Correlation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Correlation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Correlation measures the strength and direction of the linear relationship between two quantitative variables, ranging from -1 to +1.

Do two things go up and down together? r = +1 means perfectly together, r = -1 means perfectly opposite.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Correlation r near \pm 1 means a strong linear relationship; r near 0 means little linear association. Sign indicates direction (positive or negative slope).

Common stuck point: Correlation does not imply causation. Ice cream sales and drownings both correlate with summer.

Sense of Study hint: Draw a scatter plot first. If points trend upward, r is positive; downward, r is negative; no trend, r is near zero.

Worked Examples

Example 1

medium
Given five data points (1,2), (2,4), (3,5), (4,4), (5,5), compute the Pearson correlation coefficient r.

Solution

  1. 1
    Compute means: \bar{x} = 3, \bar{y} = 4.
  2. 2
    Compute \sum(x_i - \bar{x})(y_i - \bar{y}): (-2)(-2) + (-1)(0) + (0)(1) + (1)(0) + (2)(1) = 4 + 0 + 0 + 0 + 2 = 6.
  3. 3
    Compute \sum(x_i - \bar{x})^2 = 4 + 1 + 0 + 1 + 4 = 10 and \sum(y_i - \bar{y})^2 = 4 + 0 + 1 + 0 + 1 = 6.
  4. 4
    r = \frac{6}{\sqrt{10 \times 6}} = \frac{6}{\sqrt{60}} = \frac{6}{7.746} \approx 0.775.

Answer

r \approx 0.775
The Pearson correlation coefficient r ranges from -1 to 1. A value of 0.775 indicates a strong positive linear relationship between x and y.

Example 2

easy
A study finds r = -0.85 between hours of TV watched per day and exam scores. Interpret this value.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
If r = 0.92, what is the coefficient of determination r^2, and what does it mean?

Example 2

easy
A study finds a correlation coefficient of r = 0.88 between height and arm span. Describe the direction and strength of the linear relationship.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

meanstandard deviation