Construct scatter plots, calculate and interpret Pearson's correlation coefficient, and fit and use the least-squares regression line

What this dot point is asking

NESA wants you to take a paired dataset, draw or read a scatter plot, calculate the Pearson correlation coefficient $r$, fit the least-squares regression line of $y$ on $x$, and use it to predict values. You also need to interpret what $r$ and the line do and do not tell you. The three skills form one pipeline: the scatter plot is the picture, $r$ puts a single number on how linear and how tight that picture is, and the regression line turns the relationship into a formula you can predict with. Read the plot well and the calculator work that follows is almost automatic.

The answer

Scatter plots

A scatter plot displays paired data $(x_i, y_i)$ as points in the plane, one dot per observation, and is the first thing you draw because your eyes catch structure that a single number hides. By convention the independent (explanatory) variable goes on the $x$-axis and the dependent (response) variable on the $y$-axis. Never join the dots: a scatter plot is a cloud of separate observations, not a quantity changing over time, and joining them turns it into a line graph.

Read every scatter plot for four things:

• Direction. Positive (the cloud rises to the right), negative (it falls to the right), or none.
• Form. Linear (the points hug a straight band), curved, or no clear pattern.
• Strength. How tightly the points cluster around the pattern. Strength is about scatter, not steepness: a gentle but tight band is strong, a steep but loose one is weak.
• Outliers. Single points sitting far from the bulk.

The reason the picture comes first is that the correlation coefficient and the regression line both assume the relationship is a straight line. The scatter plot is the check that this assumption is reasonable before you let the calculator fit anything.

Pearson's correlation coefficient

The Pearson correlation coefficient $r$ measures the strength and direction of the linear relationship between two variables. It is defined by

where $s_x$ and $s_y$ are the sample standard deviations. The formula is a sum of products of $z$-scores: each point contributes a positive term when both coordinates sit on the same side of their means and a negative term when they sit on opposite sides, so the total is large and positive for a tidy upward cloud and large and negative for a tidy downward one. In practice you compute $r$ on your calculator from the data list, not by hand, but knowing what the formula is doing tells you why the value behaves as it does.

Key properties:

• $-1 \le r \le 1$.
• $r > 0$ means positive linear association, $r < 0$ means negative.
• $|r|$ close to $1$ means a strong linear relationship, $|r|$ close to $0$ means weak or no linear relationship.
• $r$ measures only linear association. A perfect parabola can have $r = 0$.
• $r$ is unitless and unchanged by linear rescaling of either variable, so converting hours to minutes or marks to percentages leaves $r$ exactly the same.

Rough verbal scale: $|r| \ge 0.9$ very strong, $0.7 \le |r| < 0.9$ strong, $0.5 \le |r| < 0.7$ moderate, $|r| < 0.5$ weak.

The least-squares regression line

The least-squares regression line of $y$ on $x$ is the line $y = a + b x$ that minimises the sum of squared vertical residuals $\sum (y_i - (a + b x_i))^2$. The residuals are the vertical gaps from each point to the line; squaring them before adding penalises big misses heavily and refuses to let the line drift, so the result is the single straight line that fits the cloud best overall. The solution is

Two facts fall straight out of these formulas and are worth committing to memory. First, the line always passes through the point of means $(\bar{x}, \bar{y})$, because substituting $x = \bar{x}$ gives $y = \bar{y}$. Second, the slope $b$ carries the sign of $r$ (since $s_x, s_y > 0$), so a positive correlation gives an upward line and a negative correlation a downward one. The slope itself is the predicted change in $y$ per one-unit increase in $x$.

Prediction, interpolation and extrapolation

Once you have $y = a + b x$, substitute any $x$ to predict $y$. The prediction is the model's best estimate; the actual value usually differs a little because the line is a best fit over the whole dataset, not a guarantee for any single case. Prediction inside the observed range of $x$ is called interpolation and is usually safe. Prediction outside the observed range is extrapolation and is risky: nothing guarantees the linear pattern continues, and the further out you go the worse it can be (a study-hours model predicting a mark above $100$, or a negative weight, has clearly broken down).

Correlation is not causation

A strong $r$ tells you the two variables move together. It does not establish that one causes the other. Lurking variables, reverse causation, and pure coincidence can all produce strong correlations: ice-cream sales and shark attacks rise together every summer, but hot weather drives both. Use cautious language ("is associated with", "tends to") unless the question hands you an actual mechanism.

Build the analysis, stage by stage

Here is the full routine on one dataset: weekly study hours ($x$) against an HSC trial mark out of $100$ ($y$) for a class of $12$ students. Watch it go from a blank grid to a fitted line and an interpreted correlation.

Stage 1, set up and scale the axes, then plot the first points. Put the independent variable (study hours) on the horizontal axis and the response (mark) on the vertical axis. Choose scales that spread the data across the whole plot: hours run $0$ to about $24$, marks from the mid $40$s to the mid $90$s, so the mark axis need not start at zero. Then plot the first few pairs, here $(2, 48)$, $(4, 56)$ and $(5, 50)$, as single dots.

Stage 2, plot every data pair. Continue until all $12$ points are on the plot, each pair one dot, left unconnected. The shape now emerges: a cloud running from the lower left to the upper right, which says the direction is positive and the form looks linear.

Stage 3, fit the least-squares line. Enter the pairs into the calculator and read off the slope and intercept; here $b \approx 2.4$ and $a \approx 43$, so the line is $y = 43 + 2.4 x$. It passes through the point of means and is drawn straight through the middle of the cloud. The short dashed stubs are the residuals, the vertical gaps from each point to the line, and least squares is precisely the line that makes the total of their squares as small as possible.

Stage 4, read the correlation from the scatter. The same stubs tell you the strength. They are all short, so the points sit close to the line, which means $|r|$ is close to $1$; the cloud rises, so $r$ is positive. The calculator confirms $r \approx 0.95$, a strong positive linear relationship. Strength and direction together are exactly what $r$ packages into one number.

How exam questions ask about bivariate data

The wording varies but the task is almost always one of these, and the verb tells you the method:

• "Describe the relationship shown in the scatter plot." Give direction, form and strength, each with a one-line justification from the plot.
• "Calculate the correlation coefficient" or "comment on the value of $r$." Read $r$ off the calculator, state its sign and magnitude, and translate it into words ("strong negative linear association").
• "Find the equation of the least-squares regression line." Use $b = r \cdot s_y / s_x$ and $a = \bar{y} - b\bar{x}$ if given the summary statistics, or read $a$ and $b$ from the calculator if given the data, then write $y = a + b x$ explicitly.
• "Use the line to predict $y$ when $x = \dots$" Substitute and state the result with units; note whether it is interpolation (safe) or extrapolation (flag it).
• "Interpret the slope / intercept." The slope is the average change in $y$ per one-unit increase in $x$ with units; the intercept is the predicted $y$ at $x = 0$, with an extrapolation caveat if $x = 0$ sits outside the data.
• "Does this show that $x$ causes $y$?" No: correlation is association, not cause. Mention a possible lurking variable and answer in cautious language.

Edge cases worth knowing

• A low $r$ with an obvious pattern. A clear curve (a U-shape or a hump) can give $r \approx 0$, because $r$ only measures the linear part. The scatter plot, not $r$, is what tells you the relationship is real but non-linear, and a straight-line model would mislead.
• A subgroup effect. Two distinct clouds plotted together (weekday and weekend data, say) can produce a misleading single $r$ and a meaningless single line. Describe or model the groups separately.
• A prediction that is impossible. A model predicting a mark above $100$ or a negative quantity has been extrapolated past where it makes sense; report the value but state that the model has broken down there.
• Reversing the variables. The least-squares line of $y$ on $x$ is not the line of $x$ on $y$. Fit with the response as $y$, matching what the question asks to predict.