STATISTICSDescriptive StatisticsStatistics Calculator
📊

MSE Calculator (Mean Squared Error)

Free MSE calculator. Compute MSE, RMSE, MAE, MAPE, R², adjusted R², MBE. Actual vs predicted. Regres

Run CalculatorExplore data analysis and statistical calculations

Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

MSE
STATISTICSRegression & Forecast Metrics

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.

Linear Regression →R² Calculator →

Real-World Scenarios — Click to Load

#Actual (y)Predicted (ŷ)
1
2
3
4
5
mse_results.sh
CALCULATED
$ mse_calc --n=5 --p=1
MSE
0.022000
RMSE
0.148324
MAE
0.140000
MAPE
3.09%
R²
0.9962
Adj. R²
0.9950
MBE
0.020000
SS_res
0.1100
SS_tot
29.2000
Share:
MSE Calculator Result
Regression Error Metrics (n = 5)
RMSE = 0.1483
MSE: 0.0220MAE: 0.1400MAPE: 3.09%R²: 0.9962
numbervibe.com/calculators/statistics/mse-calculator

Actual vs Predicted (green line = perfect fit y=x)

Error Metrics Comparison

Residuals Plot (actual − predicted)

Calculation Breakdown

INPUT
Sample size n
5
Number of (actual, predicted) pairs
COMPUTATION
SS_res (residual sum of squares)
0.110000
Σ(yᵢ − ŷᵢ)² = 0.01 + 0.01 + 0.04 + ...
COMPUTATION
SS_tot (total sum of squares)
29.200000
Σ(yᵢ − ȳ)²
MSE
0.022000
MSE = SS_res/n = 0.1100/5
RMSE
0.148324
RMSE = √MSE = √0.022000
MAE
0.140000
MAE = Σ|yᵢ−ŷᵢ|/n = 0.7000/5
MAPE
3.09%
(100/n) Σ |yᵢ−ŷᵢ|/|yᵢ|
R²
0.9962
R² = 1 − SS_res/SS_tot = 1 − 0.1100/29.2000
ADDITIONAL
Adjusted R²
0.9950
1 − (1−R²)(n−1)/(n−p−1), p=1
ADDITIONAL
MBE (Mean Bias Error)
0.020000
Σ(ŷᵢ−yᵢ)/n — Over-prediction

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?

📊RMSE is in the same units as your target. If predicting dollars, RMSE is in dollars.Source: NIST Handbook
📐R² = 1 means perfect fit. R² = 0 means the model explains no variance. R² can be negative for bad models.Source: Wikipedia
📈MAPE is undefined when actual values are zero. Use sMAPE or other metrics for such data.Source: Forecasting literature
🎯MBE > 0: model systematically over-predicts. MBE < 0: under-predicts.Source: Bias-variance tradeoff
📉MSE is the loss function minimized by ordinary least squares (OLS) regression.Source: Gauss-Markov theorem
🔢Adjusted R² = 1 − (1−R²)(n−1)/(n−p−1). Always ≤ R² when p ≥ 0.Source: Penn State STAT 462

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.

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.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Relative Error Calculator

Compute relative error, percentage error, absolute error, MAE, MRE, RMSE, and symmetric relative error. Used in science, engineering, and numerical analysis.

Statistics

Pearson Correlation Calculator

Dedicated Pearson r calculator with step-by-step computation, hypothesis test, confidence interval, scatter plot, and interpretation.

Statistics

Constant of Proportionality Calculator

Find the constant k in y = kx (direct) or y = k/x (inverse) from data. Test whether data follows a proportional relationship with R² goodness of fit.

Statistics

Correlation Coefficient Calculator

Compute Pearson's r, Spearman's ρ, and Kendall's τ from paired data. Scatter plot, regression line, residuals, p-value, 95% confidence interval, and R².

Statistics

Average Rating Calculator

Compute weighted average rating from star ratings. Bayesian average, Wilson score confidence interval, distribution, mode, median, and standard deviation.

Statistics

Box Plot Calculator

Generate box-and-whisker plots from data. Quartiles, IQR, fences, outlier detection with Tukey method.

Statistics