Correlation Coefficient Formula

In statistics, correlation is a way of establishing the relationship/association between two variables. In other words, the correlation coefficient formula helps in calculating the correlation coefficient which measures the dependency of one variable on the other variable. Correlation is measured numerically using the correlation coefficient. The correlation coefficient lies between -1 and 1. A negative correlation coefficient indicates that the relationship between two variables is inverse. A positive correlation coefficient indicates that the value of one variable depends on the other variable directly. A zero-correlation coefficient indicates that there is no correlation between both variables. There are many types of correlation coefficients, among them, the Pearson Correlation Coefficient (PCC) is the most common one. Let us explore how to calculate the correlation coefficient formula for a given population or sample below.

What Is the Correlation Coefficient Formula?

The correlation coefficient is a statistical concept. It establishes a relation between predicted and actual values obtained at the end of a statistical experiment. The correlation coefficient formula helps to calculate the relationship between two variables and thus the result so obtained explains the exactness between the predicted and actual values. 

Pearson Correlation Coefficient Formula:

1. Sample Correlation Coefficient

The formula for pearson correlation coefficient for population of size N (written as ρ_{X, Y}) is given as:

\(\rho_{X, Y}=\frac{\operatorname{cov}(X, Y)}{\sigma_{X} \sigma_{Y}}=\dfrac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right)}{\sqrt{{\Sigma}_{\Sigma_{i}=1}^{n}\left(X_{i}-\bar{X}\right)^{2}} \sqrt{{\sum}_{\Sigma_{i}=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}}}\)

where cov is the covariance and (cov(X,Y)= \(\frac{\sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right)}{N}\), σ_X is standard deviation of X and σ_Y is standard deviation of Y.

Given X and Y are two random variables.

2. Population Correlation Coefficient

The formula for pearson correlation coefficient for sample of size n (written as r_xy) is given as:

\(r_{x, y}=\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\Sigma}_{\Sigma_{i}=1}^{n}\left(x_{i}-\bar{x}\right)^{2} \sqrt{\sum}_{\Sigma_{i}=1}^{n}\left(y_{i}-\bar{y}\right)^{2}}\)

where n is the sample size, x_i & y_i are the i^th sample points and x̄ & ȳ are the sample means for the random variables X and Y respectively.

Given X and Y are two random variables.

3. Linear Correlation Coefficient

It uses pearson's correlation coefficient to determine the linear relationship between two variables. Its value lies between -1 and 1. It is given as:

\(r=\frac{n(\Sigma x y)-(\Sigma x)(\Sigma y)}{\sqrt{\left[n \Sigma x^{2}-(\Sigma x)^{2}\right]\left[n \Sigma y^{2}-(\Sigma y)^{2}\right]}}\) 

where n is the sample size, x_i & y_i are the i^th sample points and x̄ & ȳ are the sample means for the random variables x and y respectively.

The sign of r indicates the strength of the linear relationship between the variables.

• If r is near 1, then the two variables have a strong linear relationship.
• If r is near 0, then the two variables have no linear relation.
• If r is near -1, then the two variables have a weak (negative) linear relationship.

Let us see the applications of the correlation coefficient formula in the following section.

Examples using Correlation Coefficient Formula

Example 1. Given the following population data. Find the Pearson correlation coefficient between x and y for this data. (Take 1√7 as 0.378)

Solution:

To simplify the calculation, we divide both x and y by 100.

Using the correlation coefficient formula,

Pearson correlation coefficient for population = \(\frac{\Sigma\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\Sigma\left(x_{i}-\bar{x}\right)^{2} \sum\left(y_{i}-\bar{y}\right)^{2}}}\) = \(\frac{16}{\sqrt{8} \sqrt{56}}\) = \(\frac{2}{\sqrt{7}}\) = 0.756

Answer: Pearson correlation coefficient = 0.756

Example 2. A survey was conducted in your city. Given is the following sample data containing a person's age and their corresponding income. Find out whether the increase in age has an effect on income using the correlation coefficient formula. (Use 1√181 as 0.074 and 1√2091 as 0.07)

Solution:

To simplify the calculation, we divide y by 1000.

Pearson correlation coefficient for sample = \(\frac{\Sigma\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\Sigma\left(x_{i}-\bar{x}\right)^{2} \sum\left(y_{i}-\bar{y}\right)^{2}}}\) = \(\frac{386}{\sqrt{181} \sqrt{836}}\) = \(\frac{193}{\sqrt{181} \sqrt{209}}\) = 0.99

Answer: Yes, with the increase in age a person's income increases as well, since the Pearson correlation coefficient between age and income is very close to 1.

Example 3: Calculate the Correlation coefficient of given data.

Solution:

Here n = 5

Let us find ∑x , ∑y, ∑xy, ∑x ², ∑y²

 values:

∑x = 215

∑x² = 9255

x̄  = 43

∑(x - x̄)² = σσ_x =10  

Y values:

∑y = 17

∑y² = 57.9

∑(y - ȳ)² =σσ_y = 0.1

X and Y combined

N = 5

∑((x - x̄)(y - ȳ)) = 1

∑xy = 732

R calculation:

r = ∑((x - x̄)(y - ȳ))/√((σσ_x)(σσ_y))

r = 1/√((10)(0.1)) = 1

Since r = 1, this indicates significant relation between x and y.