How To Calculate Coefficient Of Correlation? (Explanation)

Statistics have extensive implications in every field and industry, from pharmaceuticals to sports. On one side, the probability is used to find the chances of the desired outcome in sports; the statistical analysis guides the data collection and analysis in a census. Similarly, the financial management of any business entity is incomplete without statistical analysis.

Different statistical techniques are used in investment analysis, project planning, and other financial matters. The statistical techniques guide risk management in financial institutions.

Statistics have a significant role in the stock exchange tradings and share prices. Many statistical techniques range from basic analysis to complex. However, the implication of statistical techniques in studying relationships between different variables is the most significant and commonly used.

The standard deviation tells about the dispersion of data. Variance also tells how much the data vary across the axis. Similarly, covariance, regression analysis, and correlation are the statistical tools to study relationships and interdependence of different variables.

For instance, the increase in inflation is directly related to an increase in commodity prices. This statement clearly shows that these two variables are interdependent. However, the statistical techniques help to quantify the relationships.

This article will talk about correlation, one of the most popular statistical tools that study the relationship and interdependence of different variables.

What Is Correlation?

Correlation is a statistical technique that is extensively used in financial management and investment analysis. It can be defined as,

Correlation measures the relationship between two or more variables. It calculates how one variable move or changes with the change in the other variable.

For instance, let’s take an example from the stock exchange trading as an investor. You want to invest in different securities, and therefore, diversification of your investment is necessary. You want a diversified portfolio where you can minimize the risk.

He will analyze how the movement of stock prices in the pharmaceutical industry changes with the stock prices in the automobile industry.

What Can Correlation Tell You?

Let’s understand the correlation and how the relationship of the variables is studied. In the correlation analysis, the strength of the relationship between two variables is expressed numerically. The measure of correlation is called the coefficient of correlation.

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Let’s explain the concept of correlation from the above example.

The relationship between the stock prices of different industries is measured by using correlation analysis. It measures how much change in the stock price of the pharmaceutical industry is driven by the change in the stock prices of the automobile industry. It is also possible that the two stocks are independent of each other.

In essence, the correlation measures if two variables are directly proportional, indirectly proportional, or no relationship exists between the two variables. The coefficient of the correlation quantifies the extent of the relationship.

Coefficient Of Correlation

The coefficient of correlation is the numerical measure of the strength of the relationship between the two variables. The correlation coefficient is denoted by r. The investors and financial managers use the coefficient of correlation to define if the linear relationship of the two variables is strong enough to use to model the relationship for the whole population.

The correlation coefficient is always between -1 or +1. If the correlation coefficient value is close to 0, the two variables have a weak correlation.

Since the correlation coefficient measures the relationship strength of two variables, the variables are named X and Y. The relationship and interdependence are shown as a scatter diagram by taking X and Y variables on the x-axis and y-axis, respectively.

How To Calculate Coefficient Of Correlation?

The correlation can be calculated by comparing two datasets corresponding to the two variables of the study. Most commonly, Pearson’s Correlation Coefficient is used to measure the linear interdependency of the two variables. As discussed before, the correlation value is always between -1 or +1.

Here is the formula for Pearson’s correlation coefficient.

The equation gives the correlation coefficient. The variables used in the formula are as follow:

n= Number of observations

∑x = Summation of the first variable values

∑y = Summation of the second variable values

∑xy = Summation of the product of first and second values

∑x2 = Summation of the squares of the first variable

∑y2 = Summation of the squares of the second variable

The correlation can also be found by using the sample formula. In this formula, the standard deviation of data is used to measure the correlation coefficient.

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Here is the formula for calculating the correlation coefficient.

In this formula, Sx and Sy are the sample standard deviations. SxSy is the covariance of the sample.


Before we illustrate the correlation coefficient calculation with the help of the formulas discussed above, let’s understand its interpretation. The correlation can be subdivided into three types: positive, negative, and zero.

The interpretation of the correlation values is also based on these correlation types. We will discuss each one by one. The value of the correlation coefficient always remains between -1 and +1.

The numerical value of the coefficient represents the strength of the relationship. However, the sign with the coefficient determines the direction of the relationship.   


If the value of the correlation coefficient is +1, it is called a perfect positive correlation. However, if the correlation is less than +1 but very near to it, we will call the variables to have a strong positive correlation. If the correlation coefficient is less than and away from 1 and close to 0, we will call the variables to have a weak positive correlation.

The positive correlation signifies that the two variables are moving in the same direction. The higher the value near to 1, the greater the interdependence of the two variables.


If the value of the correlation coefficient is -1, it is called a perfect negative correlation. If the correlation coefficient value for two variables is near to but less than -1, it is called a strong negative correlation. Similarly, if the value is near to but less than 0, the variables will have a weak negative correlation.

The negative correlation signifies the indirect relationship of the variables. If the value of one variable increases, the value of the negatively correlated variable will decrease.


After applying the correlation formula, if the value of the correlation coefficient is equal to zero, the two variables will not be correlated. It means that the movement or change in the values of two variables is independent of each other.


Let’s understand the example of correlation to elaborate the concept further. We will use Pearson’s correlation coefficient to find the value. We will take a hypothetical example of security prices in two different industries.

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There are two data sets available:

X = 42, 20, 24, 41, 56, 58, 34

Y = 95, 61, 75, 71, 82, 77, 61

n= 7

The formula for the correlation coefficient is as follow:

Let’s find out ∑X, ∑Y, ∑X2, ∑Y2, and ∑XY.

∑X =275∑Y=522∑X2=12,077∑Y2=39,786∑XY=21,053

The value of the correlation coefficient signifies that the two variables are positively correlated. However, the strength of the correlation is between strong and weak. Therefore, we can say that there exists a mild positive correlation between the two variables.

Conditions Of Correlation

Certain correlation conditions signify when you can apply the correlation formula on the data sets and when to not apply. 

Quantitative Variables Condition

The quantitative variable condition signifies the following implications:

  • You can apply correlation on the quantitative variables only.
  • The categorical data sets masquerading as quantitative data cannot be analyzed with correlation.
  • It is important to define the type and nature of data sets before applying correlation.

Straight Enough Condition

The second condition of correlation implies that:

  • The correlation coefficient can be calculated for any pair of two quantitative variables.
  • Since the correlation also determines the strength of the linear relationship. Using correlation can be misleading if there exists a non-linear relationship between two variables. 


There are certain limitations of the correlation analysis that can distort and affect the results.

  • Outliers in the data set can drastically and dramatically change the results. The impact of outliers can understate or overstate the value of the correlation coefficient. Besides, the negative correlation might appear to be a positive correlation and vice versa. Therefore, if the data sets have outliers, using correlation with and without a point is a better idea.
  • The results of the correlation show the two-dimensional relationship. Therefore, it is hard to identify the causative relationship between the two variables. We cannot say with certainty if one variable is causing a change in the other variable or not.


The correlational analysis has many implications and uses in different studies and industries. Most popularly, the correlational analysis predicts the relationship between two variables, concurrent validity, reliability analysis, and assessing predictive validity. However, there are certain limitations, too, that we have also discussed in the article.