ccxxviii Tables for Statisticians and Biometricians [XLVII XL VIII 



Hence unless x x > 3er to 4<r, '12 to '16 say, or x lie outside the range '10 to '30, 

 or even "06 to '34, we cannot anticipate very good results from formula (viii) or 

 (xii). It is for " tail " frequencies that these formulae will be found valuable. 



We need accordingly formulae which will give areas on either side of the mode 

 with reasonable approximation. One fairly good formula exists for finding such 

 areas near the mode and on either side. It is the expansion in incomplete 

 normal moment functions. If this gave satisfactory results for all cases up to a 

 range of three to four times the standard deviation, we could by aid of it and of 

 formula (viii) obtain the total area on one side (and therefore on the other) of the 

 mode. Hence by subtraction we could from the normal moment formula find 

 any area from terminal up to mode, i.e. in our notation \ (1 a x }. 



The formula in question, if we take as before 



e = l/m + m/l 

 and u = V(e + 2) n (x - x) = V(e + 2) n (x ) 



\ Iv / 



P ( l 

 "V fa\*~n 



is as follows : 



i x (I + 1, m + 1) = k [mo (u) - k 3 m s (u) - & 4 ra 4 (u) - k 5 m & (u) + k 6 m 6 (u) 



+ & 7 rn 7 (w) + k B m 8 (u) - k 9 m 9 (u) - k w m 1Q (u) + & 12 ra 12 (w)] (xiii), 



where i x (l + 1, m + 1) is the ratio to the total area of the area of the frequency 

 curve from mode to x, and 



2 /e-2 _ 3 6 - 1 _8e /e-2 



3 ~3V n ' 4 ~4 n ' 5 ~5wV n ' 



5 (6^-2 _ 3(6 2 -6-])) _ 4(6-1) /e^2 



^"Gl n n 2 [' ^ 7 ~ n V w ' 



7 47e 2 - 94e + 15 _ 64 6 - 2 /e - 2 



*k = 53 ~ n ~~ i "" ~ sv7 ~ \/ ~^ > 



32 ?i 2 27 



105(6-1) (e-2) 385 (e-2) 



12 = 



8 n* 72 



It is to be noted that when a; is < - , the quantity u becomes negative, and all 



even order normal moment functions remain of the same sign, but odd normal 

 moment functions change sign, i.e. 



wias ( u) = m 2g (u), but 7n 2s +i ( u) = m^+i (u). 



E =1 J? + 2^0 + 3^9, as provided for in Table XL VI. 

 2 



