Correlation: Line of Best Fit, Spearman's and Pearson's
Read a scatter diagram, draw a line of best fit, and interpret the two correlation coefficients - without ever calculating Pearson's, which this board never asks you to do.
Work through it, step by step
Work through it free and interactively, with each step checked before the next.
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Do they move together?
Correlation is about whether two things move together. Plot pairs of values on a scatter diagram and a pattern may appear: as one goes up, does the other go up, go down, or do nothing? This module covers reading that pattern, drawing a line of best fit to make predictions, and interpreting the two correlation coefficients - Spearman's and Pearson's. One rule matters for this course: you may be asked to interpret Pearson's, but you are never asked to calculate it.
The language of correlation
Five terms carry the topic. Learn them before the checks lean on them.
Name the correlation
On a scatter diagram, as the number of hours revised increases, exam marks also increase. What kind of correlation is this?
- Positive correlation
- Negative correlation
- Zero correlation
- Causation, because revising must cause better marks
Match the pattern
- Positive correlation
- Negative correlation
- Zero correlation
- Strong correlation
- As one variable increases, the other increases
- As one variable increases, the other decreases
- No pattern between the two variables
- The points lie very close to the line of best fit
Drawing the line
A line of best fit is a single straight line through the middle of the points, with roughly as many points above as below it. Drawn by eye, it should pass close to the mean point (the average x, average y). Once drawn, you can use it to predict: read across from a known value to the line, then down to the other axis. But where you read matters, as the next step shows.
How far can you trust it?
You use a line of best fit to make a prediction. Which prediction is more reliable?
- Interpolation - predicting within the range of the data you have
- Extrapolation - predicting far beyond the range of the data
- Both are equally reliable in every case
- Neither can ever be used for prediction
Two coefficients
A correlation coefficient puts a number on the strength and direction of correlation. Both of these run from -1 to +1: the sign gives the direction, and how close to 1 gives the strength.
Read the coefficient
A study reports a Spearman's coefficient of rs = 0.9. What does this tell you?
- A strong positive correlation
- A strong negative correlation
- No correlation at all
- That one variable definitely causes the other
Correlation is not proof
Here is the trap the examiner loves. A strong correlation shows two things vary together - it does not prove one causes the other. There may be a hidden third factor driving both (ice-cream sales and sunburn rise together, but ice cream does not cause sunburn - hot weather causes both), or it may be coincidence. Always call it an association, and resist the leap to causation unless a controlled experiment stands behind it.
Match each value to its meaning
- rs = 0.85
- r = -0.8
- rs = 0.05
- r = -0.3
- A strong positive correlation
- A strong negative correlation
- Almost no correlation
- A weak negative correlation
Correlation on trial
Three claims about correlation. Each time, choose the response that shows real statistical thinking.
- A newspaper says: "Towns with more libraries have less crime, so libraries cut crime." What is wrong?
- A line of best fit is drawn for children aged 5 to 11. Someone uses it to predict a 40-year-old's value. What is the problem?
- A researcher reports Pearson's r = 0.95 and starts to calculate it again by hand. What should you tell them?
Use a line of best fit
Put the steps of using a line of best fit into order.
- Plot the pairs of data as points on a scatter diagram
- Draw a straight line through the middle, with about as many points above as below
- To predict, read across from a known value to the line, then down to the other axis
- Check the prediction is within the data range (interpolation), not far beyond it
Positive, negative or none
When both variables increase together, the correlation is _____. A line of best fit is drawn through the _____ of the points. Predicting within the data is called _____, which is more reliable than extrapolation. Both Spearman's and Pearson's coefficients run from _____ to +1. And a correlation never proves _____ on its own.
Read the screen time correlation
A researcher plots students' screen time against their sleep hours and finds a Spearman's coefficient of rs = -0.7. Interpret this result and explain what it does and does not prove.
- State whether the correlation is positive or negative, and how strong it is
- Explain what the value -0.7 means about screen time and sleep
- Explain why this does NOT prove that screen time causes less sleep
- Say whether it would be safe to predict sleep for a screen time far outside the data (extrapolation)