Bivariate data, correlation and regressionEdexcel A-Level Maths: Revision notes

Bivariate data are pairs of values $(x,y)$ for each item. A scatter diagram shows whether there is a relationship.

• Positive correlation: $y$ tends to increase as $x$ increases; negative correlation: $y$ tends to decrease as $x$ increases; zero correlation: no linear pattern.
• Strong correlation means the points lie close to a line; weak means they are widely scattered. A scatter diagram may show distinct sections of the population (for example two groups with different patterns). Then a single regression line may be inappropriate, and the groups should be considered separately.

The explanatory (independent) variable is the one you choose or that you think explains the other; it goes on the horizontal axis. The response (dependent) variable is the one that is measured to see its response; it goes on the vertical axis. A regression line of $y$ on $x$ has the form $y=a+bx$. The gradient $b$ is the average change in $y$ for a one-unit increase in $x$; the intercept $a$ is the predicted $y$ when $x=0$, which may have no meaning in context. Example: $h=75+5.2a$ means height is predicted to rise by 5.2 cm per year of age on average.

Using the regression line to predict $y$ for an $x$ within the range of the data is interpolation and is usually reliable. Predicting for an $x$ outside the range is extrapolation and is unreliable, because the relationship may not continue. Example: for ages 5 to 12, $h=75+5.2a$ at $a=8$ gives $116.6$ cm (reliable), but at $a=30$ gives $231$ cm (unrealistic). Always give the reason: the value is outside the range of the data.

If $y=ax^n$, take logs of both sides: $\log y=\log a+n\log x.$ Plotting $\log y$ (vertical) against $\log x$ (horizontal) gives a straight line with gradient $n$ and vertical intercept $\log a$. Example: gradient $1.5$ and intercept $0.477$ give $n=1.5$ and $a=10^{0.477}=3.0$. Then $x=100$ gives $y=3\times100^{1.5}=3000$.

If $y=kb^x$, then $\log y=\log k+x\log b.$ Plotting $\log y$ against $x$ gives a straight line with gradient $\log b$ and intercept $\log k$. Example: the points $(0,1.20)$ and $(4,2.00)$ give gradient $\frac{0.8}{4}=0.2$, so $b=10^{0.2}=1.58$, and $k=10^{1.2}=15.8$. Use the laws $\log(ab)=\log a+\log b$ and $\log b^x=x\log b$.

Correlation does not imply causation. Two variables can be correlated because a third variable affects both, because of coincidence, or because one does cause the other. Only a controlled experiment (with random allocation of the explanatory variable) can establish causation. When evaluating a claim, state the association found, say it does not prove causation, suggest another factor, and recommend how to test it.