OCR Rendition - approximate

TRANSACTIONS OF SECTION A. 639 Sums. Next, three shows and counts of hands are to be made : (1) for less than A ; (2) for more than A, but less than B ; (3) for more than B. The results are a per cent. vote for less than A ; 100-a vote for more than A. b per cent. vote for less than B; 100-b vote for more than B. Numerous analogies amply justify the assumption that the estimates will be distributed on either side of their (unknown) median, rn, with an (unknown) quartile, q, in approximate accordance with the normal law of frequency of error. The following table of centiles (a better word than 'Per-centiles,' which I originally used), having a quartile = 1, is founded upon that law. It is extracted from my 'Natural Inheritance ' (Macmillan, 1889, p. 205) to serve the present purpose. Gentiles to the Grades 0°-100°. Grades 0~ 2= 3 5' ; 9 0° -inf: -3.45 -3.05 -2-79 -2.60 --2.44 -2.31.-2.19'-2'CS -1.99 100 -1.90 -182 -1.74 -1-67 -1•G'-) -1.54 -1.17 -1.42'-1.36 -1.30!. 20° -1.25 --1.20 -1.15 -1.10 -1.05 -1.00 -0.95 -0.91 0.86'-0152 30° --0.78 -0`74 -0.69 -0.65 -0.61 -0'57 -0,53 -0.49 -0.45 ,-0.41 40° -0'38 -0.34 -0'30 -0.26 -0.22 -0.19 -0-15 -0.11 -0.07 -0-0411 50° .0.00 + 0.04 + 0.07 + 0.11 +0.15 +0.19 +0,22'+0.26 +0.30'+0.34 60° +0-38 + 0-41 +0-45 +0-49 +0.53 +0.57'+0.61-+0.65 +0.69'+0.74: 7011 ±0.78"+082 +0.86'+0.91 +0.95 -1.00,+1.051+1.10 +1.15 -1.20 80° + 1.25 ' + 1.30 +1-36 ' +1-42 +1-47 +1-54 - 1.60 - 1.67 - 1-74 - I -S2 90° + 1-()0, + 1.99 +2-08 + 2.19 2.31 , 2.44 ' +2-60 .- 2.79 - 3.05 + 3.45 Let a be the tabular number inclusive of its sign, that corresponds to the grade a°, and let (3 be that which corresponds OLo b°, then vi +qa= i; -m q3=B, - } --L l3-a J l f -a j Example :-A =100, B = 500; a = d0°, b = 80°, whence a = - 0.38, ;3 = + 1.25 and in= 193. The truth of the determination of in may now, if so desired, be tested by putting two new values A' B' to the vote, in the same way as A and B, but A' and B' should not differ much from in, and it should be an honourable understanding that no member should deviate from his first opinion in giving his second vote. When about to utilise this method, A and B ought to be so selected that A shall secure not less than 5 per cent. of the votes, and B not more than 95, because thecurve of error ceases to be trustworthy near to its extremities, but a dependence upon it within the limits of 5° and 95° will seem pedantic only to those who are unfamiliar with its nature and with its numerous and successful applications. It will be easily understood that this method is a particular case of the more general problem, that in any system of normal variables which has been arrayed between the grades of 0° andv100°, if the values be given that correspond to any two specified grades, those that correspond to each and every other grade can be found. I heartily wish that when occasion offers, some Assembly may be disposed to experiment on the above method. The calculators should, of course, rehearse the work beforehand, and be well prepared to carry it through both rapidly and surely. It is worth mentioning flint when the above table is not at hand, a graphical substitute for it, that ranges between 5° and 93° and is true to the first place of decimals, may be quickly made by those who can recollect three simple factors. Thus, draw between two vertical limits, 0° and 100°, a straight line on squarely ruled paper, having a quartile equal to l. Accept this line in lieu of the eurve between 300 and 70'-, add one-t`ventietli to tl;e n`cthe of the centiless at 26 ' and 50`, I whence