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Mean absolute percentage error

The mean absolute percentage error (MAPE) is a widely used statistical metric for assessing the accuracy of forecasting models in fields such as time series analysis, econometrics, and operations research, representing the average magnitude of errors as a percentage of the actual values.^[1] It is formally defined by the formula

\text{MAPE} = \frac{100}{n} \sum_{t=1}^{n} \left| \frac{A_t - F_t}{A_t} \right|

where n denotes the number of observations, A_t is the actual value at period t, and F_t is the corresponding forecasted value.^[1] This formulation ensures that errors are normalized relative to the actual observations, yielding a scale-independent measure that facilitates comparisons across datasets with varying units or magnitudes.^[2] MAPE gained prominence through empirical studies like the M-competitions, which evaluated forecasting methods and highlighted its interpretability as a relative error in percentage terms, often ranging from 0% (perfect accuracy) to higher values indicating poorer performance.^[1] Despite its popularity in business and academic applications for its intuitive output, MAPE has notable drawbacks: it becomes undefined or infinite when actual values are zero, introduces asymmetry by penalizing over-forecasts more severely than under-forecasts of the same absolute magnitude, and can be overly sensitive to small actual values, leading to unstable results in intermittent or low-volume data scenarios.^[3] These limitations have prompted the development of alternatives, such as the symmetric mean absolute percentage error (sMAPE) or mean absolute scaled error (MASE), which address bias and scale issues while retaining relative interpretability.^[1]

Definition and Basic Concepts

Mathematical Definition

The mean absolute percentage error (MAPE) is a measure of prediction accuracy that expresses the average absolute error as a percentage of the actual values. It is defined mathematically as

\text{MAPE} = \frac{100}{n} \sum_{i=1}^{n} \left| \frac{A_i - F_i}{A_i} \right|,

where A_i represents the actual value for the i-th observation, F_i is the corresponding forecasted or predicted value, and n is the total number of observations.^[4] The absolute value in the formula ensures that the error is always non-negative, regardless of whether the forecast over- or underestimates the actual value, while the division by A_i normalizes the error relative to the actual value. The sum of these relative errors is then averaged across all n observations and scaled by 100 to express the result as a percentage. This formulation assumes all A_i > 0 to avoid division by zero.^[4] To illustrate the computation, consider a small dataset with two observations: actual values A_1 = 100, A_2 = 200; forecasted values F_1 = 110, F_2 = 180.

For the first observation: \left| \frac{100 - 110}{100} \right| = 0.10.
For the second observation: \left| \frac{200 - 180}{200} \right| = 0.10.
Sum of absolute relative errors: $0.10 + 0.10 = 0.20.
Average: \frac{0.20}{2} = 0.10.
MAPE: $100 \times 0.10 = 10\%.

This percentage represents the averaged relative error across the dataset.^[4]

Interpretation

The mean absolute percentage error (MAPE) serves as a scale-independent measure of prediction accuracy, expressing errors as percentages relative to the actual values, which facilitates comparisons across datasets or models involving different units or magnitudes.^[5] This relative scaling allows forecasters to evaluate performance without being influenced by the absolute size of the data, making it particularly valuable in diverse applications like economics or operations research.^[6] Due to its percentage-based nature, MAPE emphasizes errors proportional to the actual observations; for instance, an absolute error of 5 units represents a 50% deviation when the actual value is 10, but only a 5% deviation when the actual value is 100, highlighting how small absolute discrepancies can be more significant in low-value contexts. This focus on proportionality provides intuitive interpretability for stakeholders, as it aligns errors with practical impacts on the underlying scale of the data. Common benchmarks for interpreting MAPE values include: less than 10% indicating highly accurate predictions, 10–20% suggesting good accuracy, 20–50% denoting reasonable performance, and above 50% signaling inaccuracy, though these thresholds should be contextualized by industry standards and data characteristics such as intermittency or trend strength.^[7] These guidelines, while widely referenced, underscore the need for caution, as MAPE's sensitivity to low actual values can inflate errors in certain scenarios, potentially misrepresenting overall model quality. MAPE emerged in the mid-20th century, particularly during the 1950s, as a key metric in inventory control and sales forecasting, coinciding with advancements in statistical methods like exponential smoothing that required scalable error assessment for operational decision-making.^[8]

Properties

Consistency

In statistical estimation, the mean absolute percentage error (MAPE) serves as a consistent estimator of the population mean absolute percentage error, converging in probability to the true expected relative error as the sample size n \to \infty. This property holds under the assumption of independent and identically distributed (i.i.d.) observations where the actual values A_i > 0 for all i, ensuring the individual percentage error terms \left| \frac{A_i - F_i}{A_i} \right| are well-defined and non-negative.^[9] The proof outline leverages the law of large numbers, which applies directly to the sample average of these bounded or integrable relative error terms, yielding convergence to their expectation \mathbb{E}\left[ \left| \frac{A - F}{A} \right| \right], where F denotes the forecast or predicted value. Consistency requires that the actual values remain positive and that the errors possess finite variance to ensure the stability of the average under the i.i.d. framework. In linear regression settings, the empirical risk minimization (ERM) estimator optimized for the MAPE loss function achieves consistent estimation of the parameters that minimize the population mean absolute percentage loss, demonstrating the estimator's reliability for relative error assessment. This universal consistency of ERM for MAPE holds with minimal distributional assumptions on the input-output pairs, relying on techniques such as uniform laws of large numbers and exponential bounding for the theoretical guarantees.^[9]

Statistical Properties

The mean absolute percentage error (MAPE) displays a bias towards under-forecasting, particularly when actual values vary or are low. This bias stems from the asymmetric treatment of over- and under-forecasts in the percentage error calculation, where an absolute error of the same magnitude results in a larger percentage error for over-forecasts relative to under-forecasts, especially when actual values are small. Consequently, minimizing MAPE incentivizes conservative predictions that err on the side of underestimation to avoid severe penalties from over-predictions during periods of low actuals. This effect is theoretically demonstrated in economic forecasting contexts, where asymmetric loss functions based on MAPE lead forecasters to systematically bias point predictions downward.^[10] Hyndman and Koehler further emphasize this asymmetry, noting that MAPE penalizes over-forecasts more heavily than under-forecasts of equal absolute size, exacerbating the bias in datasets with variable scales or low values.^[11] The division by the actual value A_t in MAPE's formulation introduces additional variability compared to absolute error metrics like the mean absolute error (MAE), as fluctuations in A_t can amplify the relative impact of errors, particularly in heterogeneous datasets. This results in MAPE exhibiting higher variance overall, making it less stable for comparing accuracy across series with differing scales or intermittency. The metric's sensitivity to low A_t values contributes to this increased variability, as small denominators magnify even modest absolute errors into large percentage deviations.^[11] In comparison to other error metrics, MAPE's reliance on absolute differences renders it non-differentiable at zero error, complicating gradient-based optimization in machine learning applications where closed-form minimizers are unavailable, unlike squared error metrics. This non-differentiability necessitates subgradient methods or approximations for training models that minimize MAPE directly, potentially increasing computational demands relative to differentiable alternatives like mean squared percentage error.

Variants

Weighted MAPE

The weighted mean absolute percentage error (WMAPE) is a variant of the mean absolute percentage error that incorporates weights to adjust for the relative importance of individual observations in the dataset. It modifies the standard MAPE by applying weights w_i to each term in the summation, allowing forecasters to prioritize errors associated with more significant data points, such as those with higher actual values or business impact.^[12] The formula for WMAPE is:

\text{WMAPE} = \frac{100}{\sum_i w_i} \sum_i w_i \left| \frac{A_i - F_i}{A_i} \right|

where A_i represents the actual value, F_i the forecasted value, and w_i the weight for the i-th observation, assuming all A_i > 0.^[12]^[13] This metric addresses the imbalance inherent in the standard MAPE, which equally weighs percentage errors across all observations and can undervalue inaccuracies in high-volume items while overemphasizing minor errors in low-volume ones. By emphasizing high-volume items through appropriate weights, WMAPE provides a more balanced assessment of forecasting performance in scenarios where observation scale varies significantly, such as in demand planning.^[14]^[15] Common weighting schemes include w_i = A_i, which is proportional to the actual values and effectively computes the total absolute error divided by the total actuals (often expressed as \sum |A_i - F_i| / \sum A_i \times 100), thereby giving greater influence to larger observations; or w_i = 1/A_i for inverse weighting, which amplifies the relative impact of errors in smaller observations.^[12]^[13] In sales data forecasting, for instance, weighting by sales volume (w_i = A_i) assigns more importance to errors in predicting large transactions, ensuring the metric better reflects overall business risk from forecast inaccuracies.

Symmetric MAPE

The symmetric mean absolute percentage error (sMAPE) addresses a key limitation of the standard MAPE by providing a more balanced treatment of over- and under-forecasts of the same absolute magnitude. Unlike the standard MAPE, which exhibits asymmetry by penalizing over-forecasts more severely than under-forecasts due to its reliance solely on the actual value in the denominator—for example, an absolute error of 50 yields 33.3% when under-forecasting (actual=150, forecast=100) but 50% when over-forecasting (actual=100, forecast=150)—sMAPE symmetrizes the calculation. This variant was proposed by Spyros Makridakis in 1993 as part of efforts to improve accuracy measures for forecasting evaluations.^[16] The formula for sMAPE is given by

\text{sMAPE} = \frac{100}{n} \sum_{i=1}^{n} \frac{|A_i - F_i|}{\frac{|A_i| + |F_i|}{2}},

where A_i represents the actual value, F_i the forecast, and n the number of observations.^[16] This formulation averages the absolute values of the actual and forecast in the denominator, creating a midpoint that mitigates the directional bias present in MAPE. By doing so, sMAPE ensures that the relative error is computed relative to a neutral reference point between the two values, making it less sensitive to whether the forecast over- or under-shoots the actual—in the example above, both cases yield 40%. A primary advantage of sMAPE is its ability to handle cases where actual values are zero without becoming undefined or infinite (yielding 200% if the forecast is positive), unlike MAPE. For zero forecasts with positive actuals, sMAPE yields 200%, compared to MAPE's 100%. To illustrate the near-symmetry for proportional deviations, consider an actual value of 100 with a forecast of 110 (over-forecast): the individual error is $100 \times 10 / 105 \approx 9.52\%. For the same actual but a forecast of 90 (under-forecast): $100 \times 10 / 95 \approx 10.53\%. These values are close, highlighting how sMAPE treats comparable proportional deviations more equitably in certain contexts than MAPE, which would score both at exactly 10%. This property has made sMAPE a popular choice in forecasting competitions and applications requiring fair error assessment across directions.

Applications

Forecasting

In time series forecasting, the mean absolute percentage error (MAPE) serves as a key metric for assessing the accuracy of predictions generated by models such as ARIMA and exponential smoothing. These models, which capture trends, seasonality, and other temporal patterns in data, rely on MAPE to quantify the average relative deviation between forecasted and actual values, enabling forecasters to evaluate how well the model performs on out-of-sample data. For instance, in exponential smoothing methods, MAPE is computed alongside other errors like mean absolute error (MAE) to provide a percentage-based view of forecast reliability, particularly useful for short- to medium-term predictions in stable series.^[17]^[18] MAPE finds extensive application in industries like supply chain management and finance, where accurate predictions directly impact operations. In supply chain contexts, MAPE helps compare models like Holt-Winters exponential smoothing against ARIMA by measuring relative errors in demand estimates. Similarly, in finance, MAPE evaluates stock price forecasting models, including machine learning approaches, to gauge prediction accuracy for market indices, where low MAPE values indicate robust performance in volatile environments.^[19] One of MAPE's primary advantages in forecasting is its expression as a percentage, which facilitates intuitive communication of error magnitudes to non-technical stakeholders, such as managers in demand planning. Additionally, being scale-free, MAPE allows for consistent comparisons of model performance across diverse time series, like products with varying demand volumes, without bias from absolute units. This makes it particularly valuable for selecting models in heterogeneous datasets.^[20] Historically, MAPE gained widespread adoption in operations research for demand planning in the early 1980s, with its use solidified through forecasting competitions like the Makridakis M-competition, which benchmarked methods using percentage error metrics to advance practical applications.^[21]^[22]

Regression Analysis

In regression analysis, the mean absolute percentage error (MAPE) serves as a loss function to emphasize relative errors during model optimization, particularly in generalized linear models where the focus is on proportional accuracy for positive-valued outcomes. By minimizing MAPE, models prioritize predictions that scale appropriately with the magnitude of the target variable, such as in scenarios involving economic indicators or demand estimation. For implementation, libraries like scikit-learn allow custom loss functions incorporating MAPE, enabling its use in training linear or nonlinear regressors beyond standard mean squared error (MSE) objectives.^[23]^[24] As an evaluation metric, MAPE provides a post-fit assessment of model performance by quantifying average relative deviations, making it suitable for responses like prices, counts, or quantities where absolute errors may mislead due to varying scales. This is especially valuable in fields requiring intuitive percentage-based interpretations, as MAPE expresses errors in familiar terms without unit dependency. In practice, it complements other metrics to offer a balanced view of predictive reliability for non-negative targets.^[24]^[23] Compared to MSE, which penalizes larger absolute errors more heavily and remains scale-dependent, MAPE is preferred in contexts where relative accuracy is paramount, such as econometrics, to avoid biases from disparate data magnitudes. For instance, MSE might undervalue improvements in high-value predictions, whereas MAPE ensures equitable treatment across ranges, facilitating cross-model or cross-dataset comparisons. This relative focus aligns with econometric reporting needs, where percentage errors enhance interpretability for stakeholders.^[25]^[23] A representative case study involves housing price regression using datasets like the Kaggle Ames Housing competition, where MAPE evaluates models such as linear regression and random forest by scaling errors to property values. In one analysis, random forest achieved a lower MAPE than linear regression, highlighting its superior handling of nonlinear feature interactions in price predictions and underscoring MAPE's role in assessing practical accuracy for real estate applications.^[26]

Limitations

Undefined Cases

The mean absolute percentage error (MAPE) becomes undefined when any actual value A_i is zero, as the formula involves division by A_i in the denominator, resulting in division by zero.^[6] This computational failure renders the entire metric incalculable for the dataset, often leading to infinite values in practical implementations if not addressed.^[27] Such undefined cases are particularly prevalent in intermittent demand forecasting, where actual values frequently include zeros due to sporadic sales patterns, such as in inventory management for low-volume or seasonal products.^[6] In these scenarios, zero actuals can constitute a substantial portion of the data, making MAPE unreliable without modifications.^[6] Common workarounds include adding a small positive constant, known as epsilon (e.g., 0.1 or a value like $10^{-8}), to the denominator to prevent division by zero, though this introduces a minor bias that depends on the choice of epsilon.^[28]^[29] Alternatively, observations with zero actual values can be excluded from the MAPE calculation, with errors for those periods reported separately (e.g., using absolute error metrics), ensuring the percentage-based assessment applies only to non-zero cases.^[30] Mishandling these cases, such as treating zero actuals as infinite errors, can severely inflate the overall MAPE and skew performance evaluations, particularly in datasets dominated by intermittency.^[27] This distortion undermines MAPE's utility as a consistent accuracy measure in such contexts.^[6]

Interpretability Issues

The mean absolute percentage error (MAPE) suffers from inherent asymmetry in its error penalization, where over-forecasts (when the forecast F exceeds the actual value A) receive harsher treatment than under-forecasts of equivalent absolute magnitude. This occurs because the relative error formula |F - A| / A amplifies positive deviations more than negative ones; for an absolute error of 50 units, an over-forecast (A=100, F=150) yields 50% error, while an equivalent under-forecast (A=150, F=100) yields approximately 33.3%. Such bias incentivizes models that systematically under-forecast to minimize the metric, potentially leading to suboptimal decisions in applications like inventory management. This issue was first systematically critiqued by Makridakis (1993).^[16] MAPE also demonstrates a pronounced low-volume bias, where small actual values (A_i \approx 0) cause even trivial forecast errors to produce inflated relative errors, heightening sensitivity to outliers or near-zero observations. In datasets featuring intermittent demand—common in supply chain forecasting—this amplification distorts overall accuracy assessments, as sporadic low-activity periods dominate the average. For instance, a forecast error of 1 unit against an actual of 2 yields a 50% error, far outweighing errors in high-volume periods, thus rendering MAPE unreliable for heterogeneous or sparse series. Empirical analyses of intermittent demand patterns confirm this vulnerability, emphasizing how it skews evaluations toward conservative predictions in low-demand scenarios.^[6] Beyond these structural flaws, MAPE's expression as a percentage facilitates communication pitfalls, particularly for non-experts who may overemphasize apparent magnitudes without considering contextual scale. A 50% error might sound catastrophic but could represent a minor absolute deviation in low-base scenarios, while a 10% error on large volumes implies substantial practical impact; this disconnect can mislead stakeholders in business reporting or policy discussions. Such interpretive challenges arise because percentages imply uniformity across scales, fostering misperceptions of risk or performance.^[31] Empirical studies from the late 1990s and 2000s further illustrate how MAPE overstates errors in volatile time series, where irregular fluctuations exacerbate the metric's skewness from outliers. In evaluations of population projections, MAPE was found to inflate typical error representations due to its sensitivity to extreme values in heterogeneous datasets, leading to overly pessimistic accuracy portrayals.^[32] Post-competition analyses of the M3 forecasting competition highlighted MAPE's tendency to exaggerate discrepancies in volatile economic and demographic series, prompting calls for more robust alternatives in high-variability contexts.^[33]

Alternatives

Mean Absolute Scaled Error

The Mean Absolute Scaled Error (MASE) serves as a scale-independent metric for evaluating forecast accuracy, proposed by Hyndman and Koehler in 2006^[34] to provide a robust alternative to percentage error measures in time series forecasting. This approach scales the forecast errors relative to a simple benchmark, enabling consistent comparisons across datasets with varying units or magnitudes without the biases inherent in absolute or relative errors. The formula for MASE in non-seasonal time series is given by

\text{MASE} = \frac{\frac{1}{h} \sum_{i=1}^{h} |e_i|}{\frac{1}{n-1} \sum_{j=2}^{n} |A_j - A_{j-1}|},

where e_i represents the forecast error at time i, h is the number of forecast periods, the numerator is the mean absolute forecast error, and the denominator is the mean absolute error of a one-step naive forecast applied in-sample to the training data with n observations (where A_j for j=1 to n are the training actual values). The naive forecast uses the random walk model where each prediction equals the previous observation. For seasonal series, the scaling benchmark adjusts to the seasonal naive forecast, repeating the pattern from the prior season, though the core structure remains analogous. MASE offers key advantages over the Mean Absolute Percentage Error (MAPE), as it avoids division by actual values, thereby naturally handling zero or near-zero observations without producing undefined or infinite results. This makes it particularly suitable for intermittent demand data, where MAPE can introduce severe bias due to sporadic zeros, while MASE remains stable and unbiased by scaling against the naive benchmark. Additionally, its unit-free nature facilitates direct comparisons of forecasting methods across diverse series, promoting standardized evaluation in robust forecasting practices. In practice, a MASE value of 1 indicates forecast performance equivalent to the in-sample naive method, while values below 1 signify improvement over this baseline; for instance, in seasonal data, a MASE less than 1 demonstrates superiority to the seasonal naive forecast.

Root Mean Squared Error

The Root Mean Squared Error (RMSE) serves as a quadratic alternative to the Mean Absolute Percentage Error (MAPE), emphasizing absolute deviations in predictions rather than relative percentages, which makes it particularly suitable for evaluating forecast accuracy in scenarios where the scale of errors matters directly.^[35] The formula for RMSE is:

\text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (A_i - F_i)^2}

where A_i represents the actual values, F_i the forecasted values, and n the number of observations; this expression calculates the square root of the average squared errors, yielding a metric in the same units as the original data, unlike MAPE's unitless percentage form.^[36]^[37] Compared to MAPE, RMSE offers key advantages, including its differentiability, which supports gradient-based optimization in machine learning algorithms, and its stronger penalization of large errors through squaring, making it valuable in safety-critical applications where outliers could have severe consequences.^[27]^[38]^[39] RMSE is often preferred in machine learning regression tasks when absolute deviations are prioritized over relative ones, such as in weather forecasting, where the metric directly quantifies prediction errors in measurable units like temperature or precipitation.^[40]

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