How R2 is different from RMSE, MAE

 Let’s unpack this step-by-step — because R², RMSE, and MAE all involve “difference between actual and predicted,” but they measure different things and answer different questions.

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1. R² (Coefficient of Determination) — Variance Captured

• Think of your target values (y_actual) as having some spread (variance) around their mean.

• If you didn’t have a model and just guessed the mean for everyone, that’s your baseline.

• R² asks:

  "How much better is my model compared to just guessing the mean every time?"

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Formula

$$R^2 = 1 - \frac{\text{Sum of Squared Errors of Model}}{\text{Sum of Squared Errors of Mean Model}}$$

Where:

• SSE_model = Σ(Actual − Predicted)²

• SSE_mean = Σ(Actual − Mean)²

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Intuition

• R² = 1.0 → Model perfectly predicts all values (100% of variance explained).

• R² = 0.0 → Model is no better than guessing the mean.

• R² < 0.0 → Model is worse than guessing the mean (ouch).

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Example in Credit Limit Prediction

Let’s say actual limits for 5 customers are:

Actual:  10k, 12k, 15k, 20k, 25k
Mean:    16.4k

Variance is the spread around 16.4k.

Case A: Terrible Model

Predicted: 16.4k for everyone (mean model) →
SSE_model = SSE_mean → R² = 0.

Case B: Decent Model

Predicted: 9k, 13k, 14k, 21k, 26k →
SSE_model is much smaller than SSE_mean → R² ≈ 0.85.
This means the model explains 85% of the variation in limits between customers.

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2. RMSE & MAE — Error Magnitude

• These do not compare to a baseline — they tell you how far off predictions are, on average.

• RMSE penalizes large mistakes more heavily than MAE (because it squares the errors before averaging).

• Both are absolute accuracy metrics, not relative to variance.

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Example with Same Data

If predictions are:

Actual:    10k, 12k, 15k, 20k, 25k
Predicted: 9k, 13k, 14k, 21k, 26k

Errors: 1k, 1k, 1k, 1k, 1k

• MAE = (1k + 1k + 1k + 1k + 1k) / 5 = 1k

• RMSE = sqrt((1² + 1² + 1² + 1² + 1²) / 5) = 1k

• R² = very high, because variance explained is high.

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Key Difference

• R²: “How much of the pattern in the data did I capture?”

• RMSE / MAE: “How far off am I, in the actual unit (e.g., $)?”

You can have:

• High R² but high RMSE → You’re good at ranking & trend, but still making large dollar errors.

• Low R² but low RMSE → Everyone gets about the same prediction, close to average, but model doesn’t capture much variation between people.

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