# What is $\rho$ in case of $Cov(X,Y) = \rho \cdot SD(Y) \cdot SD(X)$

$\rho$ has been used as a symbol for rank correlation especially in the sense of Spearman $\rho$. What is the interpretation in the case indicated in the the question, where the covariance equals $\rho$ multiplied by the standard deviation $X$ multiplied by standard deviation $Y$. Suppose we are working on two random variables. The variables are independent as well as interdependent. For example, r is a composite of population correlation (in psychometric sense) and sampling error.

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What do you mean by interactive? I would take interactive to mean some type of dependence between the two variables, although that would contradict the independent part ... – Andy W Feb 9 '12 at 13:00
The interaction does not imply that the statistical independence is lost. here, I am trying to explain the idea in terms of parameteric statistics. – subhash c. davar Feb 9 '12 at 14:42
The use of "independent as well as interactive" here is puzzling, because it seems to suggest that independent random variables can "interact" in some way. Could you please provide a definition for your meaning of "interactive"? – whuber Feb 9 '12 at 14:52
Thanks dear Dr whuber Let us assume that X is a composite variable having Y as one of the ingredients. And please assume that X and Y are in the z form. And I refer interactive word as equivalent to interaction term in the ANOva.Do you agree to the approac? Thanks. – subhash c. davar Feb 9 '12 at 15:11
I do not understand what you mean by "composite variable" or "having ... [an] ingredient," but it seems that you intend for $X$ somehow to be constructed mathematically from $Y$ and other things. In what sense, then, could $X$ and $Y$ be independent? – whuber Feb 12 '12 at 22:46
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If the random variables $X$ and $Y$ are independent as you claim they are, then their covariance $\text{cov}(X,Y)$ equals $0$ and therefore so does the Pearson correlation coefficient $\displaystyle \rho = \frac{\text{cov}(X,Y)}{\sigma_X\sigma_Y}$ equal $0$.

However, given independent samples $\{(X_i,Y_i) \colon 1 \leq i \leq n\}$ of independent random variables $X$ and $Y$, the sample covariance and the sample Pearson correlation coefficient are not necessarily identically $0$, though it is every statistician's fondest hope that both these statistics will be small in magnitude, especially when $n$ is large.

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thanks.Technically, your response is good in terms of what exists generally in the statistical literature. The statistical independence does not imply that these two variable can not interact with each other, There is always a certain amount of interaction when we are working with the help of ANOVA type model. Let me clarify that we should interpret the formula of covariance given in the book written by statisticians such as Mathai and Rathie Probability and Statistics. could you please help me to move in the right direction. – subhash c. davar Feb 9 '12 at 14:28
@subhashdavar Maybe you should edit your question to include the formula for covariance given by Mathai and Rathie, especially if it is different from the "standard" formula and either present your own interpretation of it or ask how the formula should be interpreted. Not everyone has access to the Mathai and Rathie book. – Dilip Sarwate Feb 9 '12 at 14:33
Thanks for querry. please see response to edit by Andy above. – subhash c. davar Feb 9 '12 at 14:47
The formula is different in the sense that it employs standard deviation of population for X and Standard deviation of population for Y. Please mention the standard formula you may be aware of. – subhash c. davar Jul 7 '12 at 15:13

It is (by definition) the Pearson correlation coefficient. Whether the two variables are independent or "interactive" has no bearing on this definition.

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 Karl Pearson formula produces sample correlation and not the rho - a parametric estimate. Do you agree? – subhash c. davar Jul 7 '12 at 15:16

It seems that there is a little confusion in the use of terms. Here, $\rho$ is a parametric estimate i.e. $E(r)$. Mathai and Rathie (1977) have given the formula of $\rho$ in terms of $E(r)$ i.e. "linear correlation coefficient" and in any case it does not connote the sample correlation coefficient i.e. $r$. Moreover, we should interpret statistical independence between two variables as if there is no common moderator for the two random variables. And, generally, there is a real relationship (it may be small) between two independent explanatory variables that can be measured by covariance.

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Hi @subhash, please look at my edits if you are interested in seeing how equations are rendered in $\LaTeX$. – Macro Jul 5 '12 at 13:16
I've merged your two accounts, subhash. (Thanks to @Macro for pointing that out.) – chl Jul 5 '12 at 13:35
@ chiThanks for what you did. But, I am still novice in such matters ie merger of accounts. What was that really ? I wish I could understand it. – subhash c. davar Jul 11 '12 at 15:25