Scatter diagrams, correlation and regressionAQA A-Level Maths: Revision notes

A scatter diagram plots paired values of two variables, one point for each item. Correlation describes the pattern:

• Positive: as one variable increases the other tends to increase.
• Negative: as one variable increases the other tends to decrease.
• No correlation: no clear pattern. The closer the points lie to a straight line, the stronger the correlation. Describe both the direction and the strength, in context: 'the more hours of revision, the higher the test score tends to be'.

A regression line is the straight line that best fits the data, written $y=a+bx$ (or $y=bx+a$). Here $y$ is the response variable and $x$ the explanatory variable. You will not be asked to calculate the line in this course; you must be able to interpret it.

• The gradient $b$ is the average change in $y$ for each increase of 1 in $x$. For $s=34+4.5h$, each extra hour of revision is associated with 4.5 more marks on average.
• The intercept $a$ is the predicted $y$ when $x=0$. It is only meaningful if $x=0$ is in the range of the data and makes sense in context. Substitute a value of $x$ into the line to get an estimate of $y$: $s=34+4.5\times8=70$.

Interpolation is estimating $y$ for a value of $x$ within the range of the data. It is usually reasonably reliable if the correlation is strong. Extrapolation is estimating outside the range. It is unreliable because the relationship may not continue in the same way. For $s=34+4.5h$ with $h$ between 1 and 12, using $h=30$ gives $169$, impossible for a test marked out of 100. When asked whether an estimate is reliable, state the range of the data, say whether the value is inside or outside it, and comment on the strength of the correlation.

Sometimes the points form separate clusters, because the data contains different groups, for example women and men, or weekdays and weekends. A single regression line through all the points can be misleading: the overall correlation may come mainly from the difference between the groups, and the line may fit neither group well. If the clusters correspond to groups you can identify, it is better to analyse each group separately, with its own line. Comment on how the groups differ, for example by comparing gradients: a larger gradient means a steeper rise in $y$ per unit of $x$ for that group.

Two variables can be correlated without one causing the other. Possible reasons: a third (confounding) variable drives both, the link is a coincidence, or the direction of causation is the other way round. Example: ice-cream sales and beach rescues are positively correlated because hot weather increases both. Banning ice cream would not reduce rescues. To comment on a claim of causation, say the data show an association only, suggest another factor that could explain it, and describe what extra evidence (such as data on the other factor, or a wider range of data) would help.