Scatterplots and lines of best fit — Digital SAT Math practice
Trends, predictions and interpreting slope.
Key terms
- Line of Best Fit: A straight line that best represents the trend in a scatterplot, used for making predictions within the data range.
- Slope: The rate of change indicating how much y is predicted to increase or decrease per one unit increase in x.
- Residual: The vertical difference between an actual data point and its predicted value on the line of best fit.
- Positive Association: A pattern where values of y tend to increase as values of x increase across the dataset.
- Correlation vs Causation: A statistical link between variables does not prove that one variable directly causes changes in the other.
- Linear Trend: A pattern where differences between consecutive y values remain roughly constant as x increases by equal steps.
Common traps
- Extrapolating beyond the data range often produces unreliable predictions because the trend may change outside the observed interval.
- Assuming correlation proves causation ignores other variables that might influence both factors in the relationship.
- Confusing slope with the y-intercept leads to incorrect baseline values, since the intercept only shows the predicted y when x equals zero.
Key ideas to know
Predicting with Best-Fit Lines
A line of best fit for plant height (y cm) vs sunlight hours (x hrs) is y = 2.5x + 10. Predict the height after 8 hours of sun. Is this prediction reliable?
Substitute x=8 into the equation: y = 2.5(8) + 10 = 30 cm. Yes, because 8 is within the observed data range, making the prediction reliable.
Watch out: Extrapolating far outside the original data range often produces unreliable predictions because the trend may change.
Interpreting Slope on Scatterplots
What does the slope represent in a line of best fit modeling cost (y) versus weight (x)?
The predicted change in y for every 1 unit increase in x. It shows how much the dependent variable changes per single unit of the independent variable.
Watch out: Confusing slope with the y-intercept or assuming it represents total change rather than change per single unit of x.
Residual Sign Error
A data point sits above the line of best fit. The student writes: "Residual = predicted value minus actual value, so this residual is negative." What is wrong?
The formula is backwards. Residual equals actual value minus predicted value. Since the point is above the line, actual > predicted, making the residual positive.
Watch out: Memorizing "actual minus predicted" incorrectly as "predicted minus actual," which flips the sign of every residual.
Linear vs Exponential Choice
You are given a scatterplot and must choose between a linear or exponential best-fit line. What pattern in the data should you check to decide?
Check whether consecutive y-values have roughly constant differences (linear) or roughly constant ratios (exponential). Use that pattern to pick the model.
Watch out: Assuming all curved trends are exponential or all straight-looking trends are linear without verifying differences or ratios numerically.