Statistics and Probability: Scatter Plots, Lines of Best Fit, and Two-Way Tables
The statistics strand centers on bivariate data: reading scatter plots, describing association, interpreting a line of best fit, and analyzing two-way tables using joint, marginal, and conditional relative frequencies. Questions usually ask you to interpret a slope or intercept in context, or to choose the correct relative frequency.
Scatter plots
Describe an association by direction (positive, negative, or none), form (linear or nonlinear), and strength (tightly clustered or widely scattered). Clusters are groupings of points; outliers lie far from the overall pattern and can pull a line of best fit toward them.
Line of best fit
A line of best fit models a linear association. In y = mx + b for a fitted line, the slope m estimates the change in predicted y for each one-unit increase in x, and the intercept b estimates the predicted y when x = 0. Use the line for interpolation inside the observed data range; predictions far outside that range are extrapolation and unreliable.
Two-way tables
For a table of two categorical variables:
- Joint relative frequency = cell count ÷ grand total.
- Marginal relative frequency = row or column total ÷ grand total.
- Conditional relative frequency = cell count ÷ row total or column total.
Two variables show association when conditional relative frequencies differ across categories; if the conditionals are roughly equal, there is little or no association.
Worked example
A line of best fit for study hours x and test score y is y = 1.8x + 12. The slope 1.8 means each additional hour of study is associated with a predicted increase of 1.8 points. The intercept 12 is the predicted score for 0 hours of study. For 5 hours, the prediction is 1.8(5) + 12 = 21 points. Since 5 hours lies within the observed data range, this is interpolation.
Traps
- Confusing a strong association with a steep slope; strength and steepness are different properties.
- Reporting the intercept as the slope, or the slope as the value of y when x = 0.
- Dividing by the wrong total in a two-way table: check whether the question says 'of all students' (joint or marginal) or 'of the students who…' (conditional).
- Claiming that association proves causation.
- Extending a line of best fit well beyond the plotted data and treating the prediction as reliable.