# Difference between positively correlated graph and a linear relationship?

So I understand the differences between positive and negative correlation and so on. However what I don't understand are the similarities and differences between a positive correlation and a linear equation/relationship.

How are they the same and/or different?

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Have you read this Wikipedia article? en.wikipedia.org/wiki/Correlation#Correlation_and_linearity – Théophile Sep 7 '12 at 23:39
Never found it :( – Link Sep 8 '12 at 0:09

Correlation is a metric which measures the relation between random data samples. It is originally defined for random processes as $\rho_{XY}=\frac{E[(X_-\mu_x)(Y-\mu_Y)]}{\sigma_x\sigma_y}$
Whereas linear equation is given in such a form: $f(x)=ax+b$ which is in principe deterministic.
The idea is that if there is a correlation between two random variables $X$ and $Y$, $X$ will look like similar to $Y$ as much as the degree of the correlation. For example of $\rho_{X,Y}=1$ then $X$ and $Y$ will be perfectly correlated. This means $X$ and $Y$ will output the same samples $x$ and $y$. Now lets have a look at the linear equation for this specific case. Obviously we will have $f(x)=x$. If we have for example $X/2$ and $Y$ for which we have $\rho_{X/2,Y}=1$ then we will have $f(x)=x/2$. In general we can have the form $f(x)=ax+b$ if we are talking about a very correlated data. In case the data looses the correlation then it will spread around the line $f(x)=ax+b$ for some suitable parameters $a$ and $b$.
@Link Assume you have a data in a vector $x=[x_1,x_2,...,x_N]$ and another $y=[y_1,y_2,...,y_N]$.If your data samples, $x_i$ and $y_i$ are correlated, then they will look like similar right? in case this correlation is $100\%$ you will have $y_i=x_i$.E.g: $x=[1, 2, -1, 4]$ and $y=[1, 2, -1, 4]$.If you draw these values in $x-y$ coordinate you will see that they follow the line $y=x$. In particular $x=[1/2, 2/2, -1/2, 4/2]$ and $y=[1, 2, -1, 4]$ will also follow a line but it is not $y=x$ but $x=2x$.Another line is $x=[1/2+2, 2/2+2, -1/2+2, 4/2+2]$ and $y=[1, 2, -1, 4]$ here we have $y=2x-4$ – Seyhmus Güngören Sep 8 '12 at 0:34
@Link all above examples are for $100\%$ correlation. As you can see such correlations map to a linear line of the form $y=f(x)=ax+b$ for some suitable $a$ and $b$. In case you loose your correlations, your data $x$ and the corresponding data $y$ will also start to deviate from linearity... I hope this clarifies the things now. – Seyhmus Güngören Sep 8 '12 at 0:34