Mean Absolute Error (MAE) Calculator

Online calculator for computing the Mean Absolute Error (MAE)

MAE Calculator

The Mean Absolute Error

The mean absolute error (MAE) is a statistical measure used to assess the accuracy of predictions.

Enter Predicted and Observed Values
Predicted values (space separated)
Actually observed values (space separated)
MAE Results
Mean Absolute Error:
MAE Properties

Range: MAE is always ≥0, with0 indicating perfect predictions

MAE ≥0 Robust to outliers Same unit as data

MAE Concept

MAE measures the average absolute deviation between predictions and observations.
The smaller the MAE, the more accurate the predictions.

y x

● Observed values ● Predicted values ● Absolute errors


What is the Mean Absolute Error?

The mean absolute error (MAE) is a fundamental evaluation measure for predictive models:

  • Definition: Average of the absolute differences between predictions and observations
  • Range: MAE ≥0, with0 indicating perfect predictions
  • Unit: Same unit as the original data
  • Application: Machine learning, forecasting, quality control
  • Robustness: Less sensitive to outliers than RMSE
  • Interpretation: Direct meaning as average error

MAE Properties

The MAE possesses important statistical properties:

Mathematical Properties
  • Non-negativity: MAE ≥0
  • Zero point: MAE =0 ⟺ perfect predictions
  • Linearity: All errors are weighted equally
  • Symmetry: |a-b| = |b-a|
Statistical Properties
  • Robustness: Less sensitive to outliers
  • Interpretability: Direct meaning in original unit
  • Median property: Minimized by the median
  • L¹-norm: Based on the Manhattan distance

Applications of MAE

The MAE is applied in many fields:

Machine Learning
  • Regression models: Assess prediction accuracy
  • Model comparison: Select the best algorithm
  • Cross-validation: Hyperparameter tuning
  • Time series: Evaluate forecasts
Business & Finance
  • Demand forecasting: Predict sales
  • Budget planning: Deviation analysis
  • Risk models: Volatility forecasts
  • Portfolio management: Performance evaluation
Natural Sciences
  • Weather forecasting: Temperature and precipitation predictions
  • Climate models: Evaluate long-term forecasts
  • Medicine: Diagnostic tests, treatment outcomes
  • Physics: Experimental vs. theoretical values
Industry & Engineering
  • Quality control: Check manufacturing tolerances
  • Energy management: Consumption forecasts
  • Logistics: Delivery times and routing
  • Maintenance: Predictive maintenance

Formulas for MAE

Basic Formula
\[MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|\]

Average of absolute differences

L¹-Norm Representation
\[MAE = \frac{\|y - \hat{y}\|_1}{n}\]

Normalized Manhattan distance

Vector Form
\[MAE(\mathbf{y}, \hat{\mathbf{y}}) = \frac{1}{n} \sum_{i=1}^{n} |e_i|\]

With error vector e = y - ŷ

Weighted Form
\[WMAE = \frac{\sum_{i=1}^{n} w_i |y_i - \hat{y}_i|}{\sum_{i=1}^{n} w_i}\]

With weights w_i

Relation to SAD
\[MAE = \frac{SAD}{n} = \frac{\sum_{i=1}^{n} |y_i - \hat{y}_i|}{n}\]

SAD: Sum of Absolute Deviations

Percentile Property
\[MAE = \text{Median}(|y_i - \hat{y}_i|)\]

For symmetric errors

Example Calculation for MAE

Given
Predictions: [2,4,6,8,10] Observations: [1.8,4.2,5.5,8.3,9.7]

Calculate: Mean Absolute Error (MAE) of the predictions

1. Compute Absolute Errors
\[|2 -1.8| =0.2\] \[|4 -4.2| =0.2\] \[|6 -5.5| =0.5\] \[|8 -8.3| =0.3\] \[|10 -9.7| =0.3\]

Element-wise absolute differences

2. Sum of Absolute Errors
\[SAD =0.2 +0.2 +0.5 +0.3 +0.3\] \[SAD =1.5\]

Sum of Absolute Deviations

3. Compute MAE
\[MAE = \frac{SAD}{n} = \frac{1.5}{5} =0.3\]

Average of absolute errors

4. Comparison with RMSE
\[RMSE = \sqrt{\frac{0.04 +0.04 +0.25 +0.09 +0.09}{5}}\] \[RMSE = \sqrt{0.102} =0.32\]

Root Mean Square Error for comparison

5. Interpretation
MAE =0.30 Average error =30%
RMSE =0.32 Similar to MAE

The predictions have an average absolute error of0.3 units

6. Practical Meaning

Model quality: An MAE of0.3 for values between2-10 corresponds to about3-15% relative error. This indicates a reasonably good model. The closeness of MAE (0.30) and RMSE (0.32) shows that no extreme outliers are present.

Mathematical Foundations of MAE

The mean absolute error (MAE) is a fundamental evaluation measure in statistics and machine learning. It quantifies the average magnitude of errors in a set of predictions without considering their direction.

Definition and Basic Properties

MAE is characterized by its mathematical simplicity and interpretability:

  • L¹-norm basis: Based on the Manhattan distance (L¹-norm) between prediction and observation vectors
  • Linear loss: All errors are weighted equally, independent of their size
  • Robustness: Less sensitive to outliers than quadratic loss functions
  • Interpretability: Direct meaning in original data units
  • Non-negativity: Always ≥0, with0 indicating perfect predictions

Comparison with Other Error Measures

MAE is related to other common evaluation measures:

Root Mean Square Error (RMSE)

RMSE = √(MSE) penalizes large errors more heavily than MAE. The RMSE/MAE ratio indicates error distribution.

Mean Squared Error (MSE)

MSE = (1/n)Σ(y-ŷ)² is quadratic and therefore more sensitive to outliers than MAE.

Mean Absolute Percentage Error (MAPE)

MAPE = (100/n)Σ|((y-ŷ)/y)| normalizes errors relative to observed values.

Huber Loss

Combines MAE and MSE: quadratic for small, linear for large errors. Compromise between both.

Statistical Properties

MAE has important statistical properties:

Median property

MAE is minimized by the median of residuals, not the mean (as with MSE). This makes it robust to outliers.

Convexity

MAE is a convex function, which ensures optimization algorithms can find global minima.

Scaling behavior

MAE scales linearly with the data: doubling all values doubles the MAE.

Non-differentiability

MAE is non-differentiable at y=ŷ, which requires specialized optimization methods.

Applications in Machine Learning

MAE plays a central role in various ML areas:

Regression

As a loss function in robust regression models, especially with outlier-prone data.

Time series analysis

Evaluate forecasting methods, particularly for seasonal or trend-based data.

Model selection

Compare different algorithms, especially for heterogeneous error distributions.

Cross-validation

Robust assessment of model generalization ability.

Advantages and Disadvantages

Advantages
  • Interpretability: Direct meaning in original units
  • Robustness: Less sensitive to outliers
  • Linearity: All errors are treated equally
  • Simplicity: Easy to compute and understand
  • Scalability: Works well for different data sizes
Disadvantages
  • Non-differentiability: Complicates gradient-based optimization
  • Large errors: Penalizes large errors less than RMSE
  • Median tendency: Tends to predict the median
  • No direction information: Positive and negative errors cancel in sign-insensitive measure

Practical Considerations

Choice of error measure

Use MAE when robustness to outliers is more important than penalizing large errors.

Data preprocessing

For highly skewed distributions, transforming data can improve MAE performance.

Optimization

Use subgradient methods or smoothed MAE variants for gradient-based optimization.

Combined approaches

Huber Loss or other combinations of MAE and MSE for balanced error treatment.

Summary

The mean absolute error (MAE) is a robust and interpretable evaluation measure that is particularly suitable when outlier resistance matters more than heavily penalizing large errors. Its mathematical simplicity combined with practical relevance makes it a standard tool in statistics and machine learning. The choice between MAE and other error measures should be made in the context of the specific application and the desired model properties.

Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?

More Statistics Functions

Dice Coefficient  •  Hellinger Distance  •  Jaccard Index  •  MAE - mean absolute error  •  MSE - mean squared error  •  SAD - sum of absolute difference  •  SSD - sum of squared difference  •