Correlation Coefficient
Interpret the correlation coefficient r to measure linear relationship strength: r near plus or minus 1 signals strong correlation; r near 0 signals weak or no linear relationship.
Key Concepts
Property The strength and direction of a linear correlation is measured by the correlation coefficient, $r$. The values of $r$ can range from $ 1$ to $1$. The further $r$ is from $0$, in either direction, the closer the points are to a straight line. When $r$ is zero, there is no linear correlation.
If $r = 0.95$, the data has a strong positive correlation, with points forming a nearly straight line sloping upwards. If $r = 0.15$, the data has a very weak negative correlation; the points are widely scattered but have a slight downward trend. An $r$ value of $1$ means all data points lie perfectly on a single line that slopes upwards.
Think of the correlation coefficient, $r$, as a score from $ 1$ to $1$ that grades how well your data points form a straight line. A perfect $+1$ or $ 1$ means they are perfectly aligned in a positive or negative direction. A score of $0$ means the points are just a random mess with no line.
Common Questions
What does the correlation coefficient r measure?
The correlation coefficient r measures the strength and direction of the linear relationship between two variables. Values range from -1 to 1. Near +1 means strong positive correlation, near -1 means strong negative correlation, and near 0 means little or no linear relationship.
How do you interpret r=0.85 vs r=-0.3 in a data set?
An r of 0.85 indicates a strong positive linear relationship: as one variable increases, the other tends to increase proportionally. An r of -0.3 indicates a weak negative relationship with a slight downward trend that is not reliable for predictions.
Does a high correlation coefficient prove causation?
No. A high r only means the two variables move together in a linear pattern. Correlation does not establish that one variable causes changes in the other. Confounding variables or coincidence can produce high correlation without any causal link.