4.4 Correlation and linear regressionIB Maths: Applications and Interpretation SL: Revision notes

Bivariate data are pairs of values $(x,y)$. A scatter diagram plots them as points. Correlation describes the linear association between the variables: positive (as $x$ increases, $y$ tends to increase), negative ($y$ tends to decrease) or zero (no linear pattern). It is strong if the points lie close to a straight line and weak if they are widely scattered; with no pattern there is no correlation. A curved pattern can be strong without being linear. Example: hours of revision against test score usually shows positive correlation; the age of a car against its value usually shows negative correlation.

Pearson's product-moment correlation coefficient $r$ measures the strength and direction of linear correlation, with $-1\le r\le1$. $r=1$ is perfect positive, $r=-1$ perfect negative and $r=0$ no linear correlation. A common guide: $|r|$ close to $1$ is strong, around $0.5$ moderate, close to $0$ weak. Use your GDC to calculate $r$ (hand calculation can help understanding). Critical values of $r$ will be given where appropriate: if $|r|$ is greater than the critical value for the sample size, the linear correlation is significant. Note that $r$ is only meaningful for linear relationships: a strong curved pattern can give an $r$ close to $0$. Example: $x=1,\dots,6$ and $y=3,5,4,7,8,9$ give $r=0.949$, strong positive linear correlation.

Correlation does not imply causation. Two variables may move together because of a third variable (ice-cream sales and sunburn both rise in hot weather), or by coincidence. A causal claim needs more evidence than a high $r$. When asked to comment, say that the data show an association and name a plausible third variable if you can.

On a scatter diagram, draw a straight line of best fit by eye so that about half the points lie on each side and the line follows the trend. It should pass through the mean point $(\bar{x},\bar{y})$. You can then read approximate values, or find its equation from two points on it. Example: for $x=1,\dots,6$, $y=3,5,4,7,8,9$ the mean point is $(3.5,6)$; any good line passes through it.

The regression line of $y$ on $x$, $y=ax+b$, is found with your GDC (linear regression); it minimises the squares of the vertical distances from the points to the line and passes through $(\bar{x},\bar{y})$. Interpret the parameters in context: $a$ (gradient) is the average change in $y$ for each one-unit increase in $x$; $b$ (intercept) is the predicted value of $y$ when $x=0$ (which may not make sense if $0$ is outside the data). Example: $y=1.2x+1.8$ for the data above. $a=1.2$: each extra unit of $x$ goes with 1.2 more units of $y$; mean point check $1.2(3.5)+1.8=6$.

Use the regression line to predict $y$ for a given $x$ by substituting. A prediction within the range of the data is interpolation and is reasonably reliable if $|r|$ is high. A prediction outside the range is extrapolation and is unreliable: the trend may not continue (a predicted negative value of a car, for example, is impossible). The line of $y$ on $x$ should only be used to predict $y$ from $x$. Predicting $x$ from a given $y$ by rearranging the equation is not always reliable; use the regression line of $x$ on $y$ instead.