The essentials of mental measurement . ttle simple algebraical arrangement this then reduces to theform g2 . This is a normal curve, with area Its centre is at a; = rcaja^, y = c, on the regression Une. Itsstandard deviation is (jj^-\/{l — r^) and is independent of c. Similar statements hold for a vertical section along a fine x = c,the constant standard deviation being crgVil ^^)- A similar procedure will show, with rather more cumbrous algebra,that any vertical section is a probabihty curve. (5) OTHER METHODS OF DETERMINING CORRELATION1. Fourfold Table. h X2 Vi a b a + b Vi c d c + d a+
The essentials of mental measurement . ttle simple algebraical arrangement this then reduces to theform g2 . This is a normal curve, with area Its centre is at a; = rcaja^, y = c, on the regression Une. Itsstandard deviation is (jj^-\/{l — r^) and is independent of c. Similar statements hold for a vertical section along a fine x = c,the constant standard deviation being crgVil ^^)- A similar procedure will show, with rather more cumbrous algebra,that any vertical section is a probabihty curve. (5) OTHER METHODS OF DETERMINING CORRELATION1. Fourfold Table. h X2 Vi a b a + b Vi c d c + d a+c b + d N 124 COREBLATION [PT. II (a) When the divisions pass through the means of both characters, TT (a — b) r = sm 2(a + b) This formula (Sheppards) is of little practical use, since the meanvalues, in cases where the fourfold table is the only method which canbe used, are generally unknown. (6) In cases where the dividing Unes do not pass through the means(means not known) the equations for determining the correlation are: and. a d Fig. 22 (g + c) - (6 + d) /2 fi dx N (the total no (the cross-hatched area on the upper curve) ?ea on the sidfrom which h and k can be obtained by Sheppards Tables; and finally (a + 6)^(c + ^)^^,^^_^.^^(the cross-hatched area on the side curve) CH. VI] MATHEMATICAL THEOEY OF COERELATION 125 ad — he Whk = r + -^U + ^ (h^ - \) {h^ - \) + J h {h^ - 2.) k {B - 3) 6 r5 where + J20 (^^ - 6^ + 3) (^ - 6F + 3) + ^h{¥- 10^2 + 15) h{¥~ lOP H- 15) + i (^* - ISA* + 45A2 - 15) (*« - 15F + 45F - 15)5040 + etc., V2m V2^ In obtaining these equations, a normal correlation surface is assumed,with the equation: N 1 12 1 /x^ 2rxy y^\ The probable error of r obtained by the fourfold table method ismuch larger than that given by the formula of p. 113. The correctformula is too complicated to insert heref. 2. Method of ContingencyJ. The following is an example of a contingency table §. Tathers Merry Melanqholy Alternating Even Totals MenyMelancholyAlternating
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