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RMSE & MAE Calculator

Paste paired predictions and observations to get RMSE, MAE, MAPE, bias, and R² — the standard error metrics for evaluating forecasts, models, and measurements.

RMSE
1.0000
MAE
1.0000
MAPE
9.11%
n = 4 · Mean bias (pred − obs) = 0.0000
R² = 0.6863
RMSE ≥ MAE always; a big gap signals outliers driving the squared error.

The Formula Behind the Result

  1. Paste two aligned lists: predicted values and observed values (comma, space, or newline separated).
  2. The tool validates that both lists have equal length and computes the residuals (predicted − observed).
  3. RMSE squares, averages, and takes the root of the residuals; MAE averages their absolute values; MAPE expresses them as percentages.
  4. R² and mean bias are included so you can tell systematic over-prediction from random noise.
Formula & Technical Standard
RMSE = √(Σ(ŷᵢ−yᵢ)²/n) · MAE = Σ|ŷᵢ−yᵢ|/n · MAPE = Σ|ŷᵢ−yᵢ|/yᵢ × 100/n · Bias = Σ(ŷᵢ−yᵢ)/n

Worked Example

Forecast evaluation
Input: Predicted 10,12,8,15 · Observed 11,11,9,14
Output: RMSE 1.22 · MAE 1.00 · Bias +0.25
RMSE (1.22) exceeds MAE (1.00) because squaring punishes the largest error most — the gap between them signals outlier sensitivity.
Detecting systematic bias
Input: Predicted consistently 2 units above observed
Output: Bias = +2.0, RMSE ≈ 2.0
When bias dominates, the model is miscalibrated rather than noisy — a constant offset you can correct directly.

Common Questions

When is RMSE better than MAE?

RMSE penalizes large errors quadratically, making it the right headline metric when big misses are disproportionately costly (energy load, flood levels). MAE is more robust to outliers and easier to explain. Report both when you can.

What does R² mean here?

The squared correlation between predictions and observations: the share of variance in the observed values your predictions track. It says nothing about bias — predictions can be perfectly correlated yet offset by a constant.

Why is MAPE infinite or absurd for my data?

MAPE divides by the observed value, so zeros or near-zeros explode it. For data crossing zero use MAE or symmetric MAPE instead; this is the most common MAPE misuse in reported dashboards.

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