Find R Squared with the coefficient of determination calculator.

An online coefficient of determination calculator allows you to determine the correlation coefficient, R-squared (coefficient of determination) value of the given variable dataset. This r value calculator displays the relationship among the given two datasets and predicts the preciseness of future outcomes. In the text, you can understand better what is R squared in statistics, and how to find coefficient of determination with the R squared formula.

What is the Coefficient of Determination?

In statistics, the coefficient of determination, commonly known as the R-squared value, is obtained through regression analysis. This metric indicates how well the regression model explains the variability of the observed data points, effectively measuring the strength of the linear relationship between the independent variable and the dependent variable. For this reason, it is sometimes referred to as a measure of model fit. Typically, the coefficient of determination is denoted as R², or simply "R-squared." Additionally, an online Coefficient of Variation Calculator can be used to calculate the coefficient of variation for a given dataset.

Coefficient of Determination Formula:

There are several formulas that can be used to calculate the coefficient of determination with an R-value calculator:

Using Correlation Coefficient:

Correlation Coefficient = Σ [(A – Ā) * (B – B̄)] / √{ [Σ (A – Ā)² * Σ (B – B̄)²] }

Where,

A represents the data points in dataset A

B represents the data points in dataset B

Ā is the mean of dataset A

B̄ is the mean of dataset B

Then,

Coefficient of Determination = (Correlation Coefficient)²

Using Regression outputs:

The coefficient of determination can also be calculated from regression outputs using the following formulas:

R² (Coefficient of Determination) = Explained Variation / Total Variation

R² (Coefficient of Determination) = MSS / TSS

R² (Coefficient of Determination) = (TSS – RSS) / TSS

Where:

Total Sum of Squares (TSS) = Σ (Y_i – Ȳ)²

Model Sum of Squares (MSS) = Σ (Ŷ – Ȳ)²

Residual Sum of Squares (RSS) = Σ (Y_i – Ŷ)²

Here, Ŷ represents the predicted value, Ȳ is the mean of observed values, and Y_i is the ith observed value.

Additionally, the Covariance Calculator can be used to measure the covariance between two variables X and Y in statistical analyses.

How to Calculate Coefficient of Determination?

Consider the datasets: (12, 13, 23, 44, 55) and (17, 10, 20, 14, 35).

By inputting these values into an R-squared calculator, you can determine the coefficient of determination for the data.

The calculator will display a detailed table showing the regression model results for these datasets, including predicted values and explained variance.

Obs.	X	Y	Xᵢ²	Yᵢ²	Xᵢ⋅Yᵢ
1	12	17	144	289	204
2	13	10	169	100	130
3	23	20	529	400	460
4	44	14	1936	196	616
5	55	35	3025	1225	1925
Sum =	147	96	5803	2210	3335

Number of values (n) = 5

Next, the coefficient of determination calculator computes \( SS_{xx} \):

\( SS_{xx} = \sum^n_{i=1} X_i^2 - \frac{1}{n} \left(\sum^n_{i=1} X_i \right)^2 \)

\( SS_{xx} = 5803 - \frac{1}{5} \cdot 21609 \)

\( SS_{xx} = 1481.2 \)

Then, compute \( SS_{yy} \):

\( SS_{yy} = \sum^n_{i=1} Y_i^2 - \frac{1}{n} \left(\sum^n_{i=1} Y_i \right)^2 \)

\( SS_{yy} = 2210 - \frac{1}{5} \cdot 9216 \)

\( SS_{yy} = 366.8 \)

Now, calculate \( SS_{xy} \):

\( SS_{xy} = \sum^n_{i=1} X_i Y_i - \frac{1}{n} \left(\sum^n_{i=1} X_i \right) \left(\sum^n_{i=1} Y_i \right) \)

\( SS_{xy} = 3335 - \frac{1}{5} \cdot 14112 \)

\( SS_{xy} = 512.6 \)

Next, the calculator finds the correlation coefficient:

\( R = \frac{ SS_{xy} }{ \sqrt{ SS_{xx} \cdot SS_{yy} } } \)

\( R = \frac{ 512.6 }{ \sqrt{ 1481.2 \cdot 366.8 } } \)

\( R = 0.6954 \)

Finally, the coefficient of determination is calculated:

\( R^2 = (0.6954)^2 \)

\( R^2 = 0.4836 \)

Correlation and R-squared:

In simple linear regression of the form Y ~ aX + b, the square of the Pearson correlation coefficient is equivalent to the coefficient of determination (R²) for the datasets x₁, x₂, ..., xₙ and y₁, y₂, ..., yₙ.

Interpret the Coefficient of Determination:

The coefficient of determination ranges from 0 to 1 and can be expressed as a percentage by multiplying by 100. It represents the proportion of the variation in the dependent variable (Y) that can be explained by the independent variable (X) through the regression model. Essentially, it measures how well the regression model fits the data. However, it is important to remember that a high R² does not imply causation—two variables can be correlated without one causing the other.

How the Coefficient of Determination Calculator Works?

The R² calculator evaluates how much variability in one variable can be explained by another variable using the following steps:

Input:

Enter your datasets in the respective fields, separated by commas. Then click the calculate button to get the results.

Output:

The R² calculator outputs the Coefficient of Determination (R²), the Correlation Coefficient (R), and a clear interpretation of the results.
It also provides step-by-step calculations using multiple methods, showing different formulas applied to the given data.

FAQ:

What is a good coefficient of determination?

A coefficient of determination (R²) close to 1.0 indicates that the model fits the data very well and can reliably predict future outcomes. Conversely, a value near 0.0 means the model does not explain the variability in the data effectively.

What are the multiple coefficients of determination?

In multiple regression, the multiple coefficient of determination (R²) measures the proportion of variation in the dependent variable that can be predicted from the set of independent variables included in the model.

What does R explain?

The correlation coefficient, r, indicates both the strength and direction of a linear relationship between two variables in a dataset. Its value ranges from -1 to +1, where -1 indicates a perfect negative correlation, +1 a perfect positive correlation, and 0 no linear correlation.

What is the non-determination coefficient?

The Non-Determination Coefficient measures the proportion of variance that is not explained by the regression model. It is calculated as 1 – R² and represents the unexplained variability in the dependent variable.

The Bottom Line:

The coefficient of determination calculator computes the correlation coefficient (r) and R² for a given regression model. It also provides an interpretation in terms of the percentage of variance explained by the model. The calculator offers results using different methods, such as the regression sum method and correlation coefficient method, giving a comprehensive view of the data relationship.

Reference:

Sources include Wikipedia: Coefficient of determination, Relation to unexplained variance, explained variance, squared correlation coefficient, Interpretation; Investopedia: Understanding the Coefficient of Determination, Graphing the Coefficient of Determination, multivariate linear model; Stat Trek: Coefficient of Determination, linear regression, standard deviation.

Coefficient of Determination Calculator (R-squared)