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Statistical Relationship — Definition, Formula & Examples

A statistical relationship is a pattern in which two variables tend to change together, but not in a perfectly predictable way. Knowing one variable helps you estimate the other, yet the data points still scatter around the overall trend.

Two variables XX and YY have a statistical relationship when the distribution of YY changes as XX changes, meaning XX and YY are not independent. This differs from a deterministic (functional) relationship, where each value of XX fixes exactly one value of YY. When the relationship is linear, its direction and strength are measured by the correlation coefficient rr, where −1≤r≤1-1 \le r \le 1.

Key Formula

r=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2∑(yi−yˉ)2r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}
Where:
  • rr = correlation coefficient, which measures the strength and direction of a linear relationship
  • xi,yix_i, y_i = the paired data values
  • xˉ,yˉ\bar{x}, \bar{y} = the means of the x-values and the y-values

Worked Example

Problem: Five students report hours studied xx and quiz score out of 10, yy: (1,2),(2,4),(3,5),(4,4),(5,5)(1, 2), (2, 4), (3, 5), (4, 4), (5, 5). Describe the statistical relationship using rr.
Step 1: Find the mean of each variable.
xˉ=1+2+3+4+55=3,yˉ=2+4+5+4+55=4\bar{x} = \frac{1+2+3+4+5}{5} = 3, \qquad \bar{y} = \frac{2+4+5+4+5}{5} = 4
Step 2: Subtract the means. The deviations are xx: −2,−1,0,1,2-2, -1, 0, 1, 2 and yy: −2,0,1,0,1-2, 0, 1, 0, 1. Multiply each pair of deviations and add the products. Then add the squared deviations for each variable.
∑(xi−xˉ)(yi−yˉ)=4+0+0+0+2=6,∑(xi−xˉ)2=10,∑(yi−yˉ)2=6\sum (x_i-\bar{x})(y_i-\bar{y}) = 4 + 0 + 0 + 0 + 2 = 6, \quad \sum (x_i-\bar{x})^2 = 10, \quad \sum (y_i-\bar{y})^2 = 6
Step 3: Substitute these sums into the formula for rr.
r=610⋅6=660≈0.77r = \frac{6}{\sqrt{10 \cdot 6}} = \frac{6}{\sqrt{60}} \approx 0.77
Answer: Since r≈0.77r \approx 0.77, there is a fairly strong positive statistical relationship. More study time tends to go with higher scores. The relationship is not deterministic, because the points do not all lie on one line. For example, 4 hours of study gave a lower score than 3 hours.

Visualization

Why It Matters

In AP Statistics, you must describe a scatterplot's relationship by direction, form, and strength before you fit a least-squares line. Medical researchers use statistical relationships, such as the link between smoking and lung disease, to flag risks long before a cause is proven by experiment.

Common Mistakes

Mistake: Concluding there is no relationship because rr is close to 0.
Correction: The value rr measures only linear relationships. Data shaped like a U can have r≈0r \approx 0 and still show a strong curved relationship. Always look at the scatterplot first.

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