14 STATISTICAL METHODS. 



rule for roughly determining the theoretical mode. The 

 mode lies on the opposite side of the median from the mean ; 

 and the abscissal distance from the median to the mode is 

 double the distance from the median to the mean; or, 

 mode=mean 3 X (mean median). More precise directions 

 for finding the mode in the different types of frequency poly- 

 gons are given in the discussion of the types. 



The median magnitude is one above which and below 

 which 50% of the variates occur. It is such a point on the 

 axis of X of the frequency polygon that an ordinate drawn 

 from it bisects the polygon of rectangles or the continuous 

 curve, but not the polygon of loaded ordinates. 



To find its position: Divide the variates into three lots: those less than 

 the middle class, i.e., the one that contains the median magnitude, of 

 which the total number is a; those of the middle class, b; and those 

 greater, c. Then a + b + c = n = the total number of variates: Let I' = 

 the lower limiting value of the middle class, and I" =the upper limiting 

 value, and let x = the abscissal distance of the median ordinate above the 

 lower limit or below the upper limit of the median class according as x 

 is positive or negative. Then \n a : b = x : I" V when x is positive, 

 or %n c : b = x : I" V when x is negative. 



Thus in the last example: (12.5-8): 7 =x : 0.5; #=.32;. the median 

 magnitude = 5.0 + .32 = 5.32. Or (12.5-10): 7 = -x : 0.5; z=-.18; 

 the median magnitude =5. 5 -.18 = 5. 32. (Cf. p. 10.) 



The probable error (E) of the determination of 



any value gives the measure of unreliability of the determina- 

 tion; and it should always be found. For, any determination 

 of a constant of a frequency polygon is only an approximation 

 to the truth. The probable error (E) is a pair of values lying 

 one above and the other below the value determined. We 

 can say that there is an even chance that the true value lies 

 between these limits. The chances that the true value lies 

 within :* 



2are 4,5:1 5E are 1,310:1 



3Eare21 :1 6E are 19,200:1 



4E are 142 :1 7E are 420,000:1 



8are 17,000,000:1 



9E are about a billion to 1. 



The probable error should be found to two significant 

 * These values are easily deduced from Table IV. 



