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# Correlation Coefficient Calculator

Correlation Coefficient Calculator helps to compute the value of the correlation coefficient that is used to establish a relationship between the two given data sets. The correlation coefficient will always lie between -1 and 1.

## What is the Correlation Coefficient Calculator?

Correlation Coefficient Calculator is an online tool that helps to calculate the value of the correlation coefficient for the two given data sets. The strength of the relationship between two variables or data sets is given by the correlation coefficient. To use the * correlation coefficient calculator*, enter the values in the input boxes.

### Correlation Coefficient Calculator

NOTE: The length of data set values of both x and y should be equal.

## How to Use Correlation Coefficient Calculator?

Please follow the steps below to find the correlation coefficient using the correlation coefficient calculator:

**Step 1:**Go to Cuemath's online correlation coefficient calculator.**Step 2:**Enter the numbers, within brackets, separated by commas in the given input boxes.**Step 3:**Click on the**"Calculate"**button to find the value of the correlation coefficient for the given data sets.**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does Correlation Coefficient Calculator Work?

The correlation coefficient gives us a means of establishing a relationship between the actual and predicted values that are obtained at the end of a statistical experiment. It helps us to gauge the exactness between the actual and predicted values. We can analyze the dependency of one variable on another using this coefficient.

- If the correlation coefficient is negative it indicates that the two variables have an inverse relationship.
- If the correlation coefficient is positive it shows that the two variables have a direct relationship.
- If the correlation coefficient is 0 it means that the two data sets or variables do not have any correlation.

There are many formulas available to calculate the correlation coefficient. However, the Pearson Correlation Coefficient (PCC) is the most commonly used. PCC is used to measure the direction and strength of the linear relationship between the two variables. The formula to calculate the correlation coefficient is given as follows:

Correlation Coefficient = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\)

n = sample size.

Σx = Total of all values in data set x.

Σy = Total of all values in data set y.

Σxy = sum of products of corresponding values in data sets x and y.

Σx^{2} = sum of squares of values in data set x.

Σy^{2} = sum of squares of values in data set y.

## Solved Examples on Correlation Coefficient

**Example 1:** Find correlation coefficient for given data set x = {4, 8 ,12, 16} and y = {7, 14, 21, 28}. Verify it using the correlation coefficient calculator.

**Solution:**

Correlation Coefficient = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\)

n = 4.

Σx = 4 + 8 + 12 +16 = 40

Σy = 7 + 14 + 21 + 28 = 70

Σx^{2} = 16 + 64 + 144 + 256 = 480

Σy^{2} = 49 + 196 + 441 + 784 = 1470

Σxy = 28 + 112 + 252 + 448 = 840

Substituting these values in the given equation we get

Correlation Coefficient = 1.

The correlation coefficient value is positive indicating that the variables have a direct relationship.

**Example 2:** Find correlation coefficient for given data set x = {43, 44, 45, 46} and y = {2.5, 2.4, 2.3, 2.2}. Verify it using the correlation coefficient calculator.

**Solution:**

Correlation Coefficient = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\)

n = 5.

Σx = 43 + 44 + 45 + 46 = 178

Σy = 2.5 + 2.4 + 2.3 + 2.2 = 9.4

Σx^{2} = 1849 + 1936 + 2025 + 2116 = 7926

Σy^{2} = 6.25 + 5.76 + 5.29 + 4.84 = 22.14

Σxy = 107.5 + 105.6 + 103.5 + 101.2 = 417.8

Substituting these values in the given equation we get

Correlation Coefficient = -1

The correlation coefficient value is negative indicating that the variables have an inverse relationship.

Similarly, you can try the correlation coefficient calculator to find the value of the correlation coefficient for the following data sets:

- x = {5, 8, 11, 15} and y = {10.5, 9.5, 8.5, 7.5}
- x = {12.8, 13.1, 15.4, 17.9} and y = {-2, -3, -4, -5}

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