25.4 Scatter graphs and correlation — Learn

Prerequisites: plot coordinate pairs and describe a trend using everyday language.

Objectives: plot paired data, describe correlation, and use a line of best fit for estimates.

A scatter graph plots pairs of measurements as points. Correlation describes how two variables change together. Positive correlation means both tend to increase; negative correlation means one tends to decrease as the other increases. No correlation means there is no clear pattern.

Example 1. A class records hours of revision and test scores. Plot each pupil as $(\text{hours},\text{score})$. The points rise from left to right, so the data show positive correlation.

Correlation may be strong when points lie close to a line, or weak when they lie more widely around it. A line of best fit follows the overall pattern, with points balanced around it.

Example 2. A line of best fit passes through $(0,30)$ and $(10,70)$. Estimate the score for 6 hours.

The line rises $70-30=40$ points over 10 hours.

The estimated rise per hour is $40/10=4$ points.

At 6 hours, the estimate is $30+6\times4=54$ points.

This is interpolation because 6 hours lies within the plotted range. An estimate outside the data range is extrapolation and may be less reliable.

Watch out. Correlation does not prove that one variable causes the other. Another factor may affect both; check whether the graph shows association only.

Progress check

More hours of travel might be associated with later arrival times. Name a possible third factor that could influence both measurements.

You should now be able to

• plot paired data and describe positive, negative or no correlation;
• describe the strength of a relationship from its pattern;
• draw a line of best fit and use it for interpolation;
• explain why correlation does not prove causation.