Does Pearson population correlation(rho) formula differ from sample correlation formula?

Question

Pearson populationcorrelation (rho), measures a linear dependence between two variables (x and y). It’s also known as a parametric correlation test because it depends to the distribution of the data. Further, I find the term - sample correlation(r). My question - do these terms differ? and in what sense? The formulas ?

Answer 1

The Pearson correlation can be calculated, with the same formula, on any dataset of paired numeric values or other types of values that might be considered numeric (e.g., ordinal). As it's calculated from a sample drawn from some underlying population it is a sample correlation coefficient, often denoted $r_{xy}$. That's distinguished from the "true" population correlation coefficient, often denoted $\rho_{xy}$. The sample value $r_{xy}$ is an estimate of the population value $\rho_{xy}$, similar to the way that a sample mean $\bar x$ is an estimate of a population mean $\mu_x$.

The multiple ways to write the formula for a Pearson correlation can lead to some confusion. All the formulas for the sample estimates are related to corresponding formulas for the population value. A combination of formulas culled from this page and the Wikipedia page make this clear.

For the population correlation coefficient some formulas are:

$$\rho_{X,Y}= \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}=\frac{\operatorname{E}[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X\sigma_Y}=\frac{\operatorname{E}[\,X\,Y\,]-\operatorname{E}[\,X\,]\operatorname{E}[\,Y\,]}{\sqrt{\operatorname{E}[\,X^2\,]-\left(\operatorname{E}[\,X\,] \right)^2} ~ \sqrt{\operatorname{E}[\,Y^2\,]- \left(\operatorname{E}[\,Y\,] \right)^2}}.$$

Correspondingly, for the sample correlation coefficient we have:

$$r_{xy}= \frac{s_{xy}^2}{s_x s_y} =\frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}(y_i - \bar{y})^2}} =\frac{n\sum x_iy_i-\sum x_i\sum y_i} {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.$$

That last form is the one you cite from byjus.com in a comment. Its relationship to the corresponding formula for the population coefficient becomes even clearer if you divide its numerator and denominator by $n^2$.

Although there are non-linear factors involved in the calculations, both the sample and the population coefficients solely capture the linear association between $x$ and $y$, not any nonlinear relationships. See the figures on the Wikipedia page for examples.

A significance test on the sample Pearson correlation typically estimates the probability that you would get so large a value if the true correlation within the underlying population was actually 0. The assumption behind some forms of that test is that the data have a bivariate normal distribution. Otherwise you can still calculate a sample correlation that way, but inference about significance relies on methods like bootstrapping. Even then special care must be taken to avoid the problems that arise from the bias and skew in sample estimates of correlation coefficients.

Spearman and Kendall rank-base correlation coefficients are available, with associated tests not depending on an assumption of bivariate normality. All that matters for those tests is the relative ranks within the 2 groups.