Determining the way one variable changes when another changes: (1) by the use of averages
The problem stated in Chapter 3 was to determine how many feet automobiles traveling at a given speed require for stopping. It involves determining the average extent to which one variable changes when another variable changes. Stated mathematically, the problem is to find the functioipal relation between speed and distance--the probable distance required to stop with any given initial speed. Of the many different ways of doing this, the simplest, and the one which would suggest itself most naturally, would be to classify the record into groups, placing all of one speed in one group, all of another speed in another group, making as many groups as there are different rates of speed recorded, and then averaging the different distances for all the cases in each group. This would then give an average distance for stopping for each given rate of speed in the series of records. Table 4.1 shows this operation carried out.
Where there were only single observations, this fact has been indicated by. placing the average--the single report--in parentheses.
The averages in the last column of Table 4.1 show quite specifically how the distance required for stopping tends to increase with speed. But the increase is not uniform. The cars at 13 miles per hour averaged a greater distance than those at either 12 or 14, and the cars at 26 a shorter distance than those at 25.
If the successive averages from Table 4.1 are plotted and connected by lines, both the general increasing tendency and the irregular change from group to group are easily seen. Figure 4.1 shows this comparison.
Do these differences between the different group averages have any real significance? Is there any reason to think that this very jagged line is the true average relation between speed and distance? We can consider