# Why is correlation formula the way it is? Or Say how it formed? [duplicate]

I am really being confused by why the correlation formula is called the correlation of two variables $$X$$ and $$Y$$. Also how is it derived?

The part where we divide covariance by product of standard derivation of $$X$$ and standard derivation of $$Y$$ is the most confusing for me. Please explain the reasons or provide some good source for such things.

• One way of thinking about the correlation and it's specific representation is that it is a unitless quantity. So by dividing the product moment estimator $E(XY)$ by the SD of X and Y, you get something that doesn't depend on the scale of either variable (in a sense). Jan 7 '19 at 17:22
• Correlation equals covariance of standardized variables. Jan 7 '19 at 17:23
• Have a look here maybe that helps: stats.stackexchange.com/questions/256344/… Or here: stats.stackexchange.com/questions/70969/… Jan 7 '19 at 17:24

Maybe going back to the notion of covariance would help.

Say we have two random variables $$X$$ and $$Y$$, with a certain number $$n$$ of independent realizations $$x_1,x_2,\dots x_n$$ and $$y_1,y_2,\dots y_n$$. We know that the formula for the sample covariance is

$$\sigma_{xy} =\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})$$

where $$\bar{x}$$ and $$\bar{y}$$ are respectively sample means for $$X$$ and $$Y$$.

Now, thanks to the Cauchy-Schwarz inequality, we have that the sample covariance is bounded by the product of the standard deviations of the two random variables, which I will denote with $$\sigma_x$$ and $$\sigma_y$$. We have then that

$$-\sigma_x\sigma_y \leq \sigma_{xy}\leq \sigma_x\sigma_y$$

Now divide all terms in the inequality by $$\sigma_x\sigma_y$$ (which are, by construction, non-negative) and you have the formula for correlation

$$-\frac{\sigma_x\sigma_y}{\sigma_x\sigma_y} \leq \frac{\sigma_{xy}}{\sigma_x\sigma_y}\leq \frac{\sigma_x\sigma_y}{\sigma_x\sigma_y}$$

$$-1 \leq \frac{\sigma_{xy}}{\sigma_x\sigma_y} \leq 1$$

with the correct bounds, $$-1$$ and $$1$$. If you grasp the notion of covariance, then you'll surely see that it is simply a standardized version of the latter.

• standardized version means? Jan 7 '19 at 18:02
• It means that it gives you the same information as the covariance, but on a scale that varies from $-1$ to $1$. As others have put it, it is indeed unitless, while covariance is expressed in the units in which the variables were measured. Jan 7 '19 at 18:06
• So how does that division ensures that resultant amount (i.e. corr.) will still increase in magnitude as there will be a more linear relationship? Jan 7 '19 at 19:46
• This behavior is already an inherent quality of covariance. CS ensures the inequality, division only gives a measure of linear relationship that is not dependent on units of measure (which can be a problem, especially when dealing with big magnitudes) Jan 7 '19 at 20:01
• One last thing; what makes it pretty sure that when the relationship tends to linearity; the cov. tends to be equal to magnitude(σ X σ Y) ?? Jan 7 '19 at 20:35