Using the structure of the residuals to evaluate the goodness of fit: a pedagogical case study with Newton’s law of cooling

Abstract Introductions to data analysis in university physics often focus on fitting (typically to linearised functions) and uncertainty propagation, but residuals are less commonly introduced. Structured residuals indicate that the function (theoretical description) used to fit data does not capture all the relevant elements (i.e. the theoretical description is incomplete), and can be present even when the fit line is within the error bars of all data points. Residuals are therefore resilient to strategies such as forcing numerical agreement between data and theory (fit) through increasing uncertainty estimates. In this paper, we argue that introducing residuals alongside fitting could be an effective strategy for encouraging critical scientific thinking, particularly about trusting and understanding one’s data, and a useful tool for identifying—and realising—laboratory tasks that students can develop into genuine investigations. We illustrate this with a simple experiment of cooling water where the structured residuals of the fitted temperature data clearly indicate that the theoretical description (Newton’s Law of Cooling) is insufficient. This demonstrates the overlooked potential for residuals to be introduced as one of the earliest elements of data analysis.