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What is a correlation?

What is a correlation? Here are some definitions.

Noun
  1. A reciprocal, parallel or complementary relationship between two or more comparable objects
  2. (statistics) One of the several measures of the linear statistical relationship between two random variables, indicating both the strength and direction of the relationship.
  3. (algebra) An isomorphism from a projective space to the dual of a projective space, often to the dual of itself.
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Examples
Gyan et al. found a positive correlation between pediatric hospital admissions for respiratory distress in Trinidad and African dust events.
Is there a correlation between your biome and your BMI? Can your biome make you fat?
I reordered the values so that there was a maximum positive correlation between the two variables.
In cucumber, a correlation between fruit abortion and vegetative growth rate was also observed.
No definite correlation was found between the number of acidophil bodies and grade or stage in this sample.
A correlation technique was developed for estimating subsurface temperature from altimetric data.

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