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nitudes differing from the mean value by such and such multiples of the probable error, will occur with such and such degrees of frequency." My proposal is to reverse the process, and to say, "since such and such magnitudes occur with such and such degrees of frequency, therefore the differences between them and the mean value are so and so, as expressed in units of probable error." According to this process, the positions of the first divisions of the scale of divergence, which are those of the mean value plus or minus one unit of probable error, are of course p and q, lying at the and points of the ogive, or, if the base consist of 1000 units, at the 250th point from the appropriate end. The second divisions being those of mean value plus or minus two units of probable error, will, according to the usual Tables, be found at the 82nd point from the appropriate end, the third divisions will be at the 17th, and the fourth at the 3rd. If we wished to pursue the scale further, we should require a base long enough to include very many more than 1000 units.
Remarks on the Law of Frequency of Error.
Considering the importance of the results which admit of being derived whenever the law of frequency of error can be shown to apply, I will give some reasons why its applicability is more general than might have been expected from the highly artificial hypotheses upon which the law is based. It will be remembered that these are to the effect that individual errors of observation, or individual differences in objects belonging to the same generic group, are entirely due to the aggregate action of variable influences in different combinations, and that these influences must be (1) all independent in their effects, (2) all equal, (3) all admitting of being treated as simple alternatives "above average" or " below average ;" and (4) the usual Tables are calculated on the further supposition that the variable influences are infinitely numerous.
As I shall lay much stress on matters connected with the last condition, it will save reiteration if I be permitted the use of a phrase to distinguish between calculations based on the supposition of a moderate number (r) of elements (in which case the frequency of error or the divergence is expressed by the coefficients of the expansion of the binomial (a+b)) and one based on the supposition of the number being infinite (which is expressed by the exponential e), by calling the one the binomial and the other the exponential process, the latter being the process to be understood whenever the "law of frequency of error" is spoken of without further qualification. When the results of
these two processes have to be protracted, as in figure 2, the unit of vertical measurement in the case of a series of bino
mial grades will be a single grade, or, what comes to the same thing, the difference of the effect produced by the plus and minus phase of any one of the alternative elements, upon the value of the whole. The unit of the exponential curve will be q-m of fig. 1, or the probable error. This latter unit is equally applicable to what we may call the binomial ogive, which is the curve drawn with a free hand through the grades. The justification for such a conception as a binomial ogive will be fully established further on. Suffice it for the present to remark that, by the adoption of a unit of this kind, the middle portion of a binomial ogive of 999 elements is compared in the figure with one of 17.
The first three of the above-mentioned conditions may occur in games of chance, but they assuredly do not occur in vital and social phenomena; nevertheless it has been found in numerous
instances, where measurement was possible, that the latter conform very fairly, within the limits of ordinary statistical inquiry, to calculations based on the (exponential) law of frequency of error. It is a curious fact, which I shall endeavour to explain, that in this case a false hypothesis, which is undoubtedly a very convenient one to work upon, yields true results.
In illustration of what occurs in nature, let us consider the causes which determine the size of fruit. Some are important, the chief of which is the Aspect, whose range of influence is too wide to permit us to consider it in one of the simple alternatives good" or "bad." It is no satisfactory argument to say that variations in aspect are partly due to a multitude of petty causes, such as the interposition of leaves and boughs, because, so far as they depend on well-known functions of altitude and azimuth, they cannot be reduced to a multitude of elementary causes. There has been much confusion of ideas on this subject, and also a forgetfulness of another fact-namely, that when we once arrive at a simple alternative, there our subdivision of causes must stop, and we must accept that alternative, however great may be its influence, as one of the primary elements in our calculation.
In addition to important elements, there are others of small, but still of a recognizable value, such as exposure to prevalent winds, the pedigree of the tree, the particular quality of the soil on which it stands, the accident of drains running near to its root, &c. Again, there are a multitude of smaller influences, to the second, third, and fourth orders of minuteness.
I shall proceed to define what I mean by "small;" then I shall show how this medley of causes may admit of being theoretically sorted into a moderate number of small influences of equal value, giving a first approximation to the truth; then how, by a second approximation, the grades of the binomial expansion thence derived become smoothed into a flowing curve. Lastly, I shall show by quite a different line of argument that the exponential view contains inherent contradictions when nature is appealed to, that the binomial of a moderate power is the truer one, and that we have means of ascertaining a limit which the number of its elements cannot exceed. My conclusion, so far as this source of difficulty is concerned, is that the exponential law applies because it nearly resembles the curve based on a binomial of moderate power, within the limits between which comparisons are usually made.
We observe in fig. 2 how closely the outline of an exponential ogive resembles that of a binomial of a very moderate number of elements, within the narrow limits chiefly used by statisticians. The figure expresses a series of 1000 objects marshalled accord
ing to their magnitudes. In the one case the magnitudes are supposed to be wholly due to the various combinations of 17 alternatives, and the elements of the drawing are obtained from the several terms of the expansion of (1+1)1, all multiplied 1000 These form the following series, reckoning to the 217 nearest integer; and their sum, of course, 1000:-0, 0, 1, 5, 18, 47, 95, 148, 186, 186, 148, 95, 47, 18, 5, 1, 0, 0. In the figure these proportions are protracted so far as possible; but the numbers even in the fourth grade are barely capable of being represented on its small scale; after the fourth, the several grades are manifest until we reach the corresponding point at the opposite end of the series. Then, with a free hand, a curve is drawn through them, which gives as their mean value 8.5, as it ought to do. Now, referring to our p and q at the 250th division from either end, I measure the value of q―m (or m-p), which is the unit to which I must reduce any other ogive that I may desire to compare with the present one. Also I can find the values for m+2(q-m) and m+3(q-m), which is going as far as a figure on this small scale admits. I now protract the central portion of an exponential ogive to the same scale, horizontally and vertically. Not knowing its base, I start from its middle point, placing it arbitrarily at a convenient position in the prolongation of the m of the binomial; and I lay off, in the prolongation of p and q, points that are respectively 1 unit of probable error less and greater than m. The Tables of the law of error tell me where to lay off the other points; and so the curve is determined. It must be clearly understood that whereas in the figure both the ogive and the base are given for the binomial series of 17 elements, it is only the ogive that is given for the exponential, there being no data to determine the position of its base. The comparison is simply between the middle portions of the ogives. To speak correctly, I have not actually used the exponential Tables to draw the exponential curve, but have used Quetelet's expansion of a binomial of 999 elements, the results of which are identical, as he has shown, with those of the exponential to within extremely minute fractions, utterly insensible in a scale more than a hundred times as great as the present one.
I find the position of the various points in the two ogives, measured from the appropriate end of the base, to be as is expressed in the following Table:
The closeness of the resemblance is striking. It rapidly increases and extends in its range as the number of elements in the binomial increases; there need therefore be no hesitation in recognizing the fact that a binomial of, say, 30 elements or upwards is just as conformable to ordinary statistical observation as is the exponential. If one agrees, the other does, because they agree with one another.
The fewest number of elements that suffice to form a binomial having the above-mentioned conformity is a criterion of the meaning of the word "small," which was lately employed, because each of those elements would be just entitled to rank as small. I obtain the value of any one of them in an ogive by protracting the series and noticing how many grades are included in the interval q-m. It will be found that in a binomial of 17 elements q-m is equal to eight fifths of one grade. Thence I conclude that in any generic series an influence the range of whose mean effects in the two alternatives of above and below average is not greater than, say, one half of the probable error of the series, is entitled to be considered "small."
I now proceed to show how a medley of small and minute causes may, as a first approximation to the truth, be looked upon as an aggregate of a moderate number of "small" and equal influences. In doing this, we may accept without hesitation, the usual assumption that all small, and à fortiori all minute influences, may be dealt with as simple alternatives of excess or deficiency-the values of this excess and deficiency_being the mean of all the values in each of these two phases. The way in which I propose to build up the fictitious groups may be exactly illustrated by a game of odd and even, in which it might be agreed that the predominance of "heads" in a throw of three fourpenny pieces, shall count the same as the simple "head" of a shilling. The three fourpenny pieces may fall all heads, 2 heads and 1 tail, 1 head and 2 tails, or all tails-the relative frequency of these events being, as is well known, 1, 3, 3, 1. But by our hypothesis we need not concern ourselves about these minute peculiarities; the question for us is simply the alternative one, are the "heads" in a majority or not? We may therefore treat a ternary system of the third order of smallness exactly as a simple alternative of the first order of smallness. Or, again,