11.4 Scatter diagrams, regression lines and correlation — Learn

Prerequisites: plotting coordinates from Chapter 3; the gradient and equation of a straight line from Sections 3.1 and 3.2.

By the end of this section you can: plot and read a scatter diagram; describe the correlation in words; interpret the gradient and intercept of a regression line; predict from a line of best fit; explain why correlation does not imply causation; and recognise a scatter diagram that contains distinct sections of the population.

1. Bivariate data and the scatter diagram

Bivariate data has two values for each item. A scatter diagram plots one point for each item, with one variable on each axis.

One variable is usually the explanatory variable, which you choose or control, and the other is the response variable, which you measure. Put the explanatory variable on the horizontal axis. For journey time against distance, distance is the explanatory variable and time is the response.

2. Describing correlation

Correlation describes how the two variables move together.

• Positive correlation: as one variable increases, the other tends to increase. Points rise from left to right.
• Negative correlation: as one variable increases, the other tends to decrease. Points fall from left to right.
• No correlation: there is no clear pattern.

Correlation also has strength. Points that lie close to a straight line show strong correlation. Points that are widely scattered show weak correlation.

A regression line, or line of best fit, is a straight line drawn through the middle of the points. It is used to summarise the trend and to make predictions.

Example 1. A scatter diagram plots the distance of a house from a city centre against its price. The points fall from left to right, fairly close to a straight line. Describe the correlation.

The correlation is negative and fairly strong. Houses further from the centre tend to cost less, and the points lie near a straight line.

3. Interpreting a regression line

A regression line often has the form

where $b$ is the gradient and $a$ is the intercept on the $y$-axis. The gradient says how much the response changes for each extra unit of the explanatory variable. The intercept is the predicted response when the explanatory variable is zero.

Example 2. A line of best fit for hours of study $x$ and test score $y$ is $y = 4x + 30$. Interpret the gradient and the intercept, and predict the score for 6 hours of study.

The gradient is $4$. Each extra hour of study is associated with about 4 extra marks. The intercept is $30$: a student who does no study has a predicted score of 30.

For $x = 6$,

The predicted score is 54. A prediction inside the range of the data is called interpolation. A prediction outside it is extrapolation, and it is less reliable, because the trend may not continue.

4. Distinct sections of the population

A single scatter diagram can contain two or more groups that behave differently. This is called distinct sections of the population. One line of best fit then describes neither group well, and the overall correlation can be misleading.

Example 3. A scatter of weekly income against age contains one group of part-time workers and one group of full-time workers. The full-time group sits higher. A single line through all the points has a weak fit, because the two groups sit on two different levels. The population should be split, and each group described separately.

5. Correlation does not imply causation

A strong correlation between $x$ and $y$ does not prove that $x$ causes $y$. Both variables may be affected by a third variable, called a lurking variable, or the relationship may be a coincidence.

Example 4. In a group of children, shoe size is strongly correlated with reading ability. Shoe size does not cause reading ability. Both increase with age, so age is the lurking variable.

6. Try it: what one point does to a line

Predict first. The figure shows a scatter diagram and its line of best fit. Before you move the slider, predict what happens to the line when you move the far point upwards by a long way. Write down whether the line will tilt more or less, and whether the prediction at the right-hand side will change.

Test. Move the slider and watch the line. Note how much the line tilts for a small move and for a large move of the same point.

Explain. Write two sentences. The first explains why a point far from the others can pull the line a long way. The second explains why a line pulled by one point makes predictions less trustworthy.

Watch out. Three mistakes are common.

Reading causation into a correlation. A scatter diagram shows that two variables move together, not that one causes the other. Look for a lurking variable.

Extrapolating far beyond the data. A line of best fit describes the range of the data it came from. A prediction far outside that range may be wrong.

Drawing one line through two groups. When a scatter has distinct sections, describe each section separately. One line through both can hide the pattern.

Re-check: before using a line of best fit, ask whether the scatter is a single group and whether your prediction lies inside the data range.

You should now be able to

• plot bivariate data and identify the explanatory and response variables;
• describe correlation as positive, negative or none, and strong or weak;
• interpret the gradient and intercept of a regression line;
• predict a value from a line of best fit and say whether it is an interpolation or an extrapolation;
• recognise distinct sections of a population in a scatter diagram;
• explain why correlation does not imply causation.