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Coefficient of Determination Formula

The Coefficient of Determination is used to analyze how difference in one variable can be explained by a difference in a second variable. In statistics, the coefficient of determination is denoted as R2 or r2 and pronounced as R square.

The Coefficient of Determination is one of the most important tools to statistics that is widely used in data analysis including economics, physics, chemistry among other fields. The Coefficient of Determination is used to forecast or predict the possible outcomes.

The value of Coefficient of Determination comes between 0 and 1. The higher the value of R2, the better the prediction!

The formula of correlation coefficient is given below:

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

Where,
r = Correlation coefficient
x = Values in first set of data
y = Values in second set of data
n = Total number of values.

Solved Examples

Question 1: The number of roses and the number of lily flowers are noted every week at a garden during an observation. The observation readings of 4 successive weeks are as follows:

Compute the coefficient of determination.

Rose 2 4 3 1
Lily 6 2 5 3
Solution:
$x$ $y$ $x^2$ $y^2$ $xy$
2 6 4 36 12
4 2 16 4 8
3 5 9 25 15
1 3 1 9 3
$\sum x$=10 $\sum x$ =16 $\sum x^2$ =30 $\sum y^2$ =74 $\sum xy$ =38

Formula for correlation coefficient:

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

$r$ = $\large\frac{4\times 38-10\times 16}{\sqrt{[4\times30-10^{2}][4\times 74-16^{2}]}}$

$r$ = $\large\frac{4\times 38-10\times 16}{\sqrt{[4\times30-100][4\times 74-256]}}$

$r$ = $\large\frac{152-160}{\sqrt{[120-100][296-256]}}$

$r$ = $\large\frac{-8}{\sqrt{[20][40]}}$

$r$ = $\large\frac{-8}{\sqrt{800}}$

$r$ = $\large\frac{-8}{28}$

$r$ = -0.28