STATISTICAL METHODOLOGY
Having taken measurements with the rigorous technical precision that
is to-day demanded by anthropometry, we should know how to extract
from these figures certain _laws_, or at least certain statistical
conclusions.
There are two principal methods of regrouping the figures:--_mean_
_averages_ and _seriations_.
=Mean Averages.=--Averages are obtained, as is a matter of common
knowledge and practice, by taking the sum of all the figures and
dividing the result by the number of data. The general formula is as
follows:
(a+b+c+d)/(1+1+1+1)
When comparative figures are given, as, for example, those recorded by
Quetelet for the stature, the diameters of the head, etc., such figures
are always mean averages.
Such averages may be more or less general. We might, for example,
obtain a mean average of the stature of Italians, and this would be
more general than the mean stature for a single region of Italy, and
this again more general than the mean stature for a city, or for some
specified social class, etc.
It is interesting to know how the mean will be affected, according to
the number of individuals examined, because it is obvious that the
mean stature of Italians cannot be based upon measurements of _all_
Italians, but upon a larger or smaller number of individuals. Now,
if we take various different numbers of individuals, shall we obtain
different mean statures? And if so, what number of subjects must we
have at our disposal in order to obtain a constant medial figure, and
hence the one that represents the _real mean average_? It has been
determined that a relatively small number will suffice to give the
mean, if the measurements are taken with uniform method and from the
same class of subjects (sex, age, race, etc.); for the cranium, 25
subjects are sufficient, and for the stature, 100 subjects.
This method furnishes us with an abstract number, insofar as it does
not correspond to any _real individual_, but it serves to give us the
synthetic idea of an entirety. In anthropology we need this sort of
fundamental synthesis before proceeding to individual analysis for the
purpose of interpreting a specified person.
Now, it is evident that the figures representing the mean stature for
each region in Italy give us a basis for judging of the distribution
of this important datum, while an accumulation of a hundred thousand
individual figures would lead to nothing more profitable than confusion
and weariness.
The following table, however, is quite clear and instructive:
MEAN STATURE IN ITALY
(According to Departments)
-------------------------------
Departments |Stature in
|centimetres
-------------------+-----------
Piedmont | 162.7
Liguria | 163.7
Lombardy | 163.6
Venetia | 165.4
Emilia | 164.0
Tuscany | 164.3
Marches | 162.4
Umbria | 162.7
Latium | 162.5
Abruzzi and Molise | 160.6
Campania | 161.3
Apulia | 160.4
Basilicata | 158.9
Calabria | 159.4
Sicily | 161.1
Sardinia | 158.9
-------------------------------
Yet the interpretation of such a table is not simple; it is necessary
to read the numbers, to remember them in their reciprocal relation; and
it demands effort and time to acquire a _clear and_ _synthetic_ idea of
the distribution in Italy of this one datum, _stature_.
On the other hand, we must lose as little time and spare our forces as
far as possible. The value of positive methodology lies in the extent
to which it accomplishes these two subjects.
Geographical charts serve the purpose of this desired simplification.
Let us take an outline map of Italy, divide it into regions, and
_colour_ these different regions darker or lighter, in proportion as
the stature is higher or lower.
The gradations and shadings in colour will tell us at a single glance,
and without any fatigue on our part, what the table of figures reveals
at the cost of a very perceptible effort. Little squares must be
added on the margin of the chart, corresponding to the gradations
in colour, and opposite them the figures which they respectively
indicate--after the fashion in which the scale of reduction is given in
every geographical map. In this way we may _study_ these charts, and
their examination is pleasant and interesting, while it successfully
associates the two ideas of an "anthropometric datum" and of a
"region," a result which a series of figures, pure and simple, could
not achieve.
We have seen Livi's charts of Italy, both for stature and for the
cephalic index. Analogous charts may be constructed for all the
different data, for example, the colour of the hair, the shape of
the nose, the facial index, etc. In the same manner we may proceed
to a still more analytical distribution of anthropometric data among
the different provinces of a single _region_. For example, I myself
prepared charts of this sort for the stature, the cephalic index and
the pigmentation of the population of Latium.
Sometimes we want to see in one single, comprehensive glance, the
_progress_ of some anthropological datum; for instance, in its
development through different ages. Quetelet's series of figures for
growth in stature, in weight, in the diameters of the head, the cranial
circumference, etc., offer when read the same difficulty as the similar
tables of distribution according to regions. On the contrary, we get a
synthetic, sweeping glance in _diagrams_, such as the one which shows
the growth of stature in the two sexes. The method of constructing
such diagrams is very simple, and is widely employed. When we wish to
represent in physics certain phenomena and laws; or in hygiene, the
progress of mortality through successive years, etc., we make use of
the method of diagrams.
Let us draw two fundamental lines meeting in a right angle at _A_ (Fig.
151): _AS_ is known as the _axis of the abscissae_; _AO_, the _axis of
the ordinates_. We divide each of these lines into equal parts. Let
us assume that the divisions of _AS_ represent the years of age, and
those of _AO_ the measurements of stature in centimetres; and since the
new-born child has an average height of 50 cm., we may place 50 as the
initial figure. From the figure _O_ (age) and from 50 cm. (measure),
we erect perpendiculars meeting at _a_, where we mark the point. At
the age of one year the average stature is about 70 cm., accordingly
we erect perpendiculars from 1 (age) and from 70 (measure), obtaining
the point _c_. Since the stature at two years is about 80 cm. the same
procedure gives us the point _e_. Since the stature at the age of three
is about 86 cm., I erect the perpendicular from a level slightly higher
than half-way between 80 and 90, obtaining the point _i_; and so on,
for the rest. Meanwhile we begin to be able to see at a glance that the
stature increases greatly in the first year and that thereafter the
intensity of its growth steadily diminishes.
[Illustration: FIG. 151]
If we unite the points thus constructed, the line of representation is
completed.
The verticals 0_a_, 1_c_, 2_e_, etc., are the _ordinates_, and the
horizontals 50_a_, 70_c_, etc., are the abscissae of the line of
representation; and since it is constructed along the intersections
of these lines, they are for that reason collectively called
_coordinates_. It is usual in constructing these diagrams to mark the
coordinates in such a way that they will not be apparent, instead of
which only the axes and the line representing the development of the
phenomenon are shown (Fig. 152).
Sometimes a different method of representing the phenomenon graphically
is followed, namely, by tracing the successive series of distances
developed on the ordinates (Fig. 153); in which case the characteristic
arrangement of the lines causes this to be known as the _organ-pipe_
method.
[Illustration: FIG. 152.]
[Illustration: FIG. 153.]
The diagram for the growth in stature, given earlier in this volume, is
constructed according to the method shown in Fig. 151. When there are a
great number of data to represent, which overlap and interweave, this
method of graphic representation still lends itself admirably to the
purpose; in such a case we shall have a number of broken lines, either
parallel or intersecting, which may be distinguished by different
colours or different methods of tracing (dots, stars, etc.), so that
they may interweave without becoming confused, thus giving us at a
glance the development of several phenomena at once (for example, total
stature and sitting stature, length of upper and lower limbs, in one
and the same diagram).
For the purpose of practice, a graphic representation of the changes in
ponderal weight through the different ages may be constructed in class.
The figures for stature and weight at each age should be read aloud;
one student can find the corresponding _ponderal index_ in the tables,
while another constructs the graphic line upon the blackboard.
In this manner we can see better than by reading the figures, how the
ponderal index increases during the first year and becomes much higher
during early infancy; and then how it diminishes up to the age of
puberty, holding its ground with slight oscillations during the puberal
period; after which it again increases when the individual begins to
_fill out_ after the seventeenth year, and once again later when he
takes on flesh, to fall off again during the closing years, when old
age brings lean and shrunken limbs.
Seriation.--Another method of rearranging the figures is that of
_seriation_. Let us assume that we are taking the average of a thousand
statures, or of hundreds of thousands. We will try to find some means
of simplifying the calculation. Since the individual oscillations of
stature are contained within a few centimetres and the individuals
amount to thousands, large numbers will be found to have the same
_identical_ statures. Accordingly, let us rearrange the individuals
according to their stature, obtaining the following result:
-----------------------------------------------
Stature in metres | Number of individuals
---------------------+-------------------------
1.50 | 20
1.55 | 80
1.60 | 140
1.61 | 200
1.62 | 300
1.63 | 450
1.70 | 100
1.75 | 80
1.80 | 10
------------------------------------------------
By multiplying the 1.50 by 20, 1.55 by 80, etc., and by adding the
results, we shall have simplified the process for obtaining the sum
total which must then be divided by the number of individuals.
Well, while doing this for the purpose of simplifying the calculation,
we have hit upon the method of distributing the individuals in a
_series_, that is, we have regrouped the corresponding figures
according to _seriation_.
Seriation has been discovered as a method of _analysing_ the mean
average, and it demonstrates three things: first, the extent of
oscillations of anthropologic data, a thing which the mean average
completely hides,--indeed, we have seen in the case of the cephalic
index the mean averages oscillate between 75 and 85, when calculated
for the separate regions, while, in the case of individuals, the
oscillations extend from 70 to 90; secondly, it shows the numerical
prevalence of individuals for the one or the other measurement;
third, and finally, seriation reveals a law, to us, namely, that the
distribution of individuals, according to anthropological data, is not
a matter of chance; there is a prevalence of individuals corresponding
to certain average figures, and the number of individuals diminishes in
proportion as the measurements depart from the mean average, equally
whether they increase or diminish.
I take from Livi certain numerical examples of serial distribution:
------------------------------------------------
Stature in inches | Number of observations
---------------------+--------------------------
60 | 6
61 | 26
62 | 32
63 | 26
64 | 160
65 | 154
66 | 191
67 | 128
68 | 160
69 | 89
70 | 45
71 | 7
72 | 6
73 | 3
74 | 1
------------------------------------------------
Although these figures are not rigorously exact, there is a certain
numerical prevalence of individuals in relation to the stature of
66 inches, and above and below this point the number of individuals
diminishes, becoming very few toward the extremes.
The lack of exactness and of agreement in serial distribution is due
to the numerical scarcity of individuals. If this number were doubled,
if it were centupled, we should see the serial distribution become
systematised to the point of producing, for example, such symmetrical
series as the following:
1 1 1
12 16 15
66 120 105
220 560 455
495 1,820 1,365
792 3,368 3,003
924 8,008 5,005
--- 11,440 6,435
792 12,870 -----
495 ------ 6,435
220 11,440 -----
66 8,008 5,005
12 3,368 3,003
1 1,820 1,365
560 455
120 105
16 15
1 1
This law of distribution is one of the most widespread laws; it ordains
the way in which the characteristics of animals and plants alike must
behave; and the statistical method which is beginning to be introduced
into botany sheds much light upon it.
[Illustration: FIG. 154.]
This law may be represented graphically by arranging the anthropologic
data on the abscissae (_e.g._, those of stature), and the number of
individuals on the ordinates.
In such cases we have a curve with a maximum central height and a
symmetrical bilateral diminution (Fig. 121): this is the curve of
Quetelet.
Or better yet, it is known as _Quetlet's binomial curve_, because this
anthropologist was the first to represent the law graphically and to
perceive that its development was the same as that so well known in
mathematics for the coefficients in Newton's binomial theorem.
Newton's binomial theorem is the law for raising any binomial to the
_n_th power, and is expanded in algebra as follows:
(_a_+_b_)^{_n_} = _a_^{_n_}+
_na_^{(_n_-1)}_b_+
(_n_(_n_-1)/2)_a_^{(_n_-2)}_b_^{2}+
((_n_(_n_-1)(_n_-2))/(2.3))_a_^{(_n_-3)}_b_^{3}+
((_n_(_n_-1)(_n_-2)(_n_-3))/(2.3.4))_a_^{(_n_-4)}_b_^{4}+
((_n_(_n_-1)(_n_-2)(_n_-3)(_n_-4))/(2.3.4.5))_a_^{(_n_-5)}_b_^{5}+ ... +
_b_^{n}
substituting for _n_ some determined coefficient, for example, 10,
the binomial would develop, in regard to its coefficients, after the
following fashion:
(_a_+_b_)^{10} = _a_^{10}+10x_a_^{9}_b_+
((10.9)/2)_a_^{8}_b_^{2}+
((10.9.8)/(2.3))_a_^{7}_b_^{3}+
((10.9.8.7)/(2.3.4))_a_^{6}_b_^{4}+
((10.9.8.7.6)/(2.3.4.5))_a_^{5}_b_^{5}+
((10.9.8.7.6.5)/(2.3.4.5.6))_a_^{4}_b_^{6}+
((10.9.8.7.6.5.4)/(2.3.4.5.6.7))_a_^{3}_b_^{7}+
((10.9.8.7.6.5.4.3)/(2.3.4.5.6.7.8))_a_^{2}_b_^{8}+
((10.9.8.7.6.5.4.3.2)/(2.3.4.5.6.7.8.9))_ab_^{9}+
_b_^{10}.
Whence it appears that, after performing the necessary reductions, the
coefficients following the central one diminish symmetrically in the
same manner as they increased: that is, according to the selfsame law
that we meet in the anthropological statistics of seriations.
Indeed, here is the binomial theorem with the reductions made:
(_a_+_b_)^{10} = _a_^{10}+10x_a_^{9}_b_+
((10.9)/(2))_a_^{8}_b_^{2}+
((10.9.8)/(2.3))_a_^{7}_b_^{3}+
((10.9.8.7)/(2.3.4))_a_^{6}_b_^{4}+
((10.9.8.7.6)/(2.3.4.5))_a_^{5}_b_^{5}+
((10.9.8.7)/(2.3.4))_a_^{4}_b_^{6}+
((10.9.8)/(2.3))_a_^{2}_b_^{7}+
((10.9)/(2))_a_^{2}_b_^{8}+10x_ab_^{9}+_b_^{10}.
And after calculating the coefficients, we obtain the following numbers
in a symmetrical series:
10
45
120
210
252
210
120
45
10
This is why the curve of Quetelet is called _binomial_.
Let us assume that we wish to represent by means of Quetelet's curves,
two seriations, for instance in regard to the stature of children of
the same race, sex and age, but of opposite social conditions: the poor
and the rich.
These two curves of Quetelet's, provided that they are based upon an
equal and very large number of individuals, will be identical, because
the law itself is universal. Only, the curve for the rich children will
be shifted along toward the figures for high statures, and that for the
poor children toward the low statures.
[Illustration: FIG. 155.]
At a certain point _A_ the two curves meet and intersect, each
invading the field of the other: so that within the space _ABC_ there
are individual rich children who are shorter than some of the poor,
and individual poor children who are taller than some of the rich:
_i.e._, the conditions are contrary to those generally established
by the curve as a whole. This rule also, of the intersection of
binomial curves, is of broad application; whenever a general principle
is stated, _e.g._ that the rich are taller than the poor, it is
necessary to understand it in a liberal sense, knowing that wherever
we should descend to details, the opposite conditions could be found
(superimposed area _ABC_). For all that, the principle as a whole does
not alter its characteristic, which is a differentiation of diverse
types (for example, the tall rich and the short poor). The same would
hold true if we made a comparison of the stature of men and women;
the curve for men would be shifted toward the higher figures and that
for women toward the lower, but there would be a point where the two
curves would intersect, and in the triangle _ABC_ there would be
women taller than some of the men, and men shorter than some of the
women. The differences have reference to the numerical _majority_ (the
high portions of the curves) which are clearly separated from each
other, like the tops of cypress trees which have roots interlacing
in the earth. Now, it is the _numerical prevalence_ of individuals,
in any mixed community, that gives that community its distinctive
type, whether of class or of race. If we see gathered together in
a socialistic assemblage a proletarian crowd, suffering from the
effects of pauperism, the majority of the individuals have stooping
shoulders, ugly faces and pallid complexions; all this gives to the
crowd a general aspect, one might say, of physical inferiority. And
we say that this is the type of the labouring class of our epoch in
which labour is proletarian--a type of caste. On the other hand, if
we go to a court ball, what strikes us is the numerical prevalence
of tall, distinguished persons, finely shaped, with velvety skin and
delicate and beautiful facial lineaments, so that we recognise that the
assemblage is composed of privileged persons, constituting the type of
the aristocratic class. But this does not alter the fact that among the
proletariat there may be some handsome persons, well developed, robust
and quite worthy of being confounded with the privileged class; and
conversely, among the aristocrats, certain undersized individuals, sad
and emaciated, with stooping shoulders and features of inferior type,
who seem to belong to the lower social classes.
For this same reason it is difficult to give _clear-cut_ limits to any
law and any distinction that we meet in our study of life. This is why
it is difficult in zoology and in botany to establish a system, because
although every species differs from the others, in the salience of
its characteristics and the numerical prevalence of individuals very
much alike, none the less every species grades off so insensibly into
others, through individuals of intermediate characteristics, that it
is difficult to separate the various species sharply from one another.
It is only the treetops that are separate, but at their bases life is
intertwined; and in the roots there is an inseparable unity. The same
may be said when we wish to differentiate normality from pathology and
degeneration. The man who is clearly sane differs beyond doubt from the
one who is profoundly ill or degenerate; but certain individuals exist
whose state it would be impossible to define.
Now, while seriations analyse certain particularities of the individual
distribution, by studying the actual truth, mean averages give us
only an abstraction, which nevertheless renders distinct what was
previously nebulous and confused in its true particulars. The synthesis
of the mean average brings home to us forcibly the true nature of the
characteristics in their general effect. The analysis of the seriation
brings home to us forcibly the truth regarding this effect when we
observe it in the actuality of individual cases.
"When, from the topmost pinnacle of the Duomo of Milan or from the hill
of the Superga," says Levi in felicitous comparison, "we contemplate
the magnificent panorama of the Alpine chain, we see the zone of snow
distinguished from that free from snow by a line that is visibly
horizontal and that stretches evenly throughout the length of the
chain. But if we enter into the Alpine valleys and try to reach and
to touch the point at which the zone of snow begins, that regularity
which we previously admired disappears before our eyes; we see, at one
moment, a snow-clad peak, and at the next another free from snow that
either is or seems to be higher than the former."
Now, through the statistics of mean averages, we are able to see the
general progress of phenomena, like the spectator who gazes from a
distance at the Alpine chain and concludes that the zone of snow is
above and the open ground is below; while, by means of seriation,
we are in the position of the person who has entered the valley and
discovers the actuality of the particular details which go to make up
the uniform aspect of the scene as a whole. Both aspects are true--just
as both of those statistical methods are useful--for they reciprocally
complete each other, concurring in revealing to us the laws and the
phenomena of anthropology.