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Wize AP Statistics Textbook > Associations in Bivariate Data and Correlation

Correlation

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Correlation

The correlation coefficient r\colorFour{r} measures the direction and strength of the linear relationship between two quantitative variables:
  1. Direction (positive, negative, neither)
  2. Strength (strong, moderate, weak, no) r=r= correlation where −1≤r≤1-1\le r\le1

r=1n−1Σ(xi−x‾Sx)(yi−y‾Sy)\displaystyle\boxed{r=\frac{1}{n-1}\Sigma\left(\frac{x_i-\overline{x}}{S_x}\right)\left(\frac{y_i-\overline{y}}{S_y}\right)}


Exam Tip
There are multiple ways to calculate rr. (See: Solving for Correlation)


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Range of Correlation Values

The correlation coefficient rr must be a value between −1-1 and +1+1, inclusive.
  • The closer rr is to −1-1, the stronger the negative linear relationship between XX and YY
  • The closer rr is to +1+1, the stronger the positive linear relationship between XX and YY
  • The closer rr is to 00, the weaker the linear relationship between XX and YY
  • When r=0r=0, that means there is no linear correlation.


Credit: https://www.mathsisfun.com/data/correlation.html


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Characteristics of Correlation
  1. We can only correlate quantitative data.
  2. If you switch X and Y, the correlation value does not change.
  3. Correlation coefficient has no units.
  4. Correlation is based on linear models only.
  5. Correlation does not imply causation!
Answer the following questions.

(i) Paul wants to find the correlation between iPhone models (i.e. iPhone 3, 4, 5, 6, 7, 8, X, etc.) and battery life. What is wrong here?