why is the correlation (r) in statistics between -1 and 1?

Question

it is used to measure the strength and direction of the linear relationship between two quantitative variables. It is usually written as r.
i uderstand what it is but, i just don't understand why it is between -1 and 1.
could someone please clarify this point for me.

Thanks

Answer 1

Take a look at the following link. Of particular note:

"The correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables."

What's the range of cosine?

Answer 2

The correlation relationship is measured on a scale from -1 to 1. -1 means that the relationship between the data sets are perfectly negatively correlated. In simpler terms if one goes up, the other goes down. 1 means a perfectly positive correlation. In other words, if one goes up, the other goes up. A 0 means no correlation or one data set has no impact on the other.

Think of it in terms of percentages with the 1 or -1 being 100%.

Answer 3

r is the ratio between the Covariance of X and Y and the product of the standard deviations of X and Y. As such, if all the data is colinear (all points on the same line) then you have a correlation of 1. the sign give the direction of the relationship. if r > 0 then you have a positive or direct relationship, i.e., when one variable increases in value so does the other. if r < 0 then you have a negative or indirect relationship, i.e., when one variable decreases in value the other increases.