THE CLASSES OF FREQUENCY POLYGONS. 19 



This formula gives the value of any ordiuate y (or auy 

 class) at any distance x (measured aiong the base, X, X', of 

 Fig, '23) from the mode, e is a constant number, 2.71828, the 

 base of the Naperian system of logarithms, a is the total area 

 of the curve or number of variutes, and cr is the Standard 

 Deviation, which is constant for uuy curve and measures the 

 variability of the curve, or the steepness of its slope. 



To compare any observed curve with the theo- 

 retical normal curve we can make use of tables. For 

 the case of a polygon of integral variates the theoretical fre- 



/> 



quency of any class at a deviation - - from the mean can be 



taken directly from Table III. Here x is the actual deviation 

 from the mean expressed in the unit of the maximum, and cr 

 is the standard deviation. 



For the case of a polygon of graduated variates built up of 

 rectangles representing the relative frequency of the variates, 

 Table IV gives the relation of the actual to the theoretical 



2? 



number of individuals occurring between the values -| and 



x x 



. By looking up the given values of - the correspond- 

 ing theoretical percentage of variates between the limits 



/ji sp SH 



-\ and will be found directly. The ratio maybe 



a a or 



called the Index of Abmodality. 



The normal curve may preferably be employed even when 

 /?i is not exactly equal to 0, nor /? 2 exactly equal to 3, nor F 

 exactly equal to 0. Use the normal curve when 



Q v t <),, 4 



^X 7< 2 3 < 1 and - -^- = 1 .2 



?'4 



To determine the closeness of fit of a theoreti- 

 cal polygon to the observed polygon. There are 

 two methods according as the variates are (A) integral or (B) 

 graduated. 



(A.) Find for each class the percentage which the difference 

 between the theoretical value y and the observed frequency 

 /is of the frequency, and find the average of these percent- 

 ages, which is the index of closeness of fit sought. 



