Two-Variable Data: Scatterplots, Lines of Best Fit & Exponential Models

When researchers measure two quantitative variables on each observational unit—such as tree diameter and age, study hours and exam score, or time elapsed and bacterial population—they display the paired values (x, y) on a scatterplot. In the Problem-Solving and Data Analysis domain of the Digital SAT, bivariate data questions test whether you can distinguish actual data points from values predicted by a model (a line or curve of best fit) and whether a relationship grows linearly or exponentially. A line or curve of best fit summarizes the overall trend of the plotted points without necessarily passing through any specific data point. On the SAT, the most common mistake is reading a plotted dot when the question asks for the value predicted by the line of best fit, or vice versa. Equally important is recognizing the signature difference between linear and exponential change: linear models add or subtract a constant amount per unit of x, whereas exponential models multiply by a constant factor (or change by a constant percentage) per unit of x.

1. Observed Values, Predicted Values & Residuals: • Observed value (y_actual): the y-coordinate of a plotted dot on the scatterplot. • Predicted value (ŷ): the y-coordinate on the line or curve of best fit at that same x-value, calculated by plugging x into the model equation. • Residual: Residual = Observed − Predicted = y_actual − ŷ. A point ABOVE the line of best fit has a positive residual (the model underestimates the actual value); a point BELOW the line of best fit has a negative residual (the model overestimates the actual value). 2. Interpreting Slope and Intercept in Context: For a linear regression line ŷ = mx + b: • The y-intercept b is the predicted value of y when x = 0. • The slope m is the predicted change in y for each 1-unit increase in x. Always include the word 'predicted' or 'estimated' in SAT verbal interpretations. 3. Linear vs Exponential Growth and Decay: • Linear model: y = mx + b. Equal increments Δx = 1 produce a CONSTANT ADDITIVE difference Δy = m (e.g., adding 150 subscribers every month). • Exponential model: y = a · b^x = a(1 + r)^x. Equal increments Δx = 1 produce a CONSTANT MULTIPLICATIVE ratio y(x+1)/y(x) = b, or a constant percentage change r (e.g., increasing by 12% every month gives b = 1.12; decreasing by 8% every year gives b = 0.92). If time steps occur every T units (such as doubling every 5 years), the exponent is x / T: y = a · 2^(x/5).