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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.

  1. 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.

  2. 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} $

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Suppose that X, Y, and Z each have 100 values for 100 different floor-sweepers.

  1. 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.

  2. 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.

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    $\begingroup$ +1, re point #1, everyone can sweep faster on day Z and still have X & Z be uncorrelated, but the key is that the rank ordering of sweepers by speed is completely random. That is, although sweeper #17 was the fastest on day X, they are just as likely to be the slowest, or the median sweeper, as they are to be the fastest again on day Z. $\endgroup$
    – gung - Reinstate Monica
    Oct 8, 2012 at 0:15
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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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    $\begingroup$ +1, this is a nice contribution. I've noticed that 1 reason people tend to automatically infer causality is because we often describe correlations in temporal terms. EG, you state that, " if one variable gets bigger, the other variable tends to get bigger" & "if X increases then Y will also increase", etc. Would you mind rephrasing these w/ something that doesn't have that temporal connotation, like, 'higher values of X tend to go w/ higher values of Y'? (Some other phrasing may be even better.) $\endgroup$
    – gung - Reinstate Monica
    Oct 8, 2012 at 0:38
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    $\begingroup$ @gung I would really appreciate if you improve/rephrase the wordings to improve the answer. As I am not really good at English I think your contribution will be better than me in this regard. $\endgroup$
    – Blain Waan
    Oct 30, 2012 at 20:50
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    $\begingroup$ For someone who supposedly isn't very good at English, you seem to fake it rather well. I'm always impressed by people who can discuss topics in more than one language. I can barely work in English, but my wife is Dutch--she can speak 10 languages. $\endgroup$
    – gung - Reinstate Monica
    Oct 30, 2012 at 21:02

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