Mean Absolute Percentage Error (MAPE)

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

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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.

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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.

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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.

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5) Limitations

1. Division by zero: If any $y_i = 0$, the formula breaks.
   • Fixes: add a small constant, or use sMAPE (Symmetric MAPE).
2. Asymmetry: Over-predictions and under-predictions can have uneven effects.
3. Bias towards small actuals: When actual $y_i$​ is small, even tiny absolute errors produce very large percentage errors.
4. 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.

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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.

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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.

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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.