I propose now to speak of those fundamental principles of the laws of
Probability that are chiefly concerned in the newer methods of Biometry,
and consequently of Eugenics. Most persons of ordinary education seem to
know nothing about them, not even understanding their technical terms,
much less appreciating the cogency of their results. This popular
ignorance so obstructs the path of Eugenics that I venture to tax your
attention by proposing a method of partly dispelling it. Let me first
say that no one can be more conscious than myself of the large amount of
study that is required to qualify a man to deal adequately with the
mathematical methods of Biometry, or that any man can hope for much
success in that direction unless he is possessed of appropriate
faculties and a strong brain. On the other hand, I hold an opinion
likely at first sight to scandalize biometricians and which I must
justify, that the fundamental ideas on which abstruse problems of
Probability are based admit of being so presented to any intelligent
person as to be grasped by him, even though he be quite ignorant of
mathematics. The conditions of doing so are that the lessons shall be as
far as possible ‘Object lessons,’ in which real objects shall be handled
as in the Kindergarten system, and simple operations performed and not
only talked about. I am anxious to make myself so far understood, that
some teachers of science may be induced to elaborate the course that I
present now only in outline. It seems to me suitably divisible into a
course of five lessons of one hour each, which would be sufficient to
introduce the learner into a new world of ideas, extraordinarily wide in
their application. A proper notion of what is meant by Correlation
requires some knowledge of the principal features of Variation, and will
be the goal towards which the lessons lead.
To most persons Variability implies something indefinite and capricious.
They require to be taught that it, like Proteus in the old fable, can be
seized, securely bound, and utilized; that it can be defined and
measured. It was disregarded by the old methods of statistics, that
concerned themselves solely with Averages. The average amount of various
measurable faculties or events in a multitude of persons was determined
by simple methods, the individual variations being left out of account
as too difficult to deal with. A population was treated by the old
methods as a structureless atom, but the newer methods treat it as a
compound unit. It will be a considerable intellectual gain to an
otherwise educated person, to fully understand the way in which this can
be done, and this and such like matters the proposed course of lessons
is intended to make clear. It cannot be expected that in the few
available minutes more than an outline can be given here of what is
intended to be conveyed in perhaps thirty-fold as much time with the aid
of profuse illustrations by objects and diagrams. At the risk of being
wearisome, it is, however, necessary to offer the following syllabus of
what is proposed, for an outline of what teachers might fill in.
The object of the first lesson would be to explain and illustrate
Variability of Size, Weight, Number, &c., by exhibiting samples of
specimens that have been marshalled at random (Fig. 1), or arrayed in
order of their magnitude (Fig. 2). Thus when variations of length were
considered, objects of suitable size, such as chestnuts, acorns,
hazel-nuts, stones of wall fruit, might be arrayed as beads on a string.
It will be shown that an ‘Array’ of Variates of any kind falls into a
continuous series. That each variate differs little from its neighbours
about the middles of the Arrays, but that such differences increase
rapidly towards their extremities. Abundant illustration would be
required, and much handling of specimens.
Arrays of Variates of the same class strung together, differing
considerably in the number of the objects they each contain, would be
laid side by side and their middlemost variates or ‘Medians’ (Fig. 3)
would be compared. It would be shown that as a rule the Medians become
very similar to one another when the numbers in the Arrays are large. It
must then be dogmatically explained that double accuracy usually
accompanies a four-fold number, treble accuracy a nine-fold number, and
so on.
(This concludes the first lesson, during which the words and
significations of Variability, Variate, Array, and Median will have been
learnt.)
The second lesson is intended to give more precision to the idea of an
Array. The variates in any one of these strung loosely on a cord, should
be disposed at equal distances apart in front of an equal number of
compartments, like horses in the front of a row of stalls (Fig. 4), and
their tops joined. There will be one more side to the row of stalls than
there are horses, otherwise a side of one of the extreme stalls would be
wanting. Thus there are two ways of indicating the position of a
particular variate, either by its _serial number_ as ‘first,’ ‘second,’
‘third,’ or so on, or by _degrees_ like those of a thermometer. In the
latter case the sides of the stalls serve as degrees, counting the first
of them as 0°, making one more graduation than the number of objects, as
it should be. The difference between these two methods has to be made
clear, and that while the serial position of the Median object is always
the same in any two Arrays whatever be the number of variates, the
serial position of their subdivisions cannot be the same, the ignored
half interval at either end varying in width according to the number of
variates, and becoming considerable when that number is small.
Lines of proportionate length will then be drawn on a blackboard, and
the limits of the Array will be also drawn, at a half interval from
either of its ends. The base is then to be divided centesimally.
Next join the tops of the lines with a smooth curve, and wipe out
everything except the curve, the Limit at either side, and the
Centesimally divided Base (Fig. 5). This figure forms a Scheme of
Distribution of Variates. Explain clearly that its shape is independent
of the number of Variates, so long as they are sufficiently numerous to
secure statistical constancy.
Show numerous schemes of variates of different kinds, and remark on the
prevalent family likeness between the bounding curves. (Words and
meanings learnt—Schemes of Distribution, Centesimal graduation of base.)
The third lesson passes from Variates, measured upwards from the base,
to Deviates measured upwards or downwards from the Median, and treated
as positive or negative values accordingly (Fig. 6).
Draw a Scheme of Variates on the blackboard, and show that it consists
of two parts; the median which represents a constant, and the curve
which represents the variations from it. Draw a horizontal line from
limit to limit, through the top of the Median to serve as Axis to the
Curve. Divide the Axis centesimally, and wipe out everything except
Curve, Axis, and Limits. This forms a Scheme of Distribution of
Deviates. Draw ordinates from the axis to the curve at the 25th and 75th
divisions. These are the ‘Quartile’ deviates.
At this stage the Genesis of the theoretical Normal curve might be
briefly explained and the generality of its application; also some of
its beautiful properties of reproduction. Many of the diagrams already
shown would be again employed to show the prevalence of approximately
normal distributions. Exceptions of strongly marked Skew curves would be
exhibited and their genesis briefly described.
It will then be explained that while the ordinate at _any_ specified
centesimal division in two normal curves of deviation measures their
relative variability, the Quartile is commonly employed as the unit of
variability under the almost grotesque name of ‘Probable Error,’ which
is intended to signify that the length of any Deviate in the system is
as likely as not to exceed or to fall short of it. This, by
construction, is the case of either Quartile.
(New words and meanings—Scheme of Distribution of Deviates, Axis,
Normal, Skew, Quartile, and Probable Error.)
In the fourth lesson it has to be explained that the Curve of Normal
Distribution is not a direct result of calculation, neither does the
formula that expresses it lend itself so freely to further calculation,
as the curve of Frequency. Their shapes differ; the first is an Ogive,
the second (Fig. 7) is Bell-shaped. In the curve of Frequency the
Deviations are reckoned from the Mean of all the Variates, and not from
the Median. Mean and Median are the same in Normal Curves, but may
differ much in others. Either of these normal curves can be transformed
into the other, as is best exemplified by using a Polygon (Fig. 8)
instead of the Curve, consisting of a series of rectangles differing in
height by the same amounts, but having widths respectively
representative of the frequencies of 1, 3, 3, 1. (This is one of those
known as a Binomial series, whose genesis might be briefly explained.)
If these rectangles are arrayed in order of their widths, side by side,
they become the equivalents of the ogival curve of Distribution. Now if
each of these latter rectangles be slid parallel to itself up to either
limit, their bases will overlap and they become equivalent to the
bell-shaped curve of Frequency with its base vertical.
The curve of Frequency contains no easily perceived unit of variability
like the Quartile of the Curve of Distribution. It is therefore not
suited for and was not used as a first illustration, but the formula
that expresses it is by far the more suitable of the two for
calculation. Its unit of variability is what is called the ‘Standard
Deviation,’ whose genesis will admit of illustration. How the
calculations are made for finding its value is far beyond the reach of
the present lessons. The calculated ordinates of the normal curve must
be accepted by the learner much as the time of day by his watch, though
he be ignorant of the principles of its construction. Much further
beyond his reach are the formulae used to express quasi-normal and skew
curves. They require a previous knowledge of rather advanced
mathematics.
(New words and ideas—Curve of Frequency, Standard Deviation, Mean,
Binomial Series).
The fifth and last lesson deals with the measurement of Correlation,
that is, with the closeness of the relation between any two systems
whose variations are due partly to causes common to both, and partly to
causes special to each. It applies to nearly every social relation, as
to environment and health, social position and fertility, the kinship of
parent to child, of uncle to nephew, &c. It may be mechanically
illustrated by the movements of two pulleys with weights attached,
suspended from a cord held by one of the hands of three different
persons, 1, 2, and 3. No. 2 holds the middle of the cord, one half of
which then passes round one of the pulleys up to the hand of No. 1; the
other half similarly round the other pulley up to the hand of No. 3. The
hands of Nos. 1, 2, and 3 move up and down quite independently, but as
the movements of both weights are simultaneously controlled in part by
No. 2, they become ‘correlated.’
The formation of a table of correlations on paper ruled in squares, is
easily explained on the blackboard (Fig. 9). The pairs of correlated
values _A_ and _B_ have to be expressed in units of their respective
variabilities. They are then sorted into the squares of the
paper,—vertically according to the magnitudes of _A_, horizontally
according to those of _B_—, and the Mean of each partial array of _B_
values, corresponding to each grade of _A_, has to be determined. It is
found theoretically that where variability is normal, the Means of _B_
lie practically in a straight line on the face of the Table, and
observation shows they do so in most other cases. It follows that the
average deviation of a _B_ value bears a constant ratio to the deviation
of the corresponding _A_ value. This ratio is called the ‘Index of
Correlation,’ and is expressed by a single figure. For example: if the
thigh-bone of many persons deviate ‘very much’ from the usual length of
the thigh-bones of their race, the average of the lengths of the
corresponding arm-bones will differ ‘much,’ but not ‘very much,’ from
the usual length of arm-bones, and the ratio between this ‘very much’
and ‘much’ is constant and in the same direction, whatever be the
numerical value attached to the word ‘very much.’ Lastly, the
trustworthiness of the Index of Correlation, when applied to individual
cases, is readily calculable. When the closeness of correlation is
absolute, it is expressed by the number 1·0; and by 0·0, when the
correlation is nil.
(New words and ideas—Correlation and Index of Correlation.)
This concludes what I have to say on these suggested Object lessons. It
will have been tedious to follow in its necessarily much compressed
form,—but will serve, I trust, to convey its main purpose of showing
that a very brief course of lessons, copiously illustrated by diagrams
and objects to handle, would give an acceptable introduction to the
newer methods employed in Biometry and in Eugenics. Further, that when
read leisurely by experts in its printed form, it would give them
sufficient guidance for elaborating details.