MSE Calculator (Mean Squared Error)
Free MSE calculator. Compute MSE, RMSE, MAE, MAPE, R², adjusted R², MBE. Actual vs predicted. Regres
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
MSE Calculator — Mean Squared Error & Model Evaluation
Compute MSE, RMSE, MAE, MAPE, R², adjusted R², MBE. Actual vs predicted. Step-by-step breakdown with charts.
Real-World Scenarios — Click to Load
| # | Actual (y) | Predicted (ŷ) | |
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Actual vs Predicted (green line = perfect fit y=x)
Error Metrics Comparison
Residuals Plot (actual − predicted)
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- MSE: Mean Squared Error = Σ(yᵢ−ŷᵢ)²/n — penalizes large errors more than small ones
- RMSE: √MSE — same units as the target variable; commonly reported
- MAE: Mean Absolute Error = Σ|yᵢ−ŷᵢ|/n — robust to outliers
- MAPE: Mean Absolute Percentage Error — scale-invariant; use when comparing across different scales
- R²: Coefficient of determination = 1 − SS_res/SS_tot — proportion of variance explained (0 to 1)
- Adjusted R²: Penalizes extra predictors; use when comparing models with different numbers of predictors
- MBE: Mean Bias Error — indicates systematic over- or under-prediction
Did You Know?
How It Works
1. MSE and RMSE
MSE = average of squared errors. RMSE = √MSE. Both measure average magnitude of error.
2. MAE and MAPE
MAE = average of absolute errors. MAPE = (100/n)Σ|error/actual|. MAPE is scale-invariant.
3. R² and Adjusted R²
R² = 1 − SS_res/SS_tot. Adjusted R² accounts for number of predictors: 1 − (1−R²)(n−1)/(n−p−1).
4. MBE (Bias)
MBE = Σ(ŷᵢ−yᵢ)/n. Positive = over-prediction, negative = under-prediction.
Expert Tips
Report Multiple Metrics
MSE/RMSE, MAE, and R² together give a fuller picture. Don't rely on one.
Residuals Plot
Check for patterns. Random scatter suggests a good model. Trends indicate bias or missing terms.
MAPE Caveats
MAPE explodes when actuals are near zero. Avoid for data with zeros or mixed signs.
Adjusted R² for Model Selection
When comparing models with different p, use adjusted R². It penalizes complexity.
Frequently Asked Questions
What is the difference between MSE and RMSE?
MSE is in squared units. RMSE = √MSE is in the same units as the target, making it easier to interpret.
When is R² negative?
When the model fits worse than predicting the mean. SS_res > SS_tot implies R² < 0.
What is a good RMSE?
It depends on the scale of your data. Compare to the range or standard deviation of the target.
What does MBE tell us?
MBE measures systematic bias. MBE = 0 means no consistent over- or under-prediction.
Why use adjusted R²?
R² always increases when you add predictors. Adjusted R² penalizes extra predictors to avoid overfitting.
What are residuals?
Residual = actual − predicted. They are the unexplained part of the data.
How do I interpret the actual vs predicted scatter?
Points on the green y=x line mean perfect prediction. Points above = over-prediction; below = under-prediction.
Why does my R² differ from Excel or R?
Ensure you use the same formula: 1 − SS_res/SS_tot. Some software uses different formulas; this calculator uses the standard definition.
Formulas Reference
MSE = Σ(yᵢ − ŷᵢ)² / n
RMSE = √MSE
MAE = Σ|yᵢ − ŷᵢ| / n
MAPE = (100/n) Σ |yᵢ − ŷᵢ| / |yᵢ|
R² = 1 − SS_res / SS_tot
Adjusted R² = 1 − (1−R²)(n−1)/(n−p−1)
MBE = Σ(ŷᵢ − yᵢ) / n
SS_res = Σ(yᵢ−ŷᵢ)², SS_tot = Σ(yᵢ−ȳ)²
Applications
Regression Models
Evaluate linear, polynomial, or other regression fits.
Forecasting
Compare time series or demand forecasts to actuals.
Machine Learning
Evaluate ML model performance on regression tasks.
A/B Testing
Compare predicted vs observed outcomes in experiments.
Official Data Sources
Common Pitfalls
Mismatched pairs
Actual and predicted must be in the same order. Pair 1 actual with pair 1 predicted, etc.
Wrong p for adjusted R²
p = number of predictors (independent variables). Simple regression: p=1. Multiple regression: p = number of X variables.
Summary
The MSE calculator computes MSE, RMSE, MAE, MAPE, R², adjusted R², MBE, SS_res, and SS_tot from actual vs predicted pairs. Use the editable table or two textareas for input. The actual vs predicted scatter shows how well predictions match reality; the residuals plot reveals patterns; the error comparison chart visualizes all metrics together.
Disclaimer: MAPE is undefined when actual values are zero. Use with caution for such data. R² and adjusted R² assume a linear relationship structure.