Instead, think of how we might have begun our study of relationships, if we had chosen the more modest goal of finding a single number which measures the strength of the linear relationship between a pair of variables. Consider three examples, each involving a population of four individuals:.
In the first case, there is a strong upward-sloping relationship between X and Y; in the second case, no apparent relationship; in the third case, a strong downward-sloping relationship. Note the pairwise products: When X and Y are big together, a large product results; when at least one of X or Y is small, the product is not so large.
When will the mean product be largest? When the big X's are associated with big Y's and the little X's with little Y's , i. When will the mean product be smallest? When the big X's are associated with little Y's and the little X's with big Y's, i.
Of course, if X and Y are independent, the mean product is just the product of the means, i. In the examples above, the respective covariances are 1. Quite generally, positive covariances indicate upward-sloping relationships, and negative covariances indicate downward-sloping relationships. Covariance is an interesting concept in its own right. But the units of measurement of covariance are not very natural. For example, the covariance of net income and net leisure expenditures is measured in square dollars.
The more widely-scattered the X,Y pairs are about a line, the closer the correlation is to 0. Notice that the covariance of X with itself is Var X , and therefore the correlation of X with itself is 1.
Correlation is a measure of the strength of the linear relationship between two variables. Strength refers to how linear the relationship is, not to the slope of the relationship. Linear means that correlation says nothing about possible nonlinear relationships; in particular, independent random variables are uncorrelated i.
Two means that that the correlation shows only the shadows of a multivariate linear relationship among three or more variables and it is common knowledge that shadows may be severe distortions of reality.
In words: In a simple linear regression, the unadjusted coefficient of determination is the square of the correlation between the dependent and independent variables. Because correlation is a measure of the linear relationship between X and Y , other non-linear relationships e. The expression given by 3 is sometimes referred to as Pearson's correlation coefficient.
The applets in this section allow you to see how different bivariate data look under different correlation structures.
The Movie applet either creates data for a particular correlation or animates a multitude data sets ranging correlations from -1 to 1. Uncorrelatedness was a prerequisite outlined by Dilip in their answer. Also for variables centered, their vector lengths in a subject space are their standard deviations.
I believe that there's a simple intuition based on symmetry here, too. Imagine what goes on when you subtract two distributions. If the value of x is low then, on average, x - y will be a lower value than if the value of x is high. As x increases then x - y increase, on average, and thus, a positive correlation. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Why does the correlation coefficient between X and X-Y random variables tend to be 0. Asked 8 years, 8 months ago. Active 4 years, 4 months ago. Viewed 20k times. What is the theory behind this? Improve this question.
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