Pedagogical Anthropology

by Maria Montessori

Let us suppose, for instance, that the curve here described represents the distribution of the stature. If we mark upon the abscissæ the progressive measurements, 1.55; 1.56; 1.57; 1.58; 1.59; 1.60, etc.... 1.75; 1.76; 1.77; 1.78; 1.79; 1.80, and on the axis of the ordinates the number of individuals having a determined stature, the path of the curve will show that there is a majority of individuals possessing a mean central measurement; and that the number of individuals diminishes gradually and symmetrically above and below, becoming extremely few at the extremes (exceptionally tall and low statures). When the total number of individuals is sufficiently large, the curve is perfect (curve of errors): Fig. 156.

Fig. 156.—The highest part of this curve corresponds to the medial centre of density.

In such a case, the general mean coincides with the median, that is, with the number situated at the centre of the basal line, because, since all the other measurements, above and below, are perfectly symmetrical, in calculating the mean average they cancel out. There is still another centre corresponding to the mean: the centre of density of the individuals grouped there, because the maximum number corresponds to that measurement. Accordingly, if, for example, in place of half a million men whose measurements of stature, when placed in seriation, produced a perfect binomial curve, we had selected only ten men or even fewer from those corresponding to the median line; the general mean stature obtained from those half million men and that obtained from the ten individuals would be identical. For we would have selected ten individuals possessing that mean average stature which seems to represent a biological tendency, from which many persons deviate to a greater or less extent, as though they were erroneous, aberrant, for a great variety of causes; but these aberrant statures are still such that by their excess and their deficiency they perfectly compensate for each other; so that the mean average stature precisely reproduces this tendency, this centre actually attained by the maximum number of individuals. Supposing that we could see together all these individuals: those who belong at the centre being numerically most prevalent, will give a definite intonation to the whole mass. Anyone having an eye well trained to distinguish differences of stature could mentally separate those prevalent individuals and estimate them, saying that they are of mean average stature. This curve is the mathematical curve of errors; and it corresponds to that constructed upon the exponents of Newton's binomial theorem and to the calculation of probability. It corresponds to the curve of errors in mathematics: for example, to the errors committed in measuring a line; or in measuring the distance of a star, etc. Whoever takes measurements (we have already seen this in anthropometrical technique and in the calculation of personal error) commits errors, notwithstanding that the object to be measured and the individual making the measurements remain the same. But the most diverse causes; nerves, the weather, weariness, etc., causes not always determinable and perhaps actually more numerous than could be discovered or imagined, all have their share in producing errors of too much and too little, which are distributed in gradations around the real measurement of the object. But since among all these measurements taken in the same identical way we do not know which is the true one; the seriation of errors will reveal it to us, for it causes a maximum number of some one definite measurement (the true one) to fall in the centre of the aberrations that symmetrically grade off from the centre itself.

Viola gives some very enlightening examples in regard to errors. Suppose, for instance, that an artist skilled in modeling wished to reproduce in plaster a number of copies of a leaf, which he has before his eyes as a model.

The well-trained eye and hand will at one time cause him to take exactly the right quantity of plaster needed to reproduce the actual dimensions of the leaf; at another, on the contrary, he will take more and at another less than required.

By measuring or superimposing the real leaf upon the plaster copies, the sculptor will be able to satisfy himself at once which of his copies have proved successful.

But supposing, on the contrary, that the real leaf has disappeared and that a stranger wishes to discover from the plaster copies which ones faithfully reproduce the dimensions of the leaf? They will be those that are numerically most prevalent.

The same thing holds true for any attempt whatever to attain a predetermined object. For example, shooting at a mark. A skilful marksman will place the maximum number of shots in the centre, or at points quite near to the centre; he will often go astray, but the number of errors will steadily decrease in proportion as the shots are more aberrant, i.e., further from the centre. If a marksman wished to practise in like manner against some wall, for example, on which he has chosen a point that is not marked, and hence not recognisable by others, this point thought of by the marksman, may be determined by studying the cluster of shots left upon the wall.

In the same way an observer could determine the hour fixed for a collective appointment, such as a walking trip, by the manner in which the various individuals arrive in groups; some one will come much ahead of time because he has finished some task which he had expected would keep him busy up to the hour of appointment; then in increasing numbers the persons who come a few minutes ahead of time because they are provident and prompt; then a great number of people who have calculated their affairs so well as to arrive precisely on time; a few minutes later come those who are naturally improvident and a little lazy; and lastly come the exceptional procrastinators who at the moment of setting forth were delayed by some unexpected occurrence.