Exercise solution 5, page 710 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A scatter diagram is used to show the relationship between two numerical variables. Each value in the table is plotted as one point with coordinates \((x,y)\).

Here, the first variable goes on the horizontal axis and the second variable goes on the vertical axis. For example, if an athlete had a height of \(1.88\text{ m}\) and jumped \(2.29\text{ m}\), the point plotted would be \((1.88,\ 2.29)\).

B) To draw the graph correctly, you need to choose a suitable scale on each axis. The scale should:

- cover all the data values,
- increase regularly,
- use enough of the grid to make the pattern easy to see.

For instance, if heights were between \(1.84\text{ m}\) and \(2.05\text{ m}\), you might choose an axis marked from \(1.80\) to \(2.10\). If jump heights were between \(2.18\text{ m}\) and \(2.33\text{ m}\), you might choose an axis from \(2.15\) to \(2.35\).

C) When plotting points from a table, read each row as a pair of values. The first number gives the position across, and the second number gives the position up. Be careful not to swap them.

So a row such as “athlete height \(=1.97\text{ m}\), jump height \(=2.31\text{ m}\)” must be plotted at \((1.97,\ 2.31)\), not \((2.31,\ 1.97)\).

D) After drawing the scatter diagram, you look for a correlation. Correlation describes the overall pattern of the points.

Positive correlation: as one variable increases, the other tends to increase.

Negative correlation: as one variable increases, the other tends to decrease.

No correlation: there is no clear pattern.

If the points rise from left to right, the correlation is positive. If they fall from left to right, the correlation is negative.

E) Correlation is judged from the general trend, not from every single point. In real data, points do not usually lie exactly on a straight line. You should look at whether the cloud of points has an overall upward or downward direction.

For example, if taller athletes often jump slightly higher, the points may show an upward trend even if some points do not fit perfectly.

F) A scatter diagram can also show the strength of a correlation.

Strong correlation: the points lie close to an imaginary straight line.

Weak correlation: the points are more spread out.

So, when describing the graph, it is often helpful to say not only whether the correlation is positive or negative, but also whether it seems strong or weak.

G) Remember that correlation does not prove causation. Even if two variables are linked on a scatter graph, this does not automatically mean that one causes the other. In this type of question, you are only asked to describe the relationship shown by the data.