of the ?ijkmeasurement errors
fall with n.
Secondly, consider the case in which a measurement error ηjk is also al-
lowed to vary unobservably across departments and years. Let x∗
?∆ = ˆ µjk− ˆ µlm, and also suppose that the ηjkare identically and independently
mal distribution with variance
var(?∆) = n−1?var?x∗
true population means across the two distributions. Table 2 reports that, using a
ˆ µjk= n−1
(xijk+ ?ijk) + ηjk,
ijk= xjk+ ?ijk,
normally distributed with variance var(η).?∆ then follows an asymptotically nor-
Now consider the null that µjk= µlm, namely, that there is no difference in the
22We focus here on mean dominance, but similar insights apply to distribution dominance.
Mean dominance is expositionally simpler to deal with because the estimator of mean height is
linear in the observations of individual heights.
critical value of ±1.65 — which delimits the usual 5% threshold on each side of
the standard normal distribution — the null of non-dominance in means is rejected
for about 20% of the year-to-year comparisons. Thus, under the above null and
under the assumption that the ηjkfollow a common normal distribution across the
j and the k, we have that
zero and variance v. Transforming (18) into a standardized normal distribution,
1 − Φ∗
?∆(·;v) is the cumulative normal distribution function of?∆ with mean
1 − Φ∗
Using (17) and (20), we then have
Rearranging (21) and supposing for simplicity that var?x∗
Expression (22) says that, under the above assumptions, the standard deviation
of η is about√3∼= 1.7 times the standard error on the estimator of ˆ µjk, which is
itself given by n−1var?x∗
2mm) for the results of Table 2 to be accounted for.
Thirdly, note that the model in (16) could also be specified to allow for dis-
turbance terms that are specific to a department j and a year k, such as υjand ζk
?= 3var(ˆ µjk).
?. Since n is around 2000 and var?x∗
?is about 6cm
in our data, the standard deviation of η needs to be around 0.23cm (a bit more than
ηjk= υj+ ζk.
For instance, the presence of a year-specific error term, ζk, may seem to be sug-
gested by the variability of rejection rates across the rows of Table 2.
Fourthly, note that an important alternative interpretation of the term ηjkin
(16), and of ζkin (23), is that they account for true variations in the distributions
of heights within a single department from one year to the next. In such a case, if
we want our tests to exhibit power, it is certainly desirable that the null hypothesis
of non-dominance of the distributions be rejected more often than the nominal
5% level of the tests under the null of equality of mean. For this to happen around
20% of the time, as reported in Table 2 and as shown above in the context of
measurement errors, would only require that the variability of the yearly changes
in heights generate a standard error of about 2.3mm in ηjk.
All in all, for statistical prudence, we choose to follow a conservative infer-
ence procedure that assigns all year-to-year variations in distributions and means
to measurement errors. Hence, rather than concluding that one year dominates
another on the basis of rejecting non-dominance with a 5% nominal level, as
the usual statistical inference procedure would suggest doing, we will only draw
this conclusion if the proportion of rejections across departments exceeds that
observed in year-to-year comparisons, namely, roughly 20% for comparisons of
means and 1.5% for comparisons of distributions. A “super-conservative” pro-
cedure would set these last thresholds to the maximum proportion of rejections
observed — 55% for means in Table 2 and 13% for distributions in Table 4. Most
of the paper’s qualitative results are in fact robust to either of these procedures.
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Table 1: Bounds of the height classes for different years
Height (in meters)
1819-18291830 and 1836-1866
(15 classes)(16 classes)
less than 1.560
less than 1.5701.560 - 1.569
1.570 - 1.5971.570 - 1.597
1.598 - 1.6241.598 - 1.624
1.625 - 1.6511.625 - 1.651
1.652 - 1.678 1.652 - 1.678
1.679 - 1.7051.679 - 1.705
1.706 - 1.7321.706 - 1.732
1.733 - 1.7601.733 - 1.760
1.761 - 1.7871.761 - 1.787
1.788 - 1.8141.788 - 1.814
1.815 - 1.841 1.815 - 1.841
1.842 - 1.8681.842 - 1.868
1.869 - 1.895 1.869 - 1.895
1.896 - 1.9221.896 - 1.922
1.923 and more 1.923 and more
1867, 1868 and 1871
less than 1.550
1.550 - 1.579
1.580 - 1.609
1.610 - 1.639
1.640 - 1.669
1.670 - 1.699
1.700 - 1.729
1.730 - 1.759
1.760 - 1.789
1.790 - 1.819
1.820 - 1.849
1.850 - 1.879
1.880 - 1.909
1.910 - 1.939
1.940 - 1.959
1.960 and more
less than 1.540
1.540 - 1.629
1.630 - 1.639
1.640 - 1.649
1.650 - 1.659
1.660 - 1.669
1.670 - 1.699
1.700 - 1.729
1.730 and more
Figure 1: Dominance test of 1886 distribution over that of 1819 for the Ain de-
1.51.551.6 1.65 1.71.75 1.81.85
1.51.55 1.61.65 1.7 1.751.8 1.85
t−stat and critical value
Table 2: Proportions of departments statistically dominating (or dominated by)
the preceding year
% of departmental means
1822 9.7 22.2
Table3: Proportionsof departments dominating (and dominated by) the preceding
year at z = 1.652 (years 1819-1866) and z = 1.640 (1867-1900)
% of department F(z)
1823 9.7 26.4
Table 4: Proportions of departments statistically dominating (or dominated by)
the preceding year
% of departmental distributions
1820 3.8 0.0
1826 1.4 0.0
1858 0.0 2.3
1859 4.7 0.0
1860 0.0 2.3
Table 5: % of dominance of end over beginning of century
Mean height dominance
Ranges of poverty lines considered
1.53 - 1.78
1.55 - 1.78
1.53 - 1.75
1.55 - 1.75
1.57 - 1.57
1.62 - 1.62
Table 6: Stochastic dominance of end over beginning of century
of pointsbetween points
Figure 3: Estimated distribution for the Ain department in 1873
Figure 4: Estimated distribution for the Ain department in 1839
Figure 5: Mean height for each department from 1819 to 1900
Body−height average (m)
Figure 6: Mean height for all departments combined, from 1819 to 1900
Body−height average (m)
Figure 7: Mean height for the Calvados department
Body−height average (m)
Figure 8: Mean height for the Ain department
1800 18201840 186018801900
Body−height average (m)
Figure 9: Stochastic dominance of the end over the start of the century with vari-
able bounds for ranges of heights
rejection of H0
Table 7: Comparing Paris and neighboring departments
PeriodOther dept. Paris
1819 - 190069.7%17.0%
1819 - 1830 21.7%56.6%
1836 - 188682.1%
1890 - 190058.0%28.4%
Table 8: Comparing North and South
1819 - 1900
1819 - 1900 without Paris
1819 - 1828
1891 - 1900
Figure 10: Map of French departments
Figure 11: Mean heights of department 75 (Paris) versus neighboring departments Download full-text
1800 18201840 186018801900
Body−height average (m)