Physical meaning of correlation? Assume that X depicts a random variable denoting the time it takes someone to sweep the floor today and Y be the time it takes him tomorrow and Z be the time it takes him on the last day of October.


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*If X, Y and Z are assumed to be independent, it means that the person does not "learn" how to sweep faster through his first day's experience and the result of X is irrelevant to the result of Y and Z. 

*What does it mean physically when X,Y and Z have a correlation coeffecient of a certain amount (say 0.5).
I understand that $\rho_{XY} = \dfrac{cov(X,Y)}{\sigma_X \sigma_Y} = \dfrac{E[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X \sigma_Y} $
 A: Suppose that X, Y, and Z each have 100 values for 100 different floor-sweepers.


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*If X, Y, and Z are independent, it means that the rate at which a person sweeps on a later date does not depend on the rate at which he/she swept at an earlier date. However, it is possible for the rate to increase systematically even with independence. If everyone sweeps faster later, and if the increase does not depend on initial speed, this will be the case.

*The best way I've seen to visualize correlations of different magnitude is to graph them. 

x <- rnorm(100)
y <- x + rnorm(100, 0, .5)
cor(x,y)
plot(x,y)  
y <- x + rnorm(100, 0, 1)
cor(x,y)
plot(x,y)
y <- x + rnorm(100, 0, 2)
cor(x,y)
plot(x,y)


shows correlations of about .9, .7 and .5. 
A: In very basic and physical sense, a positive correlation means that higher values of one variable are associated with higher values of the other variable.
A negative correlation means that bigger values of one variable tend to co-occur with smaller values of the other variable. 
It is important to note that a correlation does not imply causation. That is 'X is a cause of Y' or 'Y is a cause of X', because they are highly correlated, is not true. A positive correlation only means that if X increases then Y will also increase. The value indicates the degree of this linear relationship. 
For your example, a positive correlation between X and Y will mean that if the time it takes someone to sweep the floor is high today then the time it takes him tomorrow will also be high. 
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