February 9, 2023

## Defining Rate of Improvement

If you were to open your old graduate level stats book, you might not find the phrase “rate of improvement,” in the index, but you would find some text on “slope.” Essentially, they are synonymous terms, one being slightly more angled toward the positive.

In algebraic terms, rate of improvement can be defined as the vertical change (y-axis) over the horizontal change (x-axis). More simply put, slope is the rise over run. Or the steepness of a line. The key word here is line. In order to calculate slope, one must first have a line. Once a line is determined, the formula for calculating slope is:

m = (y2 – y1) / (x2 – x1) m = slope
(x1, y1) = one point on the line
(x2, y2) = a second point on the line

Typically, when we plot student data, we end up looking at data points on a graph. Some commercially available systems provide a general line as a guide approximating where the student’s data points should fall. With just this information, there is no line from which we could calculate an accurate slope. Therefore, we have to create that line! It is the position of the authors of this site that linear regression is the best method for calculating an accurate line to determine rate of improvement.