Coefficient of Determination Calculator
The Coefficient of Determination Calculator is used to find the coefficient of determination (\(\mathrm{r^2}\)) between two sets of data. It also provides different calculators based on the coefficient of determination formula in different cases. You can check other calculators by selecting different calculator types.
How to Use a Coefficient of Determination Calculator
To use a coefficient of determination calculator, follow these steps:
- Input Data: Enter the observed values of the dependent variable and the predicted values from your regression model.
- Calculate: The calculator processes the data and computes the R² value.
- Interpret: Use the R² value to assess the fit of your model. A higher R² indicates a better fit.
What is the coefficient of determination
The coefficient of determination denoted by \( \mathrm{r^2} \) is used to measure the proportion of variance in the dependent variable which is predictable from the independent variable. In simpler terms, it tells us how well the data points fit a line or curve in a regression analysis.
It is important to note that the value of \( \mathrm{r^2} \) ranges from 0 to 1.
Learn more about coefficient of determination.
Coefficient of Determination vs Coefficient of Correlation
When we analyze the data it is crucial to know the relationship between variables. The coefficient of determination \( \mathrm{r^2} \) and the coefficient of correlation \( \mathrm{r} \) are the two commonly used statistics. While both are related to each other, they serve different purposes and provide distinct insights.
Coefficient of Determination
The coefficient of determination denoted by \( \mathrm{r^2} \) ranges from 0 to 1.
- 0: The model explains none of the variance.
- 1: The model explains all of the variance.
Interpretation:
If the value of \( \mathrm{r^2} \) is higher, it indicates a better fit of the model to the data.
Example:
If the value of \( \mathrm{r^2} \) is 0.8, it means that 80% of the variance in the dependent variable is explained by the independent variable.
Coefficient of Correlation:
The coefficient of correlation denoted by \( \mathrm{r} \) ranges from -1 to 1.
- -1: Perfect negative correlation
- 0: No correlation
- 1: Perfect positive correlation
Interpretation:
If the value of r is closer to -1 or 1, the stronger the linear relationship between the variables. If the value of r is equal to 0, it indicates no linear relationship.
What Does R Squared Mean
R-squared (R²) is also known as the coefficient of determination.
It is a statistical measure that provides insight into the goodness of fit of a regression model.
Formula for the Coefficient of Determination
The formula for the coefficient of determination is:
\( \mathrm{ r^{2} = \frac{SS_{xy}^{2} } { SS_{x } \times SS_{y }} } \)
In this formula:
- \( \mathrm{SS_{xy} } \) represents the sum of squares of cross-products.
- \( \mathrm{ SS_{x} } \) is the sum of squares of the independent variable.
- \( \mathrm{SS_{y} } \) represents the sum of squares of the dependent variable.
Applications of the Coefficient of Determination
The coefficient of determination is widely used across various fields:
- Economics: To predict economic trends and behaviors.
- Finance: To model stock prices and financial forecasts.
- Engineering: To optimize processes and predict outcomes.
- Medicine: To assess the effectiveness of treatments and interventions.
How to find coefficient of determination
Question:
Find the coefficient of determination of the following data set:
| \(\mathrm{x}\) | 6 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(\mathrm{y}\) | 4 | 5 | 7 | 1 |
Solution:
Calculation table:
| \(\mathrm{X}\) | \(\mathrm{Y}\) | \(\mathrm{X \times Y}\) | \(\mathrm{X^{2}}\) | \(\mathrm{Y^{2}}\) |
|---|---|---|---|---|
| 6 | 4 | 36 | 16 | 24 |
| 1 | 5 | 1 | 25 | 5 |
| 2 | 7 | 4 | 49 | 14 |
| 3 | 1 | 9 | 1 | 3 |
| \(\mathrm{\sum X = 12}\) | \(\mathrm{\sum Y = 17}\) | \(\mathrm{\sum X \times Y = 46}\) | \(\mathrm{\sum X^{2} = 50}\) | \(\mathrm{\sum Y^{2} = 91}\) |
Sum of squares for both X and Y:
\( \begin{align} \mathrm{SS_{x x}} &=\mathrm{\sum X^{2} -\frac{(\sum X)^{2}}{n}} \\ &=\text{50}-\frac{(\text{12})^{2}}{\text{4}} \\ &= \text{14} \end{align} \)
\( \begin{align} \mathrm{SS_{y y}} &=\mathrm{\sum Y^{2} -\frac{\sum Y^{2}} {n} } \\ &= \text{91}-\frac{\text{17}^2}{\text{4}} \\ &= \text{18.75} \end{align}\)
Sum of the cross-products between X and Y:
\( \begin{align} \mathrm{SS_{x y}} &=\mathrm{\sum XY - \frac{\sum X \times \sum Y}{n} } \\ &=\text{46}-\frac{\text{12} \times \text{17}}{\text{4}} \\ &= -\text{5} \end{align} \)
Coefficient of determination
\( \begin{align} \mathrm{r^{2}} & = \mathrm{\frac{SS_{xy}^{2} } { SS_{x } \times SS_{y }} } \\ &=\frac{ -\text{5}^{2} } { \text{14} \times \text{18.75} } \\ &= \text{0.0952} \end{align} \)
Choosing the Right Coefficient of Determination Calculator
Our coefficient of determination calculator has the following features:
- User-Friendly Interface: Our calculator provides a user-friendly interface so that anyone can easily understand.
- Accuracy: All the results provided by statcalculator.net are verified by a statistics professor.
- Additional Features: Most calculators available on the internet just provide the answer based on the x and y data, but we offer various calculators on the same topic.