# Correlation Coefficient Formula

Correlation Coefficient is a measure of the association between two variables. It is used to find the relationship is between data and a measure to check how strong it is. The formulas return a value between -1 and 1 wherein one shows -1 shows negative correlation and +1 shows a positive correlation.

The correlation coefficient value is positive when it shows that there is a correlation between the two values and the negative value shows the amount of diversity among the two values.

#### Types of a correlation coefficient formula

There are several types of correlation coefficient formulas.

But, one of the most commonly used formulas in statistics is Pearson’s Correlation Coefficient Formula.

#### Pearson’s Correlation Coefficient Formula

Also known as bivariate correlation is the most widely used correlation method among all the sciences.

The correlation coefficient is denoted by r.

To find r, let us suppose the two variables as x & y, then the correlation coefficinet r is calculated as:

$\large r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$

Here,

n= Quantity of information.
Σx = Total of the first variable value.
Σy = Total of the second variable value.
Σxy  =Sum of the product of first & second value.
Σx2 = Sum of the squares of the first value.
Σy2 = Sum of the squares of the second value.

#### Sample Correlation Coefficient Formula

$\large r_{xy}=\frac{S_{xy}}{S_{x}S_{y}}$

$S_{x}$ and $S_{y}$ are the sample standard deviations, and $S_{xy}$ is the sample covariance.

#### Population Correlation Coefficient Formula

$\large \rho_{xy}= \frac{\sigma_{xy}}{\sigma_{x} \sigma_{y}}$

The population correlation coefficient uses $\sigma_{x}$ and $\sigma_{y}$ as the population standard deviations, and $\sigma_{xy}$ as the population covariance.

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