Enter Data
Formula Used
For paired values \(y_i\) (measured) and \(\hat{y}_i\) (predicted), the error is \(e_i = \hat{y}_i - y_i\) (or \(e_i = y_i - \hat{y}_i\) if you choose that definition).
With optional weights \(w_i\), the weighted mean squared error is:
Common related metrics shown here:
- RMSE = √MSE (same unit as the values)
- MAE = ( Σ ( wᵢ · |eᵢ| ) ) / ( Σ wᵢ )
- Bias = ( Σ ( wᵢ · eᵢ ) ) / ( Σ wᵢ ) (signed average error)
How to Use This Calculator
- Enter your measured values and predicted values in the table.
- Optionally enable weights for repeated trials or confidence levels.
- Select the error definition that matches your convention.
- Click the calculate button to display results above the form.
- Use CSV or PDF buttons to export the summary and table.
Example Data Table
Example: comparing measured acceleration with a model prediction.
| # | Measured | Predicted | Weight |
|---|---|---|---|
| 1 | 9.81 | 9.79 | 2 |
| 2 | 3.20 | 3.40 | 1 |
| 3 | 1.50 | 1.55 | 1 |
| 4 | 0.75 | 0.70 | 3 |
Mean Squared Error in Physics Workflows
1) Why MSE is a practical accuracy score
Mean squared error summarizes how far predictions deviate from measurements. Squaring emphasizes larger deviations, which often dominate physical risk. In calibration and model tuning, lowering MSE usually improves overall fit.
2) Units, scaling, and interpretability
MSE carries squared units, such as (m/s)² or V². RMSE converts back to the original unit, aiding intuition. When comparing different signals, normalize first, or compare RMSE. As an example, an RMSE of 0.12 V on a 5 V range is 2.4%. If you rescale inputs by a factor of 10, MSE scales by 100.
3) Connecting MSE to noise variance
If a model is unbiased and errors are random, MSE approaches the error variance plus any systematic components. A small bias can inflate MSE when the same offset persists. This calculator reports bias alongside MSE for quick diagnosis.
4) When weights improve experimental summaries
Weights help represent repeated trials, confidence scores, or sensor quality. For example, two repeated measurements can use weight 2, while a low-confidence point can use weight 0.5. The weighted formula keeps the interpretation as an average squared error.
5) Sampling, outliers, and squared penalties
Squaring makes MSE sensitive to outliers. If a single point is off by 5 units, it contributes 25 units² to the total, while a 1-unit error contributes only 1 unit². Inspect the error table to spot dominating rows. If outliers come from known glitches, fix the source first. Otherwise, compare MAE, or apply robust filtering before scoring.
6) Comparing MSE with MAE in practice
MAE grows linearly with error magnitude and is more robust. Many physics datasets benefit from checking both: MSE highlights occasional large misses, while MAE tracks typical deviation under noisy conditions.
7) Model selection and parameter sweeps
During parameter sweeps, compute MSE for each setting, then choose the minimum under your constraints. If overfitting is a concern, split data into training and validation. A lower validation MSE indicates better generalization.
8) Reporting results clearly
For reports, include RMSE with units, the number of pairs, and bias. Also list the maximum absolute error when safety matters. Exporting CSV supports lab notebooks, while PDF suits quick sharing.
FAQs
1) What does a lower MSE mean?
A lower MSE means predictions stay closer to measured values on average, with larger mistakes penalized more strongly. It generally indicates improved fit, assuming the dataset and scaling remain consistent.
2) Should I report MSE or RMSE?
RMSE is often easier to interpret because it uses the same unit as your data. MSE is useful for optimization and theory, but RMSE communicates typical error magnitude more directly.
3) Why is MSE sensitive to outliers?
Because errors are squared, a few large deviations can dominate the average. Use the error table to find those points, or compare MAE to judge typical performance.
4) When should I enable weights?
Enable weights when some points represent repeated trials, higher confidence, or higher priority. Larger weights increase their influence on MSE, while smaller weights reduce the impact of uncertain measurements.
5) Do I need the same units for measured and predicted?
Yes. Measured and predicted values must represent the same quantity and unit. Otherwise, the error is meaningless and the computed MSE and RMSE will not reflect physical accuracy.
6) What does bias tell me here?
Bias is the average signed error. A nonzero bias suggests systematic offset, such as calibration drift or model misalignment, even if MSE looks acceptable.
7) How many data pairs should I use?
Use enough pairs to represent your operating range and noise conditions. More points usually stabilizes MSE, but ensure the set includes typical and edge-case regimes for the experiment.