452 RESEARCH IN PROTOZOOLOGY 



trophozoites of breadth 8 microns is 13.07 microns. Similarly, for 

 any other breadth class. 



In the same way the substitution of a specific value of x in the 

 equation 



y = 2.4014 + .3919.^ 

 will give us the mean breadth of trophozoites of any specific 

 length class. 



The term Oj^.y represents the variability in length of trophozoites 

 of given breadth and the term Oy^x represents the variability in 

 breadth of trophozoites of given length. 



The correlation coefficient, r, is an index number of the degree 

 of association between the two measurements in question, and may 

 take values only between — i and -{-i. When r is negative, it means 

 that one variable is increasing as the other decreases, while when r 

 is positive, it means that the two variables increase together. In 

 the present example, the positive value of r indicates that the 

 longer trophozoites are also the broader trophozoites. A value of 

 r =: o indicates no association between two variables, and in this 

 case, Ox.y is equal to Ox, and Oy.x is equal to Oy, so that the variability 

 within the arrays is the same as the variability in the marginal 

 columns. When r is either -\-i or — i, the correlation is perfect, 

 which means that any specific value of one variable is always asso- 

 ciated with a fixed value of the other variable. The frequencies in 

 the correlation table always fall in a straight line in this case. 



PROBABLE ERROR CONCEPT 



In the two previous sections we have been discussing certain 

 statistical constants that have been found useful in evaluating 

 quantitative material. One of the points that this discussion has 

 emphasized is the variability of the individual measurements them- 

 selves, and we have learned how to evaluate this variability and 

 make use of it to express the proportions of individuals we would 

 expect to find within given class intervals. An acknowledgment 

 of the fact that individuals are subject to variation leads to the 

 additional thought that this variability of the individuals will 

 have an effect on their arithmetic mean or on any statistical con- 

 stant that may be derived from their individual measurements. A 

 discussion of this effect is embraced under the heading of probable 

 error concept, and it may be summarized as follows : 



