Scatter graphs
Bivariate data records two variables for each item, for example the number of hours a student revised and the mark they scored. A scatter graph plots one variable along each axis and marks a point for every item, so the overall pattern of the pair becomes visible.
- Bivariate data
- Data with two values recorded for each item, such as height and shoe size for each person.
- Scatter graph
- A graph plotting one variable against the other, with a point for each pair of values.
- Correlation
- A measure of how closely the two variables follow a straight-line pattern together.
You do not join the points on a scatter graph. Instead you look at their overall drift: whether they trend upwards, downwards, or show no clear direction at all.
Types of correlation
The pattern of the points tells you the type and strength of the correlation.
- Positive correlation: as one variable increases, the other tends to increase. The points drift upwards from left to right.
- Negative correlation: as one variable increases, the other tends to decrease. The points drift downwards from left to right.
- No correlation: the points are scattered with no clear direction, so the two variables are not linked in a straight-line way.
Correlation also has a strength. If the points sit close to a straight line, the correlation is strong; if they are more loosely spread around it, the correlation is weak.
Describe both parts
A full description of correlation names the direction and the strength together, for example "strong positive correlation". One word alone rarely earns the full mark.
The line of best fit
A line of best fit is a single straight line drawn through the middle of the points, with roughly as many points above it as below. It captures the trend and lets you estimate one variable from the other. A good line of best fit passes through the mean point \((\bar{x}, \bar{y})\).
Once the line is drawn, its equation has the familiar form \(y = mx + c\), where \(m\) is the gradient and \(c\) is the \(y\)-intercept. You can read off a prediction, or use the equation to calculate one.
A scatter graph of revision hours \(x\) against test score \(y\) has a line of best fit through \((0, 10)\) and \((10, 60)\). Find its equation, then predict the score for 4 hours of revision.
Where this is assessed
Reading a graph and finding the gradient is Criterion A (knowing the method), while using the line to make a sensible prediction and judging how reliable it is falls under Criterion D (applying maths in context). Describing the correlation clearly in words is Criterion C communication.
Correlation is not causation
A strong correlation shows that two variables move together, but it does not prove that one causes the other. There may be a hidden third factor driving both, or the link may just be coincidence.
Beware predicting too far
Predicting inside the range of your data is reasonable. Predicting well beyond it, called extrapolation, is risky because the pattern may not continue. A line of best fit that predicts a test score above 100 percent is a sign you have gone too far.
Check yourself
Work each one through before opening the answer.
1. As the outside temperature rises, sales of hot chocolate fall. What type of correlation is this? +
As one variable goes up the other goes down, so this is negative correlation. Because the link is fairly reliable, you could reasonably call it strong negative correlation.
2. A line of best fit passes through \((0, 5)\) and \((4, 13)\). Find its equation and use it to predict \(y\) when \(x = 6\). +
Gradient \(m = \dfrac{13 - 5}{4 - 0} = \dfrac{8}{4} = 2\), and the intercept is \(c = 5\), so the equation is \(y = 2x + 5\). At \(x = 6\): \(y = 2(6) + 5 = 12 + 5 = \) 17.
3. Ice cream sales and the number of people swimming both rise together over the summer. Does buying ice cream cause people to swim? +
No. The two are correlated, but neither causes the other. A hidden third factor, hot weather, drives both. This shows why correlation does not prove causation.
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.