The idea, step by step
Pearson correlation measures linear association
For paired numerical data, r lies between -1 and 1. Its sign gives the linear direction and its magnitude describes linear strength. A value near zero does not rule out a curved association. Both variables need nonzero variation for r to be defined.
Use a calculator or spreadsheet with the pairs entered in corresponding rows. Verify the sign against the scatter plot. A result such as 1.07 signals an error, not exceptionally strong correlation.
Interpret r-squared in the fitted linear model
For least-squares regression with an intercept, the coefficient of determination is r². Multiplying it by 100 gives the percentage of observed response variation accounted for by the fitted linear relationship with the predictor.
This does not mean that the predictor caused that percentage of the response, nor that that percentage of individual predictions is correct. The sign of r is lost on squaring, so use the scatter plot or slope to describe direction.
A common trap
r²=0.81 means x causes 81% of y.
Instead: It describes variation explained by a fitted linear relationship, not causation or a fraction of the response itself.
See the thinking
Calculate from centred sums
A data set has Sxx=20, Syy=45 and Sxy=-24. Calculate r and r².
- .
- .
- The fitted linear relationship accounts for 64% of the observed variation in y.
Use paired measurements
Calculate the correlation for the supplied data.
| x | y |
|---|---|
| 1 | 3 |
| 2 | 4 |
| 3 | 4 |
| 4 | 6 |
| 5 | 8 |
- The means are x̄=3, ȳ=5.
- Calculate centred cross-products and sums of squares using the paired values.
- Technology or the formula gives r=0.9487, so r²=0.9.
Check your understanding
For r=-0.6, find the coefficient of determination.
Show answer
0.36, or 36%.
Square the correlation, including its sign: (-0.6)²=0.36.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- Given Sxx=16, Syy=25 and Sxy=18, calculate r and r².3 marks
- A fit has r=-0.75. State its direction and percentage of response variation explained.3 marks
- A student reports r=1.12. Explain the problem.3 marks
- A fitted model has r²=0.49 and a negative slope. Find r.3 marks
- If every x-value is converted from metres to centimetres, what happens to r?3 marks
- All five response values are 7. Why is Pearson r undefined?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Given Sxx=16, Syy=25 and Sxy=18, calculate r and r².3 marks
The denominator is √(16×25)=20.
Divide 18 by 20 and square the result.
r=0.9; r²=0.81. - A fit has r=-0.75. State its direction and percentage of response variation explained.3 marks
The sign gives negative association.
100×(-0.75)²=56.25.
Negative direction; 56.25% explained. - A student reports r=1.12. Explain the problem.3 marks
The theoretical range is -1≤r≤1.
Check data pairing, entry and the calculation.
A Pearson correlation cannot exceed 1. - A fitted model has r²=0.49 and a negative slope. Find r.3 marks
Take the square root, giving magnitude 0.7.
Use the slope direction to choose the negative sign.
r=-0.7. - If every x-value is converted from metres to centimetres, what happens to r?3 marks
Multiplication by a positive constant changes scale, not linear correlation.
It stays the same. - All five response values are 7. Why is Pearson r undefined?3 marks
A standardised measure of shared variation cannot be calculated.
The response has zero variation, so Syy=0 and the denominator is zero.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: statistical models. Original lessons and assessment practice; not an official SACE assessment.
Official curriculum source ↗