* Simple: Studies have shown that smoking correlates with an increased risk of lung cancer.
* With a specific relationship: The amount of time spent studying correlates positively with exam scores.
* With a lack of relationship: There is no evidence to correlate eating chocolate with improved mood.
* In the passive voice: The research team found that the two variables were correlated.
* With a specific measure: Researchers correlated the number of hours of sleep with levels of alertness.
Remember that "correlate" suggests a relationship between two things, but it doesn't imply causation.