4.4 Correlation and regression of y on xIB Maths: Analysis and Approaches HL: Revision notes

Bivariate data are pairs $(x, y)$ measured on the same items. Plotted as a scatter diagram, they can show:

• positive correlation: $y$ tends to increase as $x$ increases;
• negative correlation: $y$ tends to decrease as $x$ increases;
• zero / no correlation: no linear pattern.

Correlation is strong when points lie close to a straight line and weak when they are scattered. A line of best fit by eye should pass through the mean point $(\bar{x}, \bar{y})$, with roughly equal numbers of points on each side.

Pearson's product-moment correlation coefficient $r$ measures the strength and direction of a linear relationship. It is found with technology and satisfies $-1 \le r \le 1$.

• $r$ close to $1$: strong positive linear correlation; close to $-1$: strong negative.
• $r$ close to $0$: little or no linear correlation.

Rough guide: $|r| \ge 0.75$ strong, $0.5 \le |r| < 0.75$ moderate, $0.25 \le |r| < 0.5$ weak. When a question gives a critical value, the correlation is significant if $|r|$ is larger than it.

$r$ is only meaningful for linear relationships: points lying on a curve can give a small $r$ even though $x$ and $y$ are closely related.

A strong correlation shows that two variables move together, not that one causes the other. The link may come from a third variable affecting both (hot weather increases both ice cream sales and sunburn), or be a coincidence. In exam comments, say correlation does not imply causation and name a plausible third factor in context.

The regression line of $y$ on $x$, $y = ax + b$, is found with technology (it minimises the sum of the squared vertical distances from the points to the line). It always passes through $(\bar{x}, \bar{y})$.

Interpret the parameters in context:

• $a$ (gradient): the change in $y$ for each increase of 1 in $x$. For $y = -1850x + 21400$ (price against age), the price falls by about $\text{\textdollar}1850$ per year of age.
• $b$ ($y$-intercept): the predicted $y$ when $x = 0$; only meaningful if $x = 0$ makes sense and is near the data.

• Interpolation: predicting for an $x$ inside the data range; reliable if the correlation is strong.
• Extrapolation: predicting outside the data range; unreliable, because the linear pattern may not continue (a revision line predicting a score of 130 out of 100 is an obvious example).
• A $y$ on $x$ line should only be used to predict $y$ from $x$. Rearranging it to predict $x$ from $y$ is not valid.
• A prediction is also unreliable if $|r|$ is small.