July 16, 2024

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.

About Caitlin

Caitlin Flinn Bennyhoff, D.Ed., NCSP, is currently employed as a Nationally Certified School Psychologist in Pennsylvania. She completed her doctoral degree in school psychology at Indiana University of Pennsylvania. Her main interests include response-to-intervention, systems-level change, measuring student growth, and data analysis teams.


  1. Will H Burrow says:

    Have you ever heard of Og Lindsley and the Standard Celeration Chart?
    Do you understand why the difference between an equal interval chart and a semi-log chart are critical to interpreting student learning data?
    Have you read the work of Edward Tufte on the visual presentation of data?

    • Ogden certainly influenced how we graph information in a way that allows us to view progress at a glance. I know the Morningside schools use Standard Celeration Charts in line with Ogden’s Precision Teaching model. There’s a great book about that model by Johnson & Street, The Morningside Model of Generative Instruction: What it Means to Leave No Child Left Behind for those who are interested in learning how to incorporate Standard Celeration Charts into instructional practice or RTI frameworks.

      As for semi-logarithmic charts vs. equal interval – you are welcome to describe this further!

      Tufte is actually quoted in my “school psychology bible” of Best Practices in School Psychology on page 2117. It says, “Above all else show the data” (1983). There is a chapter in Volume VI of the Best Practices books that is entitled Best Practices in the Display of Data by Connor Hood and Clark Dorman. Good stuff for educators to review.

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