Sec. 7.2 DETERMINING LINEAR TREND LINES 199 



a procedure, however, obviously is somewhat subjective because it 

 depends quite a bit upon personal opinion. One of the chief purposes 

 of numerical measurement and statistical analysis of such measure- 

 ments is to free decisions based on relatively precise numbers from 

 distortions which might result from the exercise of personal tastes 

 and opinions. It is for this reason that it is desirable to be able to 

 describe such a line by a method which will produce the same result 

 no matter who uses it. 



The types of relationships between two kinds of numerical meas- 

 urements which were discussed in the preceding section are illustra- 

 tive of sampling experiences involving errors of observation and 

 measurement. The dots of the scatter diagram usually fail to fall 

 exactly on any simple curve for one of two reasons: (a) Sampling 

 errors or chance variations cause the values of Y, say, to be partially 

 inaccurate, (b) There are real variations from the general trend of 

 Y and X which, however, are of minor importance compared to the 

 general trend and should be smoothed out in order that the general 

 trend may be studied more effectively. The data of Table 7.21 and 

 the corresponding scatter diagram of Figure 7.21 help to illustrate 

 these points. The data in the table are considered to be a population 

 of pairs of observations, that is, a bivariate population. For con- 

 venience these data have been grouped by 16-week weights (X) to 

 the nearest pound in the scatter diagram of Figure 7.21. 



The bivariate population of Table 7.21 possesses several character- 

 istics which are of statistical interest and importance. These fea- 

 tures are exhibited by Figure 7.21, from which it is learned: (a) 

 There is a general upward trend of the 28-week weight, with increas- 

 ing 16-week weight of the same bird. (6) Within each 16-week- 

 weight class there is a frequency distribution of 28-week weights, 

 and this distribution is relatively symmetrical about the mean 28- 

 week weight for the class, (c) The means of the six 16-week-weight 

 classes lie perfectly on a straight line with a slope of 1/2. Thus the 

 true linear regression line passes through the points representing 

 the true average Y's for the given X's. The slope of this true trend 

 line is denoted by the Greek letter ft (beta). 



In a study based on samples the ft is unknown, as is the exact loca- 

 tion of the true linear trend line, and only the n pairs of sample 

 measurements are available as a basis for making decisions about the 

 linear trend line. For example, a random sample of 30 pairs (X, Y) 

 was drawn from the bivariate population of Table 7.21, with the 



