298 



NA TU.RE 



[Jan. 24, i\ 



series is no proper measure of the variety of the men wh^ 

 compose it. However few in number the objects in the 

 series may be, it is always possible that a giant or a dwarf, 

 so to speak, may be among them. The presence of either 

 would mislead as to the range of variety likely to be found 

 in another sample taken from the same group. The values 

 in a marshaled series run with regularity only about its 

 broad and middle part ; they never do so in the parts near 

 to either of its extremities. In a series that consists of a 

 few hundreds of individuals, the regularity usually begins 

 at about grade 5°, and continues up to about grade 95^ 

 Therefore it is out of this middle part, between 5" and 95°, 

 or better, in a still more central portion of it, that points 

 should be adopted between which variety may be 

 measured. Such points are conveniently found at the 

 2Sth and the 75th grades. Just as the grade 50" divides 

 the class into two equal parts, so the grades 25^^ and 75" 

 subdivide it into quarters, and the difference between 

 those values affords an irreproachable basis for the unit 

 of variety. 'J"he actual unit is half the value of that dif- 

 ference, because the value at 25° tends to be just as much 

 below that at 50°, as the value at 75" is above it. There- 

 fore the average of these two values is a better measure 

 than their sum. Briefly, if we distinguish the measure 

 at 25'' by the letter Qj, and that at 75^= by O3, then the 

 unit of variety is ^(Qs - Q,), and this unit we will hence- 

 forth call Q. As M measures the average, so Q measures 

 the variety, and they are independent of one another. In 

 strength, for example, the relation of Q to M in the 

 particular group of adult males on which I worked was 

 as I to 10 ; in the statures'of the same group it was as i to 

 40 ; in breathing capacity as i to9 ; in weight as I to 14. 



The arithmetic mean or average is a muddle of all the 

 values in the series ; it is by no means so clear an idea 

 as the middlemost value M. Therefore, although the 

 peculiarities of an individual are commonly considered 

 in the light of deviations from the average value, I prefer 

 to reckon them as deviations from M. Practically the 

 two methods are identical, but I find the latter more con- 

 venient to work with, and believe it to be the better of 

 the two in every way. 



Deviation is identical with variation, and the well-known 

 law of frequency of error gives data whence the relative 

 values of the deviations at the several grades may be cal- 

 culated for any normal series. If we know the deviation 

 at any one grade, then the absolute value of those at 

 every other grade can be calculated ; consequently the 

 variety of the whole series is thereby expressed. 



The small table of distribution, of which I spoke, gives 

 the values at each grade when Q is equal to i. Then 

 the value at 25' is - i, and that "at 75° is + i. If we 

 desire to determine O in any such series, the only required 

 datum is the deviation at some one known grade, since, 

 by dividing that deviation by the tabular value, we get O 

 at once. Or, conversely, if we know the O of the series^ 

 and wish to calculate the deviation at any given grade, we 

 multiply O by the tabular deviation. Thus, in stature, 

 which varies in an approximately normal manner, the 

 value of Q is about 17 inch, therefore to find the devia- 

 tion in stature at any grade, we multiply 17 inch by the 

 tabular value. 



If we know the vieasures at any two grades of a normal 

 series, we are easily able to calculate both Q and M, and 

 cm thence derive the measures at any other desired 

 grades. I have long since pointed out the possibility of 

 a traveller availing himself of this method ; but, for the 

 want of a table of distribution, the calculation would 

 probably puzzle him. With the aid of this table the 

 calculation is made most readily. Let us suppose that 

 the traveller is among savages who use the bow, and that 

 he desires to learn as much as he can about their 

 strengths. He selects two bows ; the one somewhat 

 easy to draw, and the other somewhat difficult, and at 

 leisure, either before or after the experiment, he ascertains 



exactly how many pounds weight they severally require 

 to draw them to the full. Then by exciting emulation, 

 and by offering small prizes, he induces a great many of 

 the natives to try their strengths upon them. He notes 

 how many make the attempt, and how many of them fail 

 in either test. This is all the observation requisite, 

 though common-sense would suggest the use of three and 

 not two bows, in order that the data from the third bow 

 might correct or confirm the results derived from the 

 other two. Let us work out a case, not an imaginary 

 one, but derived from tables I have already published, 

 and of which I will speak directly. Let the problem be 

 as follows : — 



30 per cent, of the men failed to exert a pulling 

 strength of 68 pounds ; 60 per cent, failed to pull •]■] pounds. 

 What is the O and the M of the group. ^^ 



Consider this 30 per cent, to be the exact equivalent 

 of grade 30"", and the 60 per cent, of grade bo\ The 

 reason why the percentage of failure, and the number of 

 the grade are always equivalent will be found in a foot- 

 note to the table, and I need not stop to speak of it. 

 Now, the tabular value at grade 30" is — 078 ; that at 

 60° is + o'38 ; the difference between them being ri6. 

 ■On the other hand, the difference between the two test 

 values of 68 pounds and ^'j pounds is 9 pounds. Therefore 

 Q is equal to 9 pounds divided by ri6; that is, to 7'8 

 pounds. M may be obtained by either of two ways, 

 which will always give the same answer. We may sub- 

 ract 0*38 X 7'8 pounds from 'j-j pounds, or we may add 

 078 X 7"8 pounds to 68 pounds. Each gives 74 pounds. 

 Observation gave precisely these values both for Q and 

 for M. The data were published in the Journal of this 

 Institute as a table of " percentiles," and were derived 

 from measures made at the International Health Exhibi- 

 tion. The value of M is given directly in the table, but 

 that of O happens not to be given there ; it may easily 

 be found by interpolation. That table affords excellent 

 material for experimental calculations on the principle 

 of this test, and for estimating its trustworthiness in 

 practice. 



It contains a variety of measures referring to eighteen 

 different series, all corresponding to the same grades — 

 namely, to 5", 10°, 20", and onwards for every tenth grade 

 upto9o"andendingwith95°. Themeasures refer to stature, 

 height sitting above seat of chair, span, weight, breathing 

 capacity, strength of pull, strength of squeeze, swiftness 

 of blow, keenness of eyesight, in each case of adult males 

 and of adult females separately. I have since found that 

 when the deviations are all reduced in terms of their 

 respective O values, by dividing each of them by its O, 

 that the average value of all the deviations at each of the 

 grades in the eighteen series closely corresponds to the 

 normal series, though individually they differ more or less 

 from it, some in one way, some in another. On the whole, 

 the error of treating an unknown series as if it were a 

 normal one can rarely be very large, always supposing that 

 we do not meddle with grades lower than 5 or higher 

 than 95^ 



It will be of interest to put the comparison on record. 

 It is as follows : — 



The "observed" are the mean values, made as above described, of the 

 eighteen series; the "normal" are taken from the table cf distribution 

 aiven further on. 



