14 STATISTICAL METHODS. 



2 indicates that the sum of the products for all classes into 

 frequency is to be got, and n is the number of variates, 



Thus in the last example : 

 M = (3.2 X 1 + 3.7 X 1 + 4.2 X 3 + 4.7 X 3 + 5.2X 7 + 5.7 X 5 + 6.2 X 3 



+ 6.7 X 1 + 7.2 X 1) -4-25 = 5.24, 

 or 



M l = (lxl+2Xl+3x3+4x3-f5x7+6x5+7x3-f-8Xl-h9xl)-*-25 = 5.08, 

 M = 5.2* + .08 (5.7 - 5.2) = 5.24 



A still shorter method of finding If is given on page 17. 



The mode is the class with the greatest frequency. 

 In the example, the mode is 5.2. 



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 t han 

 the middle cZass, of which the total number is a; those of the middle 

 class, b; and those greater, c. Then a + b -f 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 ?i - a : b = 

 x\l" V ichen x is positive, or \n c : b = x : I" I' when x is negative. 



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

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

 median magnitude = 5.5 .18 = 5.32. (Cf. p. 11.) 



Every determination of a constant ot the frequency polygon 

 is an approximation only to the true value of the constant. 

 The closeness of the approximation to the truth is measured by 

 the so-called probable error of the determination. This is a 

 pair of values lying one above and one below the value deter- 

 mined. We can say that tfrere is an even chance that the true 

 value lies between these limits ; the chances are 4 to 1 that the 

 true value lies within twice these limits, and 19 to 1 that it lies 

 within thrice these limits. 



The probable error of the mean is given by the for- 

 mula 



0.6745 X Sta " dard deviatioa [see below] = 0-674g _^ 

 |/number of variates \/n 



It will be seen that the probable error is less, that is, that 

 the result is more accurate, the greater the number of variates 

 * 5.2 is the true class magnitude corresponding to the integer 5. 



