How To Calculate Explained Variation

Explained Variation Formula:

\[ R² = \frac{\text{Explained SS}}{\text{Total SS}} \]

Coefficient of Determination (R²):

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1. What Is Explained Variation?

Explained variation, represented by R-squared (R²), measures the proportion of variance in the dependent variable that can be explained by the independent variables in a statistical model. It quantifies how well the regression model fits the observed data.

2. How Does The Calculator Work?

The calculator uses the R-squared formula:

\[ R² = \frac{\text{Explained Sum of Squares}}{\text{Total Sum of Squares}} \]

Where:

\( R² \) — Coefficient of determination (dimensionless)
\( \text{Explained SS} \) — Sum of squares explained by the model
\( \text{Total SS} \) — Total sum of squares in the data

Explanation: R-squared ranges from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect explanation of variance by the model.

3. Importance Of R-Squared Calculation

Details: R-squared is crucial for evaluating model performance in regression analysis, helping researchers understand how much of the variability in the response variable is accounted for by the model.

4. Using The Calculator

Tips: Enter the explained sum of squares and total sum of squares from your regression analysis. Both values must be positive, with total SS greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a good R-squared value?
A: This depends on the field of study. In social sciences, 0.3-0.5 may be acceptable, while in physical sciences, values above 0.8 are often expected.

Q2: Can R-squared be negative?
A: In ordinary least squares regression, R-squared ranges from 0 to 1. Negative values can occur in other contexts but indicate worse fit than a horizontal line.

Q3: What's the difference between R-squared and adjusted R-squared?
A: Adjusted R-squared accounts for the number of predictors in the model and penalizes excessive variables, providing a more accurate measure for multiple regression.

Q4: Does high R-squared mean the model is good?
A: Not necessarily. High R-squared doesn't guarantee causal relationships or good predictions. Other diagnostics like residual analysis are important.

Q5: How is explained sum of squares calculated?
A: Explained SS = Σ(ŷᵢ - ȳ)², where ŷᵢ are predicted values and ȳ is the mean of observed values.