Two variables can be perfectly dependent and still have a correlation of zero.
Introduction
Linked, Yet Zero Correlation?
Two variables can be perfectly linked, yet have zero correlation.
- Perfect link
- Zero correlation
Set Up: y = x²
Take values of x spread evenly from minus three to three. Let y equal x squared.
- x from -3 to 3
- y = x squared
A Perfect Parabola
Plot it and you get a perfect parabola. Know x, and you know y exactly. That is total dependence.
- Know x, know y
- Total dependence
Correlation Equals Zero
Now compute the correlation coefficient. It comes out at zero.
- r = 0
The Two Halves Cancel
Why? The left half slopes downward. The right half slopes upward. The two trends cancel out exactly.
- Left slopes down
- Right slopes up
- Trends cancel
Only Straight Lines
Correlation only measures straight-line association. It is blind to curves.
- Linear association only
- Blind to curves
One-Way Implication
So zero correlation does not mean independent. Independence does imply zero correlation, but never the reverse.
- Independent ⇒ zero correlation
- Reverse is false
Plot First. Lesson 38 Next
Always plot your data first. Lesson thirty-eight digs deeper into covariance and dependence. Follow to keep going.
- Always plot your data
- Lesson 38: covariance and dependence