STATISTICAL METHODS IN PROTOZOOLOGY 449 



the use of the standard deviations of the frequency distributions 

 in the margins, and this would be a proper procedure except for 

 the fact that it is very common for such tables to show a trend. 

 For example, in the table at hand, it can be seen on inspection 

 that the shorter trophozoites tend to be narrower and the longer 

 trophozoites tend to be broader. This tendency of one variable 

 to change its value with a change in the value of the other vari- 

 able introduces a concept not present in the treatment of a single 

 variable, which is embraced under the general head of correla- 

 tion. 



Let us examine some of the ideas that enter into a proper con- 

 cept of correlation. In the first place, in such a table, we should 

 like to describe the trend of one variable with the other, and 

 while this may be considered from many points of view, it is 

 perhaps most clear when we think of the changes in the mean 

 value of one variable that occur with changes in the other variable. 

 For example, the average breadth of trophozoites eleven microns 

 in length is clearly different from the average breadth of troph- 

 ozoites fourteen microns in length. We might, therefore, very 

 naturally ask the question as to how the mean breadth of the 

 trophozoite increases as we step from one length class to another. 

 Conversely, if we examine the table as to the average length of the 

 trophozoites of the same breadth, we see that the broad trophozoites 

 have a greater average length than the narrow ones, and we may 

 therefore ask the question as to the increase in average length of 

 the trophozoites that occurs as we pass from one breadth class 

 to the next. 



It is one of the functions of any study of correlation to answer 

 these two questions, and in the case of normal correlation, the 

 answer is given by determining two Hues, one giving the movement 

 of mean breadth with length and the other giving the movement 

 of mean length with breadth. These two lines are called regression 

 lines and their determination will be illustrated in the example 

 which follows. 



Another idea involved in the general subject of correlation arises 

 from the fact that it is obvious on inspection that trophozoites of 

 a specific length class, say twelve, are less variable in breadth than 

 the total group of trophozoites. Similarly, trophozoites of a specific 

 breadth class are clearly less variable as to length than trophozoites 

 in general. Thus, in studying the correlation between two variables, 



