The R^2 value:

Question

The R^2 value:

BlackTom AI · Accepted Answer

R^2 measures how much of the total variation in the dependent variable is accounted for by the regression model: R^2 = SSR / SST = 1 − SSE / SST.

Relevant concepts:
- SST (total sum of squares) = Σ(yi − ȳ)² measures total variability of observed Y around its mean.
- SSR (regression sum of squares) = Σ(ŷi − ȳ)² measures variability explained by the fitted model (predicted values ŷi relative to ȳ).
- SSE (error or residual sum of squares) = Σ(yi − ŷi)² measures variability left unexplained by the model.
- R^2 is dimensionless and ranges from 0 to 1 in ordinary least squares: 0 means model explains none of the variance, 1 means it explains all.

Step-by-step reasoning and intermediate expressions:
1. Compute ȳ = (1/n) Σ yi.
2. Compute SST = Σ(yi − ȳ)².
3. Fit model and get predicted ŷi; compute SSR = Σ(ŷi − ȳ)² and SSE = Σ(yi − ŷi)².
4. Use R^2 = SSR / SST (equivalently R^2 = 1 − SSE / SST). This fraction gives the proportion of total variance in Y that is explained by the independent variable(s). In simple linear regression, R^2 also equals the square of the Pearson correlation r between X and Y: R^2 = r².

Why the correct answer is correct:
- The expression SSR/SST is precisely the proportion of total variation accounted for by the model; therefore the statement “gives the proportion of variation in the dependent variable that is explained by the independent variable” accurately describes R^2.

Why the other options are incorrect:
- Option A: The intercept is a fixed model parameter, not something whose magnitude is indicated by R^2; R^2 does not convey how intercept changes with X.
- Option C: “Variability of observed Y-values from predicted values” describes SSE (residual variance), not R^2 itself. R^2 relates SSE to SST via a ratio.
- Option D: Transforming cumulative probability scales to make a normal CDF linear refers to probit/Q–Q plotting or probit transformation, unrelated to R^2.

Therefore, the correct choice is: “gives the proportion of variation in the dependent variable that is explained by the independent variable.” This matches the provided answer.

The R^2 value:单项选择题

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