Scatter Matters

Scatter Matters

A scatter graph plots two things about each item, such as hours revised and test score, as a single point. The *shape* of the cloud tells you whether they are related, and that relationship is called correlation. The closer the points lie to a straight line, the stronger it is; where there is no clear pattern at all, there is no correlation.

Name the correlation

The more hours a student revises, the higher their score tends to be: that is _____ correlation. The older a car gets, the less it is usually worth: that is _____ correlation. A person's shoe size and how funny they find a joke show _____ correlation at all. And the closer the points lie to a straight line, the _____ the correlation is.

Correlation is not causation

A correlation tells you two quantities move together. It does not tell you that one causes the other, and saying so is the single easiest mark to lose on this topic. Three things can produce a correlation: - A really does cause B. - B really causes A, and you have the direction backwards. - Neither causes the other, and some third factor causes both. This is the one people miss. So the safe wording is always "there is a positive correlation between X and Y", never "X causes Y". If you want to claim cause, you need more than a scatter graph.

Ice cream and sunburn

In summer, ice cream sales and cases of sunburn both rise together: a positive correlation. What can you correctly conclude?

• They are correlated, but ice cream does not cause sunburn: hot weather drives both
• Eating ice cream causes sunburn
• Getting sunburnt makes people buy ice cream
• There is no relationship at all

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 above it as below. It does not pass through every point, and it is not meant to. It summarises the trend so you can make predictions from it. An outlier is a point lying well away from that trend: on a revision graph, the student who revised for six hours and scored two. Outliers are ignored when you draw the line, because one stray point should not drag the summary of thirty others off course.

Add the data point

This scatter graph shows hours revised (across) against test score (up), with a clear positive trend. A student revised for 4 hours and scored 6. Plot that point.

Read off the line

A line of best fit on the revision graph passes through the points (2, 3) and (6, 9). Use it to estimate the score of a student who revised for 4 hours.

Predicting from the line

Once the line is drawn you can estimate a missing value, and the exam cares a great deal about WHERE you read it from:

When the line breaks

The same line of best fit is extended to predict the score of somebody who revised for 40 hours. It gives 60 marks, on a test marked out of 40. What does that tell you?

• The prediction is worthless: 40 hours is far outside the data, and the trend clearly cannot continue past the maximum mark
• The line of best fit must have been drawn incorrectly
• A student who revised for 40 hours really would score 60
• It shows the correlation is actually negative

Comparing data sets

To compare two sets of data properly, compare two things: - An average (mean, median or mode): the typical value. - A measure of spread (the range = highest minus lowest): how consistent the data is. One number is never enough. "Class A scored higher on average, but Class B was far more consistent" says something that either figure alone would hide.

Work out the range

Five test scores are 7, 4, 12, 9 and 3. What is the range?

What does it actually show?

A student looks at the revision-hours scatter graph and says: "This proves that revising more causes higher scores." Explain what the graph does show, and what it does not.

• Describe the relationship the graph shows, using the correct term for it
• Explain what a correlation on its own cannot prove
• Suggest one other factor that might explain the pattern
• Say what you could reliably predict from the line of best fit, and what you could not

The scatter rules

When one quantity increases as the other increases, the scatter graph shows _____ correlation. A line of best fit should pass through the _____ of the points, ignoring any outliers. Predicting beyond the range of the data is called _____, and is unreliable. And however strong it looks, a correlation on its own never proves _____.