Logical Line Fitting: One Step in the EDA Process by Shannon Guerrero Northern Arizona University NCTM 2008 Annual Meeting & Exposition Salt Lake City, UT April 2008

EDA (Exploratory Data Analysis)  Mostly graphical approach to data analysis  Emphasizes uncovering underlying structure of data, extract important variables, detect outliers/anomolies, test underlying assumptions, maximize insight into data set  Graph the data, graph the data, graph the data  Focus on sense-making rather than theory

Why curve fitting?  Applications in data analysis & algebra  “Analyses of the relationships between two sets of measurement data are central in high school mathematics” (p. 328 NCTM PSSM)  modeling, prediction, symbolic representation, correlation, regression, residuals

“Line of Best Fit”  Explains relationship between two variables with a straight line that “best fits” the data  Line may pass through some, none, or all of the points  Used to predict future values from existing values (interpolate vs extrapolate)

Outliers  An observation that lies outside the overall pattern of a distribution  For one variable, a convenient def’n is a point that falls more than 1.5 times the IQR above the 3 rd quartile or below the 1 st quartile  Examine outliers carefully and understand their appearance in your data set  Need to decide what to do with outliers – include or discard?

Curve Fitting vs. Regression  Power of curve fitting often lost as we revert right to regression calculations  Curve fitting is more general and an approximation  Equation found (using either method) can help uncover underlying structure of data, predict future values from past ones, model causal relationships, and maximize insight into a data set

Linear Regression  Statistical approach to finding relationship between two variables  Least squares regression attempts to minimize the squared residuals (residual – difference between observed value and value given by model)  Assumption: for a fixed value of x the value of y is normally distributed with equal variations across x

r 2 and residuals  residual – difference between an observed value and value predicted by regression line  residual plot is a scatterplot of regression residuals against the explanatory variable  helps us assess fit of regression line  r 2 is another way to assess how well the line fits the data (the closer to 1 the better the fit)