Variance Explained (Variation Explained)

Adjust the relationship between X and Y using the slider. Notice how as the effect strengthens, the data points cluster tighter around the prediction lines, increasing the proportion of Variance Explained (R² or η²).

Use the toggles below to directly observe the components: Mean (baseline deviation), Residuals (unexplained deviation), and Predicted Values (explained deviation).

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Scatterplot

Proportion Explained (R²)

R² =
SSModel SSTotal
Variance Explained (R²)
0.49
Correlation (r)
0.70
How they relate: The correlation (r) is the square root of R². Because R² is a squared value (representing variance), it is always positive. The correlation r can be negative to show the direction of the line (r = ±√R²).
Variance Breakdown (SS, df, MS)
Total
SSTotal
0
df: 0
MS (Variance)
0.0
Σ(y - ȳ)²
Sum of squared vertical distances from all blue dots to the grey mean line.
=
=
≠
Explained
(Model)
SSModel
0
df: 0
MS (Variance)
0.0
Σ(ŷ - ȳ)²
Sum of squared vertical distances from green predicted dots to the grey mean line.
+
+
+
Unexplained
(Residual)
SSError
0
df: 0
MS (Variance)
0.0
Σ(y - ŷ)²
Sum of the squared red vertical lines (distance from blue dots to predictions).
Notice the math: Both Sum of Squares (SS) and Degrees of Freedom (df) add up perfectly. However, the unbiased sample Variance, or Mean Square (MS), is calculated by dividing SS by df. Because the denominators are different for each component, Mean Squares do NOT add up (Total ≠ Explained + Unexplained).