Revise Correlation and regression: Pearson and Spearman for IB Mathematics: Applications and Interpretation (4.3)

A scatter diagram plots one variable against the other, one point per observation. Before any calculation, look at the cloud of points and describe it in two words. The DIRECTION is positive if the points drift upwards from left to right (as $x$ rises, $y$ rises) and negative if they drift downwards. The STRENGTH is how tightly the points hug an imagined trend line: points strung almost perfectly along a line show a strong correlation; a loose, fuzzy cloud shows a weak one; a shapeless scatter shows none. Direction and strength are independent — a tight downward band is a strong negative correlation, not a weak one.

The scatter also tells you WHICH coefficient to reach for. If the points hug a straight line, Pearson's $r$ is the natural measure. If they follow a curve that keeps climbing (or keeps falling) — a monotonic pattern — or if one stray point sits far from the rest, then Spearman's rank coefficient $r_s$ describes the association more faithfully.

Pearson's coefficient $r$ measures the strength and direction of a LINEAR relationship between two quantitative variables. It always satisfies $-1 \le r \le 1$: a value of $+1$ means the points lie exactly on a rising straight line, $-1$ means exactly on a falling straight line, and $0$ means no linear association at all. Read $r$ straight from the GDC's linear-regression output; you are never asked to compute it by hand in AI SL. Interpreting it takes two words — the sign gives the direction, the size gives the strength — and both must be tied to the context of the question.

Not every relationship is a straight line, and not every data set is free of outliers. Suppose two variables rise together steadily but along a curve, or suppose one extreme point would drag Pearson's $r$ away from the real trend. For these cases we use Spearman's rank correlation coefficient $r_s$. The idea is simple: replace each value by its RANK within its own variable — smallest is rank 1, next is rank 2, and so on — then find Pearson's $r$ on the two lists of ranks. Because ranks record only the ORDER of the values, not their exact size, a monotonic curve ranks as neatly as a straight line, and a wild outlier becomes just 'the largest', so $r_s$ resists it. Like $r$, it satisfies $-1 \le r_s \le 1$, and $r_s = \pm 1$ signals a perfect monotonic (order-preserving or order-reversing) relationship.

Once a scatter looks reasonably linear, we fit the least-squares regression line of $y$ on $x$, written $y = ax + b$. 'Least squares' means the line is positioned to make the total of the squared VERTICAL gaps between the points and the line as small as possible; the GDC reports the gradient $a$ and intercept $b$. This line of $y$ on $x$ is built to predict $y$ from a value of $x$. Two facts steer its safe use: the line always passes through the mean point $(\bar{x}, \bar{y})$ — a quick sanity check — and a prediction is only trustworthy INSIDE the range of the data you collected.

A strong correlation is easy to over-read. It tells you two variables move together — nothing more. It does NOT establish that changing one would change the other, because a hidden confounding variable may be driving both. The number of firefighters at a blaze correlates strongly with the damage done, but sending fewer firefighters would not reduce the damage; the size of the fire is the confounder that raises both. In an exam, describe a correlation by its strength and direction only, and reserve any claim of causation for questions that supply evidence of it, such as a controlled experiment.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and an accuracy mark depends on the method mark it follows. Follow-through (FT) means an earlier wrong value need not cost you the marks that depend on it, provided the later step is done correctly on your figure. The engine also ignores subsequent working (ISW) once a correct answer appears, and accepts any equivalent form and any correctly-rounded value. That protection exists only if the method is on the page. Study how each mark below is earned by a specific line.

Correlation and regression are how the statistics course turns description into prediction. The regression line you fit here is the simplest statistical model, and the discipline it teaches — inspect the scatter, choose the coefficient that suits the data, quote it with strength and direction, predict only where the data supports it, and never confuse association with cause — is exactly what later modelling, from exponential and quadratic fits to the reasoning behind the chi-squared test of independence, depends on. Master the two questions 'how tightly do they move together?' and 'how far can I trust a prediction?', and the rest of applied statistics becomes a set of variations on skills you already own.