700 APPENDIX 



As has been pointed out in the text, the standard deviation gives a good 

 idea of the spread of the distribution. From the accompanying footnote * 

 we are now in a position to appreciate its mathematical significance. It 



- 

 is the <r in the equation^ = j= e 2 r of the normal probability curve, 



and bears a similar relation to the probability curve that the radius of a 

 circle bears to the circle. If <r is small the probability curve is crowded 

 together so as to resemble curve A in Fig. 7, while if <r is large it is spread 

 out so as to resemble curve B in Fig. 7. 



Hence the standard deviation along with the mean completely describes 



VV^2 

 , which is thus a 

 n 



perfect measure of variability for a normal distribution, is a good measure 

 of variability when the distribution is not normal, but it is not completely 

 descriptive of the population. 



Another measure of variability is sometimes used which consists simply 

 in taking the arithmetical average of the n deviations, these deviations 

 being given the positive sign. 



1 The probability that a system of n variates fall in intervals x^ to #, + A#, x 2 to x a + Ax, 

 . . . x n to x n + A# is given by 



P= * g'Zt* ' /^... l e ~^. 



<rV27T <rV27T 0-V27T 



This may be written more briefly as 



_2f2 



P= -e 2<r2 (A*). 



<r (2 TT)? 



For a given set of deviations which occur, <r should be selected so as to make the 

 probability P a maximum. , 



Equating the first derivative to zero, we obtain 



CLl I --2 I . I M _o 



da ff^ " (r + i 



Now, by means of integral calculus, tables are formed of the area included by the curve 



the #-axis, and any two ordinates at equal distances a from the mean. Such a table with 

 the argument - is found in Davenport's Statistical Methods, second edition, pp. 119-125. 



This table shows that x 



- = 0.6745 (0 



when just one half of the area under the curve is included as described above. By definition, 

 the particular value of x given by (i) is called the probable error in a single variate, and we 

 shall represent it by E s . 



Hence, E, = .6745 <r - 



2 See Galton, Natural Inheritance, p. 62. 



