Revise Bivariate data: correlation and linear regression for IB Mathematics: Analysis and Approaches (4.2)

The first step is always a scatter diagram: plot each data pair as a point $(x, y)$. The independent (explanatory) variable goes on the horizontal axis and the dependent (response) variable on the vertical axis. From the cloud of points you read off two things at once. The direction is positive if the points trend upwards (as $x$ increases, $y$ increases), negative if they trend downwards, and there is no correlation if they scatter with no slope. The strength is how tightly the points hug a straight line — the closer they are to lying on one line, the stronger the correlation. A scatter diagram also exposes outliers and shows whether the pattern is linear at all; a strong U-shaped curve, for instance, is a real relationship that a straight-line measure will miss.

A scatter diagram is a judgement by eye; Pearson's product-moment correlation coefficient $r$ turns it into a single number. Your GDC computes it from the two data lists. It always lies in the range $-1 \le r \le 1$. The sign gives the direction — positive $r$ for a positive correlation, negative $r$ for a negative one — and the size of $|r|$ gives the strength: the closer $|r|$ is to $1$, the more nearly the points lie on a straight line, while $r$ near $0$ means little or no LINEAR relationship. Two cautions matter for full marks. First, $r$ only measures LINEAR association: a data set curving strongly can have $r$ close to $0$ yet be tightly related. Second, a large $|r|$ shows association, not cause — correlation is never, by itself, evidence of causation.

If the linear correlation is strong enough to justify it, we model the relationship with the least-squares regression line — the line for which the total of the squared VERTICAL distances from the data points to the line is as small as possible. The IB formula booklet writes it as:

y = ax + b

where $a$ is the gradient and $b$ is the $y$-intercept, both produced by your GDC. This is the regression line of $y$ ON $x$: it is built to predict $y$ from a given $x$, so substitute an $x$-value to obtain a predicted $y$. Two facts are worth carrying into every question. First, the sign of the gradient $a$ always matches the sign of $r$. Second — a guaranteed property — the line always passes through the mean point $(\bar x, \bar y)$, so if a question hands you both means, you know one exact point on the line, and you can use it as a check on the GDC output.

The whole value of the regression line is prediction, but not every prediction deserves the same trust. Interpolation means substituting an $x$-value that lies INSIDE the range of the original data; the line was established across exactly that range, so the estimate is generally reliable. Extrapolation means substituting an $x$-value OUTSIDE the data range; the line simply continues its straight trend into territory where no data was ever collected, and there is no guarantee the real relationship stays linear — or stays at all. This is why extrapolation is treated as unreliable and is a favourite examiner target. A model that fits perfectly across observed temperatures may predict, if pushed far enough, a negative number of sales, which is nonsense. Whenever you predict, look at the data range first and name which side of it your $x$-value falls on.

On Paper 2 the GDC produces $r$, $a$ and $b$ together from one linear-regression command, so the routine is always the same: enter the explanatory variable in the first list and the response variable in the second, run the regression, and read off the three numbers. Keep the full-precision $a$ and $b$ — either leave them on screen or store them in memory — and only round the values you actually write as final answers. A quick reliability check on the whole set-up: because the line passes through $(\bar x, \bar y)$, substituting the mean of your $x$-list should return (very close to) the mean of your $y$-list. If it does not, the lists were probably entered in the wrong order or a value was mistyped.

On Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) for the correct approach or an accuracy mark (A) for a correct value that DEPENDS on that method being right. Follow-through (FT) means a correct step built on an earlier slip can still score, 'ISW' (ignore subsequent working) means a correct answer is not un-done by extra scribble, and equivalent forms and correctly-rounded values are accepted. But that protection only exists if the method is on the page. Study how each of the 5 marks below is earned by a specific line — the interpretation, the substitution, the value, and the two-part reliability judgement.

Correlation and regression are your first models of a relationship between two variables, and the habits they teach carry forward: describe before you compute, let the GDC do arithmetic while you own the interpretation, and always ask whether a prediction sits inside the evidence. The same caution about extrapolation reappears wherever any model is used to forecast, and the idea that a summary number can hide a non-linear reality is a theme throughout statistics. Master the routine — scatter first, interpret $r$ with sign and strength, read the line, predict only with a reliability check — and bivariate questions become a fixed, high-scoring method you already own.