APPENDIX 



691 



It may be well to explain the chief source of this irregularity. This 

 can be seen by observing two classes, such as the 7-inch class and the 

 6. 75-inch class. As the measurements were recorded to the nearest tenth 

 inch, the 7-inch class includes the measurements recorded as 6.9, 7, and 

 7.1, while the 6.75-inch class includes only those recorded as 6.7 and 6.8. 

 This should evidently produce a biased result. Instead of making a new 

 frequency table with a different grouping, we may substitute for each 

 frequency a number derived by smoothing. This smoothing can be accom- 

 plished by substituting for each frequency, except the two extreme ones, 

 the mean of the given frequency and the one immediately before and 

 the one immediately after it. Thus, for frequency of ears of length 4.75 



2 + 4 + 13 



inches we should substitute = 6. But as this is only an approxi- 

 mation, we may as well take the nearest integral value, or 6. For an extreme 

 frequency, we substitute the mean (to nearest integer) of the extreme fre- 

 quency taken twice and the adjacent frequency taken once. Thus, for the 



frequency corresponding to length 4.5 inches we substitute = 2, 



or, in integral numbers, 3. It is sometimes desirable to apply this process 

 more than once to a given distribution in order to give it the desired 

 regularity. 



The results of the scheme for the given frequency distribution are as 

 follows : 



In general algebraic terms, if v v v v , v n are the marks of classes and 

 # 15 # 2 , , a n the corresponding frequencies, in smoothing the tf's we should 

 substitute for them the following values respectively : 



2 tf 



It can be easily seen from these algebraic expressions that the arith- 

 metic mean of the measurements is scarcely affected at all by smoothing, 

 but that the mode is sometimes considerably changed. In general, the 

 "standard deviation" (to be discussed in Section VII) is but slightly 

 affected by smoothing. 



