# Coefficient of Determination Formula

The coefficient of determination, also known as the r squared formula is generally represented by R2 or r2. It is used to calculate the number that indicates the variance in the dependent variable that is to be predicted from the independent variable. It is a statistical model that is used for making future outcomes and predictions. The coefficient of determination formula is also regarded as testing of the hypothesis. and is helpful in the determination of the linear relation between the dependent and independent variables. Let us understand the coefficient of determination formula in detail in the following section.

## What is Coefficient of Determination Formula?

The coefficient of determination formula calculates the value R^{2}, finds application in analyzing how differences in one variable can be explained by a difference in a second variable. To find the R^{2} using coefficient of correlation formula, we calculate the square of coefficient of correlation, R. The coefficient of determination 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

Let us now look at a few solved examples on the coefficient of determination to understand the concept better.

## Examples Using Coefficient of Determination

Example 1:** **Find the coefficient of determination using coefficient of determination formula for the following set of data:

X | Y |

1 | 1 |

3 | 2 |

4 | 3 |

7 | 5 |

**Solution:**

Given data is:

X | Y |

1 | 1 |

3 | 2 |

4 | 3 |

7 | 5 |

Creating table out of given scores, we get,

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

1 | 1 | 1 | 1 | 1 |

3 | 2 | 6 | 9 | 4 |

4 | 3 | 12 | 16 | 9 |

7 | 5 | 35 | 49 | 25 |

∑X=15 | ∑Y=11 | ∑XY=54 | ∑X^{2}=75 |
∑Y^{2}=39 |

Here, N = 4

The coefficient of correlation is given by

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

\(=\frac{4(54)-(15)(11)}{\sqrt{4(75)-15^{2}}\sqrt{4(39)-11^{^{2}}}}\\ = \frac{216-165}{\sqrt{75}\sqrt{35}}\\ = 0.9954\)

Using coefficient of determination formula, R^{2} = 0.9908

**Answer:** **Coefficient of determination for given data = 0.9908.**

Example 2: Calculate the coefficient of determination using the coefficient of determination formula for given data:

X = 4, 6 ,12, 16 and

Y = 7, 12, 21, 28

**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 |
---|---|---|---|---|

4 | 7 | 16 | 49 | 28 |

6 | 12 | 36 | 144 | 72 |

12 | 21 | 144 | 441 | 252 |

16 | 28 | 256 | 784 | 448 |

∑X=38 | ∑Y=68 | ∑X^{2}=452 |
∑Y^{2}=1,418 |
∑XY=800 |

The coefficient of correlation is given by,

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

\( \begin{align*} r &= \frac{ 4\times 800 - (38)(68) }{\sqrt{[4 \times 452 - (38)^2][4 \times 1,418 - (68)^2]}} \\ &= \frac{3200 - 2584}{ \sqrt{[1808 - 1444][5672 - 4624]}} \\ &= 0.9973 \end{align*}\)

Using coefficient of determination formula, R^{2} = 0.9946

**Answer: Coefficient of determination for the given data = 0.9946.**

## FAQs on Coefficient of Determination Formula

### What Is the Coefficient of Determination Formula?

The coefficient of determination 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

### What Is R in Coefficient of Determination Formula?

R in the coefficient of determination formula is the coefficient of correlation, such that

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

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

### What Are Use of Coefficient of Correlation Formula?

The coefficient of correlation(R^{2}) is a statistical measure of how close the data is to the fitted regression line. It is also known as the coefficient of multiple determination for multiple regression. R^{2 }equal to 0% indicates that the model explains none of the variability of the response data around its mean.