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Nonlinear Association — Definition, Formula & Examples

Nonlinear association is a relationship between two variables in which the points on a scatterplot follow a curved pattern instead of a straight line. As one variable increases, the other changes at a rate that is not constant.

Two quantitative variables xx and yy have a nonlinear association when yy shows a systematic dependence on xx that a linear model y^=a+bx\hat{y} = a + bx cannot adequately describe. The standard sign of this is a residual plot with a clear curved pattern rather than random scatter around zero. Common nonlinear forms include quadratic, exponential, and logarithmic patterns.

Worked Example

Problem: The data points (−2,4)(-2, 4), (−1,1)(-1, 1), (0,0)(0, 0), (1,1)(1, 1), (2,4)(2, 4) lie exactly on the curve y=x2y = x^2. Compute the correlation coefficient rr and decide whether the variables are associated.
Step 1: Find the mean of each variable.
xˉ=−2−1+0+1+25=0,yˉ=4+1+0+1+45=2\bar{x} = \frac{-2 - 1 + 0 + 1 + 2}{5} = 0, \qquad \bar{y} = \frac{4 + 1 + 0 + 1 + 4}{5} = 2
Step 2: Add up the products of the deviations. This sum is the numerator of rr.
∑(x−xˉ)(y−yˉ)=(−2)(2)+(−1)(−1)+(0)(−2)+(1)(−1)+(2)(2)=0\sum (x - \bar{x})(y - \bar{y}) = (-2)(2) + (-1)(-1) + (0)(-2) + (1)(-1) + (2)(2) = 0
Step 3: Compute rr. Here ∑(x−xˉ)2=10\sum (x-\bar{x})^2 = 10 and ∑(y−yˉ)2=14\sum (y-\bar{y})^2 = 14. The denominator is not zero, so r=0r = 0.
r=010⋅14=0r = \frac{0}{\sqrt{10 \cdot 14}} = 0
Answer: The correlation coefficient is r=0r = 0, yet yy is completely determined by xx through y=x2y = x^2. The variables have a perfect nonlinear association but no linear association.

Why It Matters

In AP Statistics you must check a scatterplot and a residual plot before you fit a least-squares line. If the pattern is curved, a linear model gives biased predictions. Applying a log or square-root transformation, or fitting a curve, often produces a far better model. Population growth, drug concentration over time, and stopping distance versus speed are all nonlinear.

Common Mistakes

Mistake: Concluding that two variables are unrelated because rr is close to 0.
Correction: The correlation coefficient measures only the strength of a linear relationship. Always graph the data first, because a strong curved pattern can still give r≈0r \approx 0, as the worked example shows.

Related Terms