# What is the effect of scaling down covariance in the formula for rho? [closed]

The rho formula divides the covariance by SD of X and SD of Y. Does it result in a valid estimate of the true correlation (i.e., in the population)?

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It is not at all clear what your question: The rho formula divides the covariance by sd of X and SD of Y. Does it result in validy coefficient (or true correlation) for the population. is asking. I'm having trouble linking this to the question posed by your title: what is the effect of scaling down covariance in the formula for rho? (this answer to this is that it will decrease $\rho$, by the way). This may be a language issue - can you please rephrase/clarify the question? –  Macro Jul 8 '12 at 4:52
scaling down here implies that we are going to divide the covariance by SD of X and SD ofY? Here we are interested in calculation of rho. If the sd is in ratio terms, it will increase the rho estimate given other things do not change. –  subhash c. davar Jul 8 '12 at 5:35
Subhash, could you please tell us what you mean by "valid"? For instance, by some notions of "valid" the estimate $0$ (made regardless of the data) is "valid." Statistical estimation is not like mathematical equality: statisticians are interested in how well an estimator can perform, understanding that in most applications any estimator will be at least slightly wrong (something that is incomprehensible in mathematics). –  whuber Jul 9 '12 at 12:52

## closed as not a real question by Macro, gung, whuber♦Aug 14 '12 at 20:36

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• Spearman $rho$ correlation is Pearson $r$ correlation computed on ranks rather than on raw data.
• "Scaling down" the covariance is normalizing it, stripping it of sensitivity to the spread of variables, leaving only sensitivity to tightness of linear association.
• This well-known $cov_{xy}/(s_xs_y)$ formula of Pearson $r$ gives the sample statistic, not the estimate of the population parameter (unless all three its terms are the true parameters, then we get the true parameter).
• The just mentioned estimate, also called shrunk or adjusted $r$, is $\sqrt{1-(1-r^2)\frac{n-1}{n-2}}$.
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please note that it is not the sample correlation coefficient. The formula uses standard deviation of population for X as well as Y.Moreover, it reflects interdependence between X and Y and not the correlation estimate by KP formula –  subhash c. davar Jul 8 '12 at 5:42
Of course, if all 3 terms, COVxy, SDx, SDy, are true population parameters, then COVxy/(SDx*SDy) gives true population r, because "population" and "sample" coincide here. –  ttnphns Jul 8 '12 at 6:05
Your answer is valid. Many a thanks. May be if you could clarify to me whether it is an outcome of interdependence on account of being a composite say Y - error (sampling)= rho or influence of a third factor on both Y (r and sampling error). –  subhash c. davar Jul 8 '12 at 10:34
can we treat a large sample of K= 28 sample correlaion coefficients as population for this purpose? does it meet the condition that "Population" and "sample" coincide. –  subhash c. davar Jul 9 '12 at 12:01
You could read about population and sample everywhere around, for example here –  ttnphns Jul 9 '12 at 13:45