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Coefficient of Determination Calculator (R-squared)

Enter your dataset values, and the calculator will instantly compute their variance (sample or population), coefficient of variation, and sum of squares, displaying detailed calculations

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An online coefficient of determination calculator helps you determine the R-squared value (coefficient of determination) for two datasets. This R value calculator evaluates the strength of the linear relationship between the independent and dependent variables and predicts how well future outcomes may be estimated. In the following sections, you will learn what R-squared is and how to calculate it using the R-squared formula.

What is the Coefficient of Determination?

The coefficient of determination, denoted as R², is a statistical measure derived from regression analysis. It shows how well the regression model explains the variation of the observed data points. An R² value close to 1 indicates that the model explains most of the variability in the dependent variable. It is a key indicator of model fit and can be calculated using either correlation coefficients or regression outputs. Additionally, a Coefficient of Variation Calculator can complement this analysis by showing variability relative to the mean.

Coefficient of Determination Formula:

Using Correlation Coefficient:

Correlation Coefficient formula:

r = Σ [(X − X̄) * (Y − Ȳ)] / √{ [Σ (X − X̄)² * Σ (Y − Ȳ)²] }

Then,

R² = (Correlation Coefficient)²

Where:

X = data points of the independent variable
Y = data points of the dependent variable
X̄ = mean of X
Ȳ = mean of Y

Using Regression Outputs:

R² can also be calculated from regression sums:

R² = Explained Variation / Total Variation
R² = MSS / TSS
R² = (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 Y, and Y_i is the ith observed value. You can also use a Covariance Calculator to compute covariance between X and Y for additional analysis.

How to Calculate Coefficient of Determination?

Consider two datasets: (10, 20, 30, 40, 50) and (12, 18, 33, 37, 55).

Using an R-squared calculator, the coefficient of determination can be calculated step by step.

Obs.	X	Y	X²	Y²	X·Y
1	10	12	100	144	120
2	20	18	400	324	360
3	30	33	900	1089	990
4	40	37	1600	1369	1480
5	50	55	2500	3025	2750
Sum	150	155	5500	5951	6660

Number of values (n) = 5

Compute the sums of squares:

SS_xx = ΣX² − (ΣX)² / n = 5500 − (150² / 5) = 5500 − 4500 = 1000

SS_yy = ΣY² − (ΣY)² / n = 5951 − (155² / 5) = 5951 − 4805 = 1146

SS_xy = Σ(X·Y) − (ΣX·ΣY) / n = 6660 − (150·155 / 5) = 6660 − 4650 = 2010

Correlation coefficient:

R = SS_xy / √(SS_xx·SS_yy) = 2010 / √(1000·1146) ≈ 2010 / 1070 = 0.941

Coefficient of determination:

R² = (0.941)² ≈ 0.886

Correlation and R-squared:

For linear regression Y ~ aX + b, the square of the correlation coefficient equals R², showing how much of Y's variability is explained by X.

Interpret the Coefficient of Determination:

R² ranges from 0 to 1. Multiply by 100 to express as a percentage. For the example above, 88.6% of the variation in Y is explained by X. A high R² indicates a strong linear relationship, but it does not imply causation.

How the Coefficient of Determination Calculator Works?

Input:

Enter your datasets in separate fields, separated by commas, and click calculate.

Output:

Coefficient of Determination (R²)
Correlation Coefficient (R)
Step-by-step calculation showing different formulas applied

FAQ:

What is a good coefficient of determination?

R² close to 1 indicates a model that fits the data well and predicts future outcomes reliably. Values near 0 indicate poor fit.

What are multiple coefficients of determination?

In multiple regression, R² measures the proportion of variance in the dependent variable explained by all independent variables combined.

What does R explain?

The correlation coefficient, r, shows both the strength and direction of a linear relationship between two variables. It ranges from -1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear correlation.

What is the non-determination coefficient?

The non-determination coefficient = 1 − R². It represents the portion of variance not explained by the regression model.

The Bottom Line:

The R² calculator computes the correlation coefficient and coefficient of determination for your data. It shows the percentage of variance explained and provides multiple calculation methods, giving a clear understanding of the relationship between variables.

References:

Wikipedia: Coefficient of determination; Investopedia: Understanding R²; Stat Trek: Linear regression.

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