# Coefficient of Determination Calculator

The coefficient of determination formula calculates the value R^{2}, which is used to analyze how differences in one variable can be explained by a difference in a second variable. The coefficient of determination is also known as the R squared formula.

## What is a Coefficient of Determination Calculator?

'Coefficient of Determination Calculator' is an online tool that helps in calculating the coefficient of determination and correlation coefficient for a given data set. Just enter the values given in the data set and find the coefficient of determination in a few seconds.

## How to Use Coefficient of Determination Calculator?

Please follow the below steps to find the coefficient of determination:

**Step 1:**Enter the values of x and y (separated by comma) in the given input boxes.**Step 2:**Click on the**"Calculate"**button to find the coefficient of determination and correlation coefficient of the given dataset.**Step 3:**Click on the**"Reset"**button to clear the fields and enter the different values.

## How to Find a Coefficient of Determination?

To find the coefficient of determination or r squared value, we calculate the square of the coefficient of correlation, R. The r squared formula is given as:

\(\large R^{2}=\left[\frac{N\sum xy-\sum x \sum y}{\sqrt{\left[N\sum x^{2}-\left(\sum x\right)^{2}\right]\left[N\sum y^{2}-\left(\sum y\right)^{2}\right]}}\right]^2\)

Where,

- R = Coefficient of correlation
- N = No of scores given
- ∑ XY = Sum of paired product
- ∑ X = X score sum
- ∑ Y = Y score sum
- ∑ X
^{2}= square of X score sum - ∑ Y
^{2}= square of Y score sum

**Solved Example:**

Calculate the coefficient of determination using the r squared formula for given data:

X = 5, 6 ,12, 15 and

Y = 7, 14, 20, 25

**Solution:**

We will first construct a table to get the required values for the coefficient of determination formula:

X | Y | X^{2} |
Y^{2} |
XY |
---|---|---|---|---|

5 | 7 | 25 | 49 | 35 |

6 | 14 | 36 | 196 | 84 |

12 | 20 | 144 | 400 | 240 |

15 | 25 | 225 | 625 | 375 |

∑X=38 | ∑Y=66 | ∑X^{2}=430 |
∑Y^{2}=1,270 |
∑XY=734 |

The coefficient of correlation is given by,

\(\large R=\frac{N\sum xy-\sum x \sum y}{\sqrt{\left[N\sum x^{2}-\left(\sum x\right)^{2}\right]\left[N\sum y^{2}-\left(\sum y\right)^{2}\right]}}\)

\( \begin{align*} r &= \frac{ 4\times 734 - (38)(66) }{\sqrt{[4 \times 430 - (38)^2][4 \times 1,270 - (66)^2]}} \\ &= \frac{2,936 - 2,508}{ \sqrt{[1,720 - 1,444][5,080 - 4,356]}} \\ &= \frac{428}{447.02} \\ &= 0.9574 \end{align*}\)

Using r squared formula, coefficient of determination = R^{2} = 0.9167

**Answer: Coefficient of determination for the given data = 0.9167**

Similarly, you can try the calculator and find the coefficient of determination for the following:

- Calculate the coefficient of determination using the r squared formula for given data:

X = 1,3, 5 ,10, 12 and

Y = 6, 15, 22, 25

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