Relationship between two data sets
Answers are for parents and teachers.
1
Find r for the points (1, 1), (2, 2), (3, 3), (4, 4) and describe the scatter plot.
The points lie on the line y = x, r = 1: a perfect positive linear relationship.
2
Compute r for x = 2, 4, 6, 8 and y = 1, 2, 2, 3.
x̄ = 5, ȳ = 2; Σ(x − x̄)(y − ȳ) = 3 + 0 + 0 + 3 = 6, Σ(x − x̄)² = 20, Σ(y − ȳ)² = 2; r = 6/√40 = 3/√10 ≈ 0.95: strong positive.
3
Which pair would you expect to have a negative relationship: (a) study time – test score, (b) outdoor temperature – heating cost, (c) distance to school – travel time?
(b): the warmer it is, the lower the heating cost. (a) and (c) are positive.
4
How would you describe the relationship if r = 0.45?
Since |r| < 0.5 it is a weak positive relationship: the points are rather far from a line.
5
Why can two quantities still have a curved (for example parabolic) relationship even if r = 0?
r measures only a linear relationship. For example for y = x² at x = −2, −1, 0, 1, 2 we get r = 0, yet y is completely determined by x.