On a Proposed Statistical Scale

Abstract

AT a lecture last Friday evening, at the Royal Institution, I spoke on a subject which happens to lie at the meeting-point of many special sciences, and therefore, as I am desirous of having it well discussed, and from many points of view, it seems to me best to state it afresh in your columns for that purpose. It refers to the definition of the estimated degree of development of any quality whatever, without reference to external standards of measurement. The scale I propose depends on two processes; the one is securely based on the law of statistical constancy, the other is doubtfully based on the law of frequency of error. (1) At present we are accustomed to deal with averages and the like, which can only be obtained by measuring every individual by a detached standard scale, and going through an arithmetical process afterwards. Now I want to deal with cases for which no external standard exists, and I propose to proceed in quite another way, on the principle that intercomparison suffices to define. We have only to range our group in a long series, beginning with the biggest and ending with the smallest; and then we know by the law of statistical constancy that, the individual who occupies the half-way point, or any other fractional position of the entire length, will be of the same size as the individual who occupies a similar position in any other statistical group of similar objects. We state his size with statistical precision by saying that his place is so and so in a series. We appeal to a standard which lies dormant in every group, and which a statistician can evoke, for temporary purposes of comparison, whenever he will. (2) What places in the series shall we select for our graduations? Equal fractions of its length will never do—I mean such as one-tenth, two-tenths, &c. —because of the great inequality of the variation in diffeient parts of the series, being insensible between those whose position is near its middle and great between those at either end. I propose to use a scale founded on the law of Frequency of Error, which gives a scale of equal parts wherever that law applies, and I use the “probable error” for the unit of tlie scale. Thus, in a row of a hundred individuals the graduations of + 2°, + 1°, 0°, − 1°, − 2°, respectively would be at the following places, in percentages of the length of the series:—2, 9, 25, 50, 75, 91, 98. We know that the law of Frequency of Error applies very closely to the linear measurements of the human form. Now suppose that I want to get the average height and “probable error” of a crowd of savages. Measuring them individually is out of the question; but it is not difficult to range them—roughly for the most part, but more carefully near the middle and one of the quarter points of the series. Then I pick out two men, and two only—the one as near the middle as may be, and the other near the quarter point, and I measure them at leisure. The height of the first man is the average of the whole series, and the difference between him and the other man gives the probable error. The question I put is, whether any more convenient subdivision of a series can be suggested for universal use than that above mentioned. Its merits are, that it applies very fitly to linear measurements of all natural groups; also to errors of observation, which are akin to many of the moral qualities, for the measurement of which the scale is especially needed. It would not apply to weight, but is less out of relation to it than most persons might think, because weights do not vary as the cubes of the heights. Tall men are often thin, and short ones are fat, and the curious fact seems thoroughly verified that the general relation between height and weight is strictly as the squares. (See Gould's “Sanitary Memoirs of the War of the Rebellion,” Cambridge, U. S., 1869, p. 408—410.) If we arrange a series and graduate it according to equal differences of the squares of the heights of the men, we are not so far astray as if we had dealt with the cubes. But I cannot imagine any quality, unless possibly music and memory, to vary so rapidly towards the large end of the series as the latter division would show. To sum up: subdivision in equal parts is of no use practically, and is therefore out of the question; the law of error will do very accurately for many large groups of cases; the law of error modified by being brought into relation to bulk will rarely, if ever, be right for other qualities. It therefore seems to me reasonable to adopt the law of error series, as the best compromise, and to accept it as “the common statistical scale.” If, for example, I estimate a soldier's energy at + 2° (S.S.), I state what everybody who cared to inquire into the subject would construe in exactly the same sense as I used the phrase, and he would also be inclined to believe, until better informed, that the difference between such a man's energy and that of a man of + 0° (S.S.) was twice as great as between him and a man of + 1° (S.S.).