What is the Correlation Coefficient?

Correlation Coefficient Formula

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Where

  • r = correlation coefficientn = number of observationsx = 1st variable in the contexty = 2nd variable

Explanation

Suppose there is any correlation or relationship between two variables. In that case, it shall indicate if one variable changes in value, then the other variable will also tend to change in value, say in specific, which could be either in the same or opposite direction. The numerator part of the equation conducts a test and the relative strength of the variables moving together. The denominator part of the equation scales the numerator by multiplying the differences between the variables from squared variables.

Examples

Example #1

Consider the following two variables, x and y. Then, you are required to calculate the correlation coefficient.

Below is given data for the calculation.

Solution:

Using the above equation, we can calculate the following:

We have all the values in the above table with n = 4.

Let’s now input the values for the calculation of the correlation coefficient.

Therefore, the calculation is as follows:

r = ( 4 * 25,032.24 ) – ( 262.55 * 317.31 ) / √[(4 * 20,855.74) – (262.55)2] * [(4 * 30,058.55) – (317.31)2]

r = 16,820.21 / 16,831.57

The coefficient will be –

Coefficient = 0.99932640

Example #2

Country X is a growing economy country, and it wants to conduct an independent analysis of the decisions taken by its central bank regarding interest rate changes, whether those have impacted inflation, and whether the Central Bank can control the same.

The following is a summary of the interest rate and the inflation rate that prevailed in the country on average for those years are below:

Below is given data for the calculation:

The President of the country has approached you to conduct an analysis and provide a presentation at the next meeting. Use correlation and determine whether the Central Bank has met its objective.

Using the formula discussed above, we can calculate the correlation coefficient. For example, treat interest rate as one variable, say x, and treat inflation rate as another as y.

We have all the values in the above table with n = 6.

r = ( 6 * 170.91 ) – (46.35 * 22.24 ) / √[(6 * 361.19) – (46.35)2] * [(6 * 82.74) – (22.24)2]

r = -5.36 / 5.88

The correlation will be –

Correlation = -0.92

Analysis: It appears that the correlation between the interest rate and the inflation rate is negative, which appears to be the correct relationship. As the interest rate rises, inflation decreases, which means they tend to move in the opposite direction from each other, and it appears from the above result that the central bank was successful in implementing the decision related to interest rate policy.

Example #3

ABC laboratory is researching height and age and wanted to know if there is any relationship between them. So, they gathered a sample of 1,000 people for each category and found an average height in that group.

Below is given data for the calculation of the correlation coefficient.

You are required to calculate the correlation coefficient and come up with a conclusion if any relationship exists.

Treating age as one variable, say x, and treating height (in cms) as another variable as y.

r =( 6 * 10,137 ) – (70 * 850) / √[(6 * 940 – (70)2] * [(6 * 1,20,834) – (850)2]

r= 1,322.00 / 1,361.23

Correlation = 0.971177099

Relevance and Use

It is used in statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more mainly to analyze the strength of the relationship between the variables under consideration. It also measures if there is any linear relationship between the given sets of data and how well they could be related. One of the common measures of correlation is the Pearson Correlation CoefficientPearson Correlation CoefficientPearson correlation coefficient measures the strength between the different variables and their relationships. Therefore, whenever any statistical test is conducted between the two variables, it is good to analyze the correlation coefficient value to know how strong the relationship between the two variables is.read more.

If variable changes in value along with that other variable changes in value, then understanding that relationship is critical as one can use the value of the former variable to predict the change in the value of the latter variable. A correlation has multiple usages today in this modern era, like in the financial industry, scientific research, and where not. However, it is important to know that correlation has three major types of relationships. The first one is a positive relationship, which states if there is a change in the value of a variable, then there will be a change in the related variable in the same direction. Similarly, the related variable will behave in the opposite direction if there is a negative relationship. Also, if there is no correlation, r will imply a zero value. See the below images to understand the concept better.

This article is a guide to the Correlation Coefficient and its definition. Here, we learn how to calculate the correlation coefficient using its formula, examples, and a downloadable Excel template. You can learn more about financing from the following articles: –

  • Formula of CorrelationCalculate the Coefficient of VariationCovariance CalculationCorrelation vs Covariance