Scatter Graphs and Correlation for ESAT Mathematics

Understanding Bivariate Data and Scatter Graphs

Bivariate data is data involving two variables. Common examples include the height and weight of pupils in a class, or the daily growth of tomato plants alongside the number of umbrellas sold in a town. A scatter graph is the set of points plotted from a bivariate data set, used to visualise any association or relationship between the two variables.

When plotting these graphs, if one variable is believed to change in response to the other, we distinguish between them:

1. The explanatory variable (the cause of variation) is placed on the horizontal axis.
2. The response variable (the one that changes in response) is placed on the vertical axis.

For instance, if investigating how a pupil's height affects their weight, height is the explanatory variable on the horizontal axis. In cases where no clear cause and effect exists, such as comparing marks in two different subjects, the variables can be placed on either axis.

Types of Correlation

Correlation describes the relationship between two variables. Linear correlation refers to straight line relationships, while non-linear correlation occurs when data points form a curve.

When describing correlation, you must comment on:

• Direction: Whether the correlation is positive (both increase together) or negative (one increases as the other decreases).
• Strength: Whether the correlation is perfect, strong, weak, or non-existent.

Perfect positive correlation occurs when all points lie exactly on a straight line with a positive gradient.

If the straight line has a negative gradient, it is a negative linear correlation. When points are not in a perfect line, drawing a rough rectangle around them helps determine strength: a narrower rectangle indicates stronger correlation.

Positive correlation:

Weak positive correlation:

Strong positive correlation:

Negative correlation:

No correlation:

Correlation vs Causation

It is vital to understand that correlation does not indicate causation. A change in one variable does not necessarily cause the change in the other. The association might be due to a third variable affecting both, or it could be a spurious correlation occurring by chance.

For example, the growth of tomato plants and the number of umbrellas sold might show correlation, but plant growth does not cause umbrella sales: the common cause is the level of rain.

Estimated Lines of Best Fit

A line of best fit represents the general trend of the data. It should be drawn roughly parallel to the sides of the imaginary rectangle surrounding the points, with approximately equal numbers of points above and below the line. It does not need to pass through any specific points.

Interpolation and Extrapolation

These are methods for estimating values not present in the original data set:

• Interpolation is estimating a value within the range of the given data. This is generally accurate because it follows the observed trend.
• Extrapolation is estimating a value outside the range of the given data by extending the line of best fit. This is dangerous because there is no evidence that the trend continues linearly beyond the observed data points.

Worked Example: Absence and Exam Marks

A teacher believes pupil absence affects test marks. She plots the marks against days of absence.

By drawing a rectangle around the points, we see it is narrow and slopes downwards, indicating strong negative correlation.

While this indicates an association, we cannot state that absence directly causes lower marks. Other factors like maths ability, illness, or nerves might be involved.

Worked Example: Physics and Chemistry Marks

The following table shows marks out of $90$ for $15$ pupils:

Chemistry	$35$	$38$	$42$	$45$	$47$	$53$	$55$	$59$	$63$	$66$	$67$	$69$	$72$	$74$	$77$
Physics	$42$	$36$	$37$	$42$	$53$	$44$	$62$	$59$	$59$	$62$	$69$	$66$	$71$	$77$	$74$

We plot Chemistry on the horizontal axis as the marks are already ordered.

The rectangle shows positive correlation. We draw the line of best fit through the middle of the trend.

Task 1: Estimate Physics mark for Chemistry $50$. We draw a line up from $50$ on the horizontal axis to the line of best fit, then across to the vertical axis. The estimate is approximately $49$. This is interpolation and is likely reasonable, though it doesn't account for individual variation in subject ability.

Task 2: Estimate Chemistry mark for Physics $85$. We extend the line of best fit to reach a height of $85$. The corresponding Chemistry mark is approximately $86$. This is extrapolation. It is less reliable because we have no guarantee the trend continues outside the $35$ to $77$ range.