# Difference Between Positive Correlation and Negative Correlation

Positive Correlation vs Negative Correlation

Correlation is a measure of the strength of the relationship between two variables. The correlation coefficient quantifies the degree of change of one variable based on the change of the other variable. In statistics, correlation is connected to the concept of dependence, which is the statistical relationship between two variables.

Pearson`s correlation coefficient or the Pearson Product-Moment Correlation Coefficient, or simply the correlation coefficient is obtained by the following formulae.

For a population:

$\rho&space;_{X,Y}&space;=&space;\frac{Covariance&space;(X,Y)}{\rho&space;_{X}\rho&space;_{Y}}&space;=&space;\frac{\epsilon&space;\left&space;[&space;\left&space;(&space;X&space;-&space;\mu&space;_{X}&space;\right&space;)\left&space;(&space;Y&space;-&space;\mu&space;_{Y}&space;\right&space;)&space;\right&space;]}{\rho&space;_{X}\rho&space;_{Y}}$

For a sample:

$r&space;=&space;\frac{\sum_{i=1}^{n}\left&space;(&space;X_{i}&space;-&space;\bar{X}&space;\right&space;)\left&space;(&space;Y_{i}&space;-&space;\bar{Y}&space;\right&space;)}{\sqrt{\sum_{i=1}^{n}\left&space;(&space;X_{i}&space;-&space;\bar{X}\right&space;)^{2}&space;.&space;\sum_{i=1}^{n}\left&space;(&space;Y_{i}&space;-&space;\bar{Y}&space;\right&space;)^{2}}}$

and the following expression is equivalent to the above expression.

$r&space;=&space;\frac{1}{n-1}\sum_{i=1}^{n}\frac{(X_{i}-\bar{X})}{s_{X}}.\frac{\left&space;(&space;Y_{i}-\bar{Y}&space;\right&space;)}{s_{Y}}$

$\frac{(X_{i}-\bar{X})}{s_{X}}$ and $\frac{(Y_{i}-\bar{Y})}{s_{Y}}$ are standard scores of X and Y respectively. $\bar{X}$ is the mean and sX and sY are the standard deviations of X and Y.

The Pearson’s correlation coefficient (or just the correlation coefficient) is the most commonly used correlation coefficient and valid only for a linear relationship between the variables. r is a value between -1 and 1 (-1 ≤ r ≤ +1). If r = 0, no relationship exists and, if r ≥ 0, the relation is directly proportional and the value of one variable increases with the other. If r ≤ 0, one variable decrease as the other increases and vice versa.

Because of the linearity condition, correlation coefficient r can also be used to establish the presence of a linear relationship between the variables.

What is the difference between Positive Correlation and Negative Correlation?

• When there’s a positive correlation (r > 0) between two random variables, one variables moves proportional to the other variable. If one variable increases the other increases. If one variables decreases, the other decreases too.

• When there’s a negative correlation (r < 0) between the two random variables, variables moves opposing each other. If one variable increases the other decreases and vice versa.

• A line approximating a positive correlation has positive gradient, and a line approximating negative correlation has a negative gradient.