A Clarification of Slope and Scale.
Use L’s new formulas to get the true slope on any behavior graph, making visual checks and cross-study matches more reliable.
01Research in Context
What this study did
Hamama (2022) wrote new math formulas. The formulas fix slope on any single-case graph.
Old slope counts only work when both axes use the same scale. These new equations work even when the x and y scales differ.
What they found
The paper gives step-by-step formulas. Plug in your axis units and data points. You get the true geometric slope every time.
Accurate slope lets you compare graphs across studies and spot real change faster.
How this fits with other research
Dowdy et al. (2024) show that stretching or squeezing the x-to-y ratio can fool the eye. L’s equations remove that visual bias by giving the real slope number.
Peltier et al. (2024) tried shrinking the y-axis and saw mixed effects on how big the change looked. Using L’s formulas after their tweaks would show if the true slope actually changed.
Kranak et al. (2022) reviewed ways to teach graph skills. They list L’s formulas as a new tool trainers should add to BCBA coursework.
Why it matters
Next time you eyeball a graph, run the quick slope math first. If the number and your visual guess clash, re-inspect. This small step guards against axis tricks and gives your team a shared ruler for deciding what works.
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02At a glance
03Original abstract
Improvements in the quantification and visual analysis of data, plotted across non-standardized graphs, are possible with the equations introduced in this paper. Equation 1 (an expression of graphic scale variability) forms part of the foundation for Equation 2 (an expansion on the traditional calculation of the tangent inverse of a line's algebraic slope). These equations provide clarification regarding aspects of "slope" and graphic scaling that have previously confused mathematicians. The apparent lack of correspondence between geometric slope (the angle of inclination) and algebraic slope (the m in y = mx + b) on "non-homogeneous" graphs (graphs where the scale values/distances on the y-axis are not the same as on the x-axis) is identified and directly resolved. This is important because nearly all behavior analytic graphs are "non-homogeneous" and problems with consistent visual inspection of such graphs have yet to be fully resolved. This paper shows how the precise geometric slope for any trend line on any non-homogeneous graph can quickly be determined-potentially improving the quantification and visual analysis of treatment effects in terms of the amount/magnitude of change in slope/variability. The equations herein may also be used to mathematically control for variability inherent in a graph's idiosyncratic construction, and thus facilitate valid comparison of data plotted on various non-standard graphs constructed with very different axes scales-both within and across single case design research studies. The implications for future research and the potential for improving effect size measures and meta-analyses in single-subject research are discussed.
Behavior modification, 2022 · doi:10.1177/0145445520953366