216 LINEAR REGRESSION AND CORRELATION Ch. 7 



ignored, the downward trend in percentage farm population is quite closely 

 represented by a straight line. Make a scatter diagram of the above data, 

 omit 1943, 1944, and 1945 from further consideration, fit a linear trend line by 

 the method of least squares, and then read from the line the approximate per- 

 centages for the years omitted. Would the discussions of estimates of the 

 standard deviation and the formulas given in this book for them be appropriate 

 here? Give reasons for your answer. 



10. Referring to problem 9, could you use the equation obtained there to 

 predict satisfactorily the percentage farm population for 1953? For 1960? 

 Justify your answers. 



7.4 COEFFICIENTS OF LINEAR CORRELATION 



It is not always desirable — or even appropriate — to obtain an equa- 

 tion for the linear relation between the two types of measurements 

 being studied, as was done earlier in this chapter. It may be better 

 to describe the relationship as linear, and to give a standard, unitless, 

 measure of its strength, or closeness. This is the purpose of a co- 

 efficient of linear correlation. 



Although correlation coefficients are widely used, and often with- 

 out attention to the satisfaction of necessary assumptions, it should 

 be kept in mind that, strictly speaking, both X and Y must be random 

 variables which follow normal frequency distributions. This will be 

 assumed to be true in the following discussion of this section. 



It has been seen that the variance of the observed F's about the 

 least-squares regression line depends on the size of S(F — Y) 2 , in 

 which Y = y + bx. Hence the magnitude of this variance depends on 

 2(y _ g _ bx) 2 = S(y - bx) 2 = 2(?/ 2 - Zbxy + b 2 x 2 ). But the last 

 summation can be computed in three parts as follows: 



S(2/ 2 - 2bxy + b 2 x 2 ) = S(t/ 2 ) - 2bX(xy) + b 2 2x 2 , 



= 2G/ 2 ) - 2 



<{xy) 



V/^2 



OT-I 



[2(^)1 



+ 



-i2 



2(* 2 )J 



v/%2 



2(3*), 



hence, 



_ 2[£Qn/)] 2 [X(xy)] 2 

 " {y) 2(.r 2 ) 2(x 2 ) 



V(xy)] 2 



(7.41) 2(F - Y ) 2 = S(, 2 ) 



Zj\X ) 



