4.10 Regression line of x on yIB Maths: Analysis and Approaches HL: Revision notes

The regression line of $y$ on $x$ is the least squares line that minimises the squared vertical distances from the points to the line. It is built to predict $y$ from a given $x$.

The regression line of $x$ on $y$, written $x=cy+d$, minimises the squared horizontal distances. It is built to predict $x$ from a given $y$.

The two lines are different unless the correlation is perfect ($r=\pm1$). The weaker the correlation, the further apart they are. So an $x$ on $y$ line cannot reliably predict $y$ from $x$, and a $y$ on $x$ line cannot reliably predict $x$ from $y$.

On a GDC, enter the data in two lists and run linear regression with the lists swapped: $y$ as the independent (explanatory) list and $x$ as the dependent list. The calculator returns an equation in its own letters. Rewrite it as $x=cy+d$.

For example, for sleep $x$ hours and reaction time $y$ ms, the calculator gives $x=-0.0624y+24.2$, while the $y$ on $x$ line is $y=-13.3x+369$. Give coefficients to 3 significant figures, but keep full accuracy in the calculator for predictions.

Every least squares regression line passes through the mean point $(\bar{x},\bar{y})$. So:

• the $y$ on $x$ and $x$ on $y$ lines intersect at the mean point, and solving them simultaneously gives $\bar{x}$ and $\bar{y}$;
• if you know one mean and one line, substitute to find the other mean;
• an unknown data value can be found from a given mean, e.g. $\frac{5+9+p+16+20}{5}=12$ gives $p=10$.

With $y=2x+10$ and $x=0.4y-2$: $x=0.4(2x+10)-2$, so $0.2x=2$, giving $x=10$ and $y=30$. The mean point is $(10,30)$.

A prediction from a regression line is more reliable when:

• the correct line is used ($x$ on $y$ to predict $x$);
• the correlation is strong ($|r|$ close to 1);
• the given value lies within the range of the data (interpolation).

Predicting outside the data range is extrapolation. The linear pattern may not continue there, so the estimate is unreliable. For example, the reaction times in the data run from 248 ms to 305 ms, so estimating sleep for a reaction time of 340 ms is extrapolation. In an exam comment, refer to both the value of $r$ and the range of the data.