Math · Introductory statistics · Worked example
Interpret a negative correlation
A used car model is worth 20, 17, 15, 11 and 7 thousand dollars at ages 0, 1, 2, 3 and 4 years. Fit a line, find r, and interpret the slope.
Find the means
Find the slope and intercept
The products of deviations add to −32 and the squared x-deviations to 10.
Find r
The squared y-deviations add to 104. The negative sign matches the downward slope.
Interpret
Each extra year lowers the predicted value by about $3,200. With r² ≈ 0.985, age accounts for about 98.5% of the variation in value in these data; mileage and condition, which travel with age, are part of that story.
Result
ŷ = 20.4 − 3.2x (thousands of dollars); r ≈ −0.992: value falls about $3,200 a year.
Your turn
If r = −0.99 for a data set, what is r², and what does it say?
Show the answer and explanation
r² = 0.9801: the line explains about 98% of the variation in y.
Square r; the sign disappears, because r² measures only how much variation the line explains.
Keep exploring
In Statistics, Regression shows r ≈ −0.992 and R² ≈ 0.985. Change the 4-year value from 7 to 12 and both weaken, because that point now sits well above the line.
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