Code of Federal Regulations Title 49. Transportation 49 CFR § 575.104, App. C Appendix C—Method of Least Squares

The method of least squares is a method of calculation by which it is possible to obtain a reliable estimate of a true physical relationship from a set of data which involve random error. The method may be used to establish a regression line that minimizes the sum of the squares of the deviations of the measured data points from the line. The regression line is consequently described as the line of “best fit” to the data points. It is described in terms of its slope and its “y” intercept.

The graph in Figure 5 depicts a regression line calculated using the least squares method from data collected from a hypothetical treadwear test of 6,400 miles, with tread depth measurements made at every 500 miles.

In this graph, x_j, y_{j(j = 0, 1, . . . 8)} are the individual data points representing the tread depth measurements (the overall average for the tire with 6 measurements in each tire groove) at the beginning of the test (after break-in) and at the end of each 800-mile segment of the test.

The absolute value of the slope of the regression line is an expression of the mils of tread worn per 1,000 miles, and is calculated by the following formula:

The “y” intercept of the regression line (a) in mils is calculated by the following formula: