1) Definition
MAPE measures the average size of prediction errors as a percentage of the actual values. It expresses model accuracy in a scale-free, intuitive way (percent error).
Formula:
$\text{MAPE} = \frac{100\%}{n} \sum_{i=1}^{n} \left| \frac{y_i – \hat{y}_i}{y_i} \right|$
- $y_i$ = actual value
- $\hat{y}_i$ = predicted value
- $n$ = number of observations
2) Interpretation
- Output: a percentage (%).
- Example: A MAPE of 8% means predictions are off by 8% on average.
- Lower is better: Smaller percentage error = better model fit.
3) Example
Suppose we predict product demand:
| Observation | Actual ($y$) | Predicted ($\hat{y}$) | Error $|y-\hat{y}|$ | % Error = $\frac{\lvert y – \hat{y} \rvert}{y} \times 100\%$ |
|---|---|---|---|---|
| 1 | 100 | 90 | 10 | 10% |
| 2 | 200 | 220 | 20 | 10% |
| 3 | 400 | 360 | 40 | 10% |
$\text{MAPE} = \frac{10\% + 10\% + 10\%}{3} = 10\%$
Interpretation: On average, the model is 10% off from actual demand.
4) Strengths
- Scale-independent: Can compare accuracy across datasets with different scales.
- Easy to interpret: Percentages are intuitive for business stakeholders.
- Directly linked to KPIs: Helps businesses understand error impact in relative terms.
5) Limitations
- Division by zero: If any $y_i = 0$, the formula breaks.
- Fixes: add a small constant, or use sMAPE (Symmetric MAPE).
- Asymmetry: Over-predictions and under-predictions can have uneven effects.
- Bias towards small actuals: When actual $y_i$ is small, even tiny absolute errors produce very large percentage errors.
- Not always aligned with business cost: A 10% error in predicting 10 units vs 10% error in predicting 10,000 units may not have the same real-world impact.
6) Variants
- sMAPE (Symmetric MAPE):
- $\text{sMAPE} = \frac{100\%}{n} \sum_{i=1}^{n} \frac{|y_i – \hat{y}_i|}{(|y_i| + |\hat{y}_i|)/2}$
- Fixes the zero-division problem and makes errors more balanced.
- WAPE (Weighted Absolute Percentage Error):
- $\text{WAPE} = \frac{\sum |y_i – \hat{y}_i|}{\sum |y_i|}$
- Weighted by actual demand volume, so big items matter more.
7) When to use
- Good for forecasting tasks (demand prediction, sales forecasting, energy consumption).
- Useful when communicating with non-technical stakeholders, since percentage errors are easy to understand.
- Best when there are no zeros or near-zeros in the data.
Summary:
MAPE is the average percentage error between predicted and actual values. It’s intuitive and scale-free but has issues with zeros and small denominators. Alternatives like sMAPE and WAPE are often used in practice to address these problems.