Analyse bivariate data using scatterplots, correlation, and least-squares regression lines.

What this dot point is asking

Bivariate data is paired data: each subject gives two numbers, such as a person's height and weight. The whole topic is about whether one variable helps predict the other, and how strongly.

Describing a scatterplot

When you read a scatterplot, comment on four features: form (linear or non-linear), direction (positive or negative), strength (how tightly the points cluster about a line), and any outliers. A positive direction means $y$ tends to rise as $x$ rises.

Correlation coefficient

Pearson's correlation coefficient $r$ measures the strength and direction of a linear relationship. It always lies between $-1$ and $+1$. Values near $\pm 1$ mean a strong linear pattern; values near $0$ mean little or no linear relationship.

Coefficient of determination

The coefficient of determination is simply $r^2$. It gives the proportion of the variation in the response variable that is explained by the linear relationship with the explanatory variable. If $r = 0.8$ then $r^2 = 0.64$, so about $64\%$ of the variation in $y$ is explained by $x$ (and $36\%$ is due to other factors).

Least-squares regression line

The least-squares line is the straight line $\hat{y} = a + bx$ that minimises the sum of the squared vertical distances from the points to the line. In a TCE exam you usually read $a$ (intercept) and $b$ (slope/gradient) from technology, then interpret them.

Interpolation and extrapolation

Predicting inside the range of the data (interpolation) is reasonably safe. Predicting outside the range (extrapolation) is risky because the linear pattern may not continue. State which you are doing whenever you make a prediction.

Reporting good practice: quote $r$ to two decimals, state the form/direction/strength of the relationship, give the equation of the line, and interpret slope and intercept in the words of the context.