Lessons 32 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
32
Investigating the relationship between two kinds of data
Textbook, Part 2: pp. 57–67
GoalLearn to use a scatter plot and the correlation coefficient to judge the direction and strength of the relationship between two data sets.
Marking pairs of values (xᵢ, yᵢ) of two quantities as points in the coordinate plane gives a scatter plot. If the points rise from left to right the relationship is positive (as x grows, y grows too), if they fall it is negative, and if they are scattered without pattern there is no relationship. The closer the points lie to a straight line, the stronger the linear relationship. The correlation coefficient measures this with a number: r = Σ(x − x̄)(y − ȳ) / √(Σ(x − x̄)² · Σ(y − ȳ)²). Its properties: −1 ≤ r ≤ 1; its sign gives the direction; |r| = 1 means all points lie exactly on one line, and r = 0 means no linear relationship. Roughly, |r| ≥ 0.87 is strong, 0.5 ≤ |r| < 0.87 moderate and |r| < 0.5 weak. Note: a relationship does not mean cause and effect — both quantities may depend on a third factor.
Worked examples
Four students: study hours x = 1, 2, 3, 4 and test score y = 2, 3, 5, 6. x̄ = 2.5, ȳ = 4. Σ(x − x̄)(y − ȳ) = 3 + 0.5 + 0.5 + 3 = 7, Σ(x − x̄)² = 5, Σ(y − ȳ)² = 10. r = 7/√50 ≈ 0.99 — a strong positive relationship.
Hours of phone use x = 1, 2, 3, 4, 5 and remaining battery charge y = 9, 7, 6, 3, 0. x̄ = 3, ȳ = 5; Σ(x − x̄)(y − ȳ) = −8 − 2 + 0 − 2 − 10 = −22, Σ(x − x̄)² = 10, Σ(y − ȳ)² = 50. r = −22/√500 ≈ −0.98: a strong negative relationship — the more use, the lower the charge.
Class activity
“Height and arm span”: measure the height and the arm span of 8 classmates, draw a scatter plot, state the direction and strength of the relationship, and then compute r with a table (use a calculator if needed).
Practice
1
If the points slope downward from left to right, what is the sign of r?
Negative
2
Find r for x = 1, 2, 3 and y = 2, 4, 6.
1
3
Find r for x = 1, 2, 3 and y = 1, 3, 2.
0.5
4
Over the summer months the correlation between ice-cream sales and the number of swimming accidents is r = 0.9. Does ice cream cause these accidents?
No. Both depend on hot weather: in the heat more ice cream is sold and more people swim. A high r shows only that they change together, not the cause.