196 LINEAR REGRESSION AND CORRELATION Ch. 7 



think in terms of general rather than precise curves as was done in 

 algebra, where all points which belonged with a certain graph fell 

 exactly on that graph. Data to which statistical analysis is applied 

 are not so well behaved as that. It will be necessary later to learn 

 how to decide which curve to choose as best describing the relation- 

 ship between X and Y suggested by a scatter diagram ; and it will not 

 be expected that all the points will fall perfectly on the line finally 

 chosen. 



The following information can be derived from a careful inspection 

 of Figures 7.11: 



From (A): Y definitely tends to increase uniformly (linearly) as 

 A' increases. On the average, Y increases about 13/6 units for each 

 unit increase in X. 



From (B) : Y decreases in proportion to the increase in X. Again 

 the relationship can be briefly described as linear. More specifically, 

 Y tends to decrease about 10 units for each unit increase in X. As 

 a result the slope of the straight line which indicates the linear trend 

 is said to be —10. 



From (C) : Y has no apparent relation to X; hence the X measure- 

 ment may as well be ignored in the statistical analysis of the meas- 

 urements, Y. 



From (D) : Y increases with X, but the increase is not uniform. In 

 fact, Y increases more rapidly for large X's than for the smaller X's. 

 This relationship between Y and X is called curvilinear. In this in- 

 stance, it follows the non-linear mathematical law: Y = 0.5e x , 

 where e is the base for natural logarithms. 



From (E) : As the measurement represented by X increases from 

 — 3 toward 0, the corresponding measurement, Y, tends to decrease 

 in a non-linear manner. Thereafter, Y increases non-uniformly. As 

 a matter of fact, the points on this scatter diagram tend to follow 

 the curve, Y = X 2 . 



From (F) : Y tends to increase non-uniformly with X, as in (D) , 

 but the curve rises more sharply here. 



From (G) : There is no apparent relationship between X and Y, 

 as in (C). 



Another point should be noted regarding the scatter diagrams of 

 Figures 7.11. If the concomitant measurement, X, were to be ignored 

 during an analysis of the data of Y corresponding to any of the sit- 

 uations except (C) and (G) , a considerable portion of the variability 

 of the Y's about their means would represent unnecessary variation 

 in this sense. We know from (B) , say, that if X = 1, the corre- 



