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Means distinguished from Averages.


the table below. * Supposing each measure exactly performed, these, therefore, may be taken as the results of nature's own measurements of her own model; and the question whether she recognises such a model ? is at once decided by inspection of the groups, in which the animus mensurandi is broadly apparent. It is equally so that such model would fall within the group of 40 inches. An exact calculation of the mean allowing to each group a weight in proportion to the number it contains, assigns 39.830 inches as the circumference of the chest of this model.

Now this result, be it observed, is a mean as distinguished from an average. The distinction is one of much importance, and is very properly insisted on by M. Quetelet, who proposes to use the word mean only for the former, and to speak of the latter as the arithmetical' mean. We prefer the term average, not only because both are truly arithmetical means, but because the latter term carries already with it that vitiated and vulgar association which renders it less fit for exact and philosophical use. An average may exist of the most different objects, as of the heights of houses in a town, or the sizes of books in a library. It may be convenient, to convey a general notion of the things averaged; but involves no conception of a natural and recognisable central magnitude, all differences from which ought to be regarded as deviations from a standard. The notion of a mean, on the other hand, does imply such a conception, standing distinguished from an average by this very feature, viz. the regular march of the groups, increasing to a maximum, and thence again diminishing. An average gives us no assurance that the future will be like the past. A mean may be reckoned on with the most implicit confidence. All the philosophical value of statistical results depends on a due appreciation of this distinction and acceptance of its consequences.

The recognition of a mean, so distinguished from a mere average, among a series of results thus grouped in order, depends on the observance of a conformity between the observed law of progression in the magnitude of the groups, and the abstract law of probability above stated, from which every consideration

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has been excluded but the reality of some central truth, and an intention of arriving at it liable to be baffled by none but purely casual causes of error. And the test to be applied in this and all similar cases, is this. Is it possible to assign such a mean value, and such a probable error as shall alone, by the simple application of the table of probabilities, reproduce the numbers under the several groups in order, with no greater deriations than shall be fairly attributable to a want of observations numerous enough to bring out the truth? In the instance before us, the answer to this inquiry is contained in the results of calculation as compared with fact in the table above referred to. The mean we have used is 39.830 inches, and our probable error 1.381 inches. Those of M. Quetelet differ somewhat from these values, which accounts for the trifling discrepancy of the results.

The coincidence admits of being placed in even a more striking light. In the complete expression, by theory, of all the groups in a statement of this kind, three elements are involved – the mean value — the maximum group having that mean for its center - and the probable error. And to determine these, it ought to suffice to have before us three terms of the series. Suppose then we take for our data the numbers corresponding to 35, 39, and 43 inches, viz. 81, 1073, and 370, given by observation. Then, by a computation of no great difficulty, there will result, for the mean value, 39.834 inches, and for the probable error 1.413 inches, both agreeing almost precisely with those already stated. For the greatest possible group of an inch in amplitude the same calculation gives 1161, which is in obvious accord with observation. No doubt, then, can remain as to the reality of a typical form, from which all deviations are to be regarded as irregularities. On this M. Quetelet observes —

"I now ask if it would be exaggerating to make an even wager that a person little practised in measuring the human body, would make a mistake of an inch in measuring a chest of more than 40 inches in circumference. Well! admitting this probable error, 5738 measurements made on the same individual, would certainly not group themselves with more regularity as to the order of magnitude than these 5738 measurements made on the Scotch soldiers; and if the two series were given us without their being particularly designated, we should be much embarrassed to state which series was taken from 5738 different soldiers, and which was obtained from one individual with less skill and ruder means of appreciation.' (P. 92. Transl.)

This is assuredly an over-statement. So far from less skill being supposed in the measurements of the individual, the probable error of nature is nearly half as much more than that as

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sumed here for the term of comparison (1 inch); and it is clearly beyond the bounds of any supposable negligence or rudeness of practice, to commit such errors as the extreme registered deviations (7 inches one way and 9 the other), in a series of such measurements however multiplied ; or even half those amounts.

We are thus led to the important and somewhat delicate question,-What we are to consider as reasonable limits, in such determinations - beyond which, if deviations from the central type be recorded, they are either to be referred to exaggeration, or regarded as monstrosities?

The answer to this question must evidently depend, first, on the probable' deviation from the mean or typical value; secondly, on the number of cases experience has offered, or within which we agree to limit our range of speculation. We have already seen that 20,000 might be betted against 1 that an observed deviation, one way or other from the type, will not exceed sixfold its probable' value; and therefore we shall have double that amount of chances against such a deviation in either direction separately. Among 40,000 individuals, therefore, we are entitled to expect to find one so far deviating from the mean type in excess, and one in defect. Beyond this the probabilities decrease with extreme rapidity. Thus, for a 7-fold deviation, we must seek our specimen among 263,000; and, for an 8, 9, 10-fold, among 4,760,000, 250,000,000, and 25,000,000,000 respectively.

We might apply these numbers to the case of giants and dwarfs, if we had any dependable data from which the mean human stature and its probable deviation could be ascertained. From an interesting discussion of the measurements of 100,000 French conscripts, taken at the age of 20 years, and arranged in groups, inch by inch, M. Quetelet concludes a mean height of 63.947 inches (English measure), with a probable deviation of 1.928 inches. The numbers in the respective groups (with certain exceptions at the lower limit, run in satisfactory accordance with the law of abstract probability, and afford complete evidence of the existence of a central type, uniform or nearly so, for the French nation. Allowing (according to the tables given by M. Quetelet in his · Essai de Physique Sociale') 0:43 inches for the growth from the 20th year to adult stature, we may take 5 ft. 44 in. (English) for the adult height of a typical Frenchman, with a probable deviation of almost exactly 2 in. Calculating on these data, we should expect to find in the existing population of France (taken at 12,000,000 adult males), one individual of 6 ft. 9 in. in height; in that of the whole world only one of 6 ft. 11 in.; and, in the whole records of the human race, not more than one of 7 ft. 1 in. The corresponding dwarfs would be respectively 4 A. IİR., 3 ft. 11 in., ad 3 ft. 9 in.

The actual limits, both in excess and defect, we need barily observe, are very much wider. M. Quetelet, on the authority of Birch, assigns 17 in, as the minimum of human stature anthentically recorded. • The celebrated dwarf Bebé, king of * Poland, was taller. The most celebrated dwarf of recent times, C. Stratton (atias Tom Thumb), exceeds this limit by 10 in. * Taking 17 for the minimum, and allowing an equal deviation in excess from the conscript type, our author fixes his gigantic limit hypothetically at 9 ft. 3 in. Even this we are disposed to extend." Disregarding such pigmies as the Swedish body guard of Frederick the Great (8 ft. 3 in.); Byrne, the celebrated · Irish "giant' (8 ft. 4 in.), whose skeleton adorns the Hunterian collection; the Dutch giant of Schoonhaven (8 ft. 6 in.), attested by Diemerbroeck and Ray; and the Emperor Maximin — we have the testimony of Pliny to an Arab, named Gabbara (9 ft. 9 in.) "the tallest man that hath been seen in our age' (Nat. Hist. book vii. Wern. CL Transl. ï. 200.), and to the preservation, as curiosities, in a vault in the Sallustian Gardens, of the bodies of “two others, named Pusio and Secundilla,' higher than Gabbara by half a foot (10 ft. 3 in.). The mummies might be counterfeited: the living Arab, exhibited by Claudius, would hardly escape some scrutiny. But, even in modern times, we have testimony, to which we cannot refuse at least the epithet of respectable, for the existence of giants who might well claim companionship with him of Basan, whose bedstead measured nine cubits after the cubit of a man,' or the Philistine, whose stature is expressly stated at six cubits and a span (11 ft. 5 in.). Thus, Dr. Thomas Molyneux,' an excellent scholar and phy“sician,' and a Fellow of the Royal Society, describes a wellformed human os frontis preserved in the School of Medicine at Leyden, from whose dimensions, carefully measured by himself, he concludes it to have belonged to an individual between 11 and 12 ft. in height: "a goodly stature,' he remarks, and such as may well deserve to be called gigantick.'t Molyneux ac

* Cardan saw a man in Italy of full age not above a cubit (21.9 in.) high. He was carried about in a parrot cage.-Wern. Club. Transl. Pliny's Hist. of Nature, ii. 200. note. Suetonius mentions a Roman knight exhibited by Augustus in the theatre · tantum ut ostenderet 'adolescentulum, honeste natum, quod erat bipedali minor, librarum

septemdecim et vocis immensæ. • f Dr. Molyneux was brother to the celebrated astronomer of that name. (See Phil. Trans. vol. xv. p. 880. and vol. xxii. 487.) He gives · engravings of this extraordinary bone, accompanied by one of a similar bone of ordinary size for comparison. Its dimensions are stated

Giants and Dwarfs.

27 companies this description with notices of several other cases, which it may perhaps be worth while to recall attention to: as, for example, that of a skeleton seen and measured by Andreas Thevet, cosmographer to Henry III., king of France and Portugal, which belonged to a man 11 ft. 5 in. in height, who died in 1559.* And again, the cases of a man nearly, and a woman quite, 10 ft. in height, are attested by Beccanus, in his Origines Antwerpianæ, 1569, as an eye-witness, the man living within ten miles of his own residence. We find mention by Dr. Degg (Phil. Trans. xxxv. p. 363.) of the exhumation, in 1686, at Repton, of a human skeleton 9. ft long.

As regards men of seven feet in stature, so many cases are recorded that they can hardly be termed gigantic; and, whatever we may think of such extreme cases as 11 or 12 feet, it seems impossible to hesitate at admitting 9 ft. 6 in. as a stature which may be exceeded, and perhaps even 10 ft. attained, without monstrosity in the proper sense of the word. We must, therefore, conclude that the probable’ deviation of nature's workmanship from her universal human type cannot possibly be less than the double of that resulting from the French measurements : a conclusion which ought to excite no surprise ; since it is impossible to reason from a single nation (and that decidedly undersized, and of remarkable uniformity as to habits of life) to the whole species. †

Practically speaking, nothing can be simpler or more easily stated than the rules for handling any given series of determinations of a single quæsitum, supposed to be arranged to our hands as follows. Coronal suture, measured along its course from orbit to orbit, 21 in. Breadth from back to front to junction of the bones of the nose, 9.1 in.; from side to side following the convexity of the skull, 12.2 in.

[N. B. Just as our last revise is going to press we are informed, on the highest authority, that this might have been a case of hydrocephalus, though it could not have been known as such to Molyneux.]

* The head was 37 in. in circumference. The leg bones measured fully 3 ft. 4 in.

† M. Quetelet, as above stated, makes the mean French conscript of 20 years 63.947 in.; but he elsewhere (Essai de Phys. Soc. ii. 14.) states it at 1:615 met. or 63.583 in., which, with a growth of 0.433 in. to the adult age, gives only 5 ft. 4.02 in. for the typical French stature in 1817. The Belgian type (Essai de Ph. Soc. ii. 42.) is 5 ft. 768 in. That of the English non-manufacturing labourers in the neighbourhood of Manchester and Stockport, he states at 1.775 met. or 5 ft. 9.88 in. at the age of 18, which gives 5 ft. 10:75 in. for the adult type in Lancashire. The mean between extremes of 17 and 120 in. is 5 ft. 8.5 in., which is by no means improbable as a general standard.

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