Why Mean Absolute Percentage Error (MAPE) is Biased and How SMAPE Attempts to Correct It

For any percentage-based forecasting metric to function mathematically, a strict binding constraint must hold: the actual values being measured must be strictly positive and meaningfully distant from zero. If a time series contains zeros—as is common in intermittent retail demand, rare event tracking, or web traffic—the denominator in the calculation vanishes, rendering the metric undefined. Despite this mathematical fragility, the Mean Absolute Percentage Error (MAPE) remains one of the most widely used accuracy measures in corporate forecasting and machine learning, prized for its scale-independence and intuitive interpretation. However, the metric carries a structural mathematical bias that systematically distorts model evaluation.[6]

The mechanism of MAPE is straightforward: it calculates the absolute difference between an actual observation and its forecasted value, divides that difference by the actual observation, and averages the result across all data points. Because the result is expressed as a percentage, it allows data scientists to compare the accuracy of models across datasets with entirely different scales—comparing a model forecasting thousands of daily website visits against one forecasting dozens of daily product sales. "MAPE is often used as the loss function in regression problems and forecasting models due to the intuitive interpretation in terms of relative error," notes Arize AI in their 2026 technical documentation.[2]

Yet the mathematics of placing the actual value in the denominator introduces a severe asymmetry. As forecasting researcher Rob J. Hyndman noted in 2014, the metric "puts a heavier penalty on negative errors... than on positive errors." Because a forecast can never be less than zero, the maximum possible MAPE for an under-prediction is 100%. Conversely, there is no mathematical upper bound for an over-prediction. If a model forecasts 1,000 units for an actual demand of 100, the MAPE explodes to 900%. However, if it forecasts 10 units for an actual demand of 100, the MAPE is only 90%.[3][5]

This asymmetry means that when MAPE is used as an optimization target during model training, it systematically rewards models that under-forecast. A model that consistently predicts values slightly lower than reality will achieve a better MAPE score than a model that occasionally overshoots, even if the absolute magnitude of their errors is identical. In supply chain and inventory management, optimizing for MAPE can therefore lead to chronic under-stocking, as the algorithm learns that predicting zero is mathematically safer than risking a high over-prediction.[6]

To correct this structural flaw, the forecasting community introduced the Symmetric Mean Absolute Percentage Error (SMAPE). Originally proposed in 1985 by J. Scott Armstrong as an "adjusted MAPE" and later formalized by Spyros Makridakis in 1993, SMAPE alters the denominator. Instead of dividing the absolute error solely by the actual value, SMAPE divides it by the average of the actual value and the forecasted value.[3][5]

This simple mechanical adjustment fundamentally changes the metric's boundaries. By incorporating the forecast into the denominator, SMAPE bounds the maximum possible error between 0% and 200%. A double overestimation and a half underestimation are treated equivalently. If an actual value is 100, a forecast of 200 yields a SMAPE of 66.67%, and a forecast of 50 also yields a SMAPE of 66.67%. This bounded nature made SMAPE highly popular, and it served as the primary evaluation metric in the influential M3 forecasting competition in 2000.[3][5]

However, the "symmetric" label is mathematically deceptive. In 1999, researchers Paul Goodwin and Richard Lawton published a paper demonstrating that SMAPE simply reverses the bias of MAPE, introducing a new asymmetry that penalizes under-forecasting more heavily than over-forecasting. The mechanism lies in how the denominator shifts. If the actual value is 100, an over-forecast of 110 produces a SMAPE of 9.52%. An under-forecast of 90—an identical absolute error of 10 units—produces a SMAPE of 10.53%.[3][5]

Because the forecasted value inflates the denominator during an over-prediction, the resulting percentage error shrinks. "SMAPE considers the underestimation to be slightly 'worse' than the overestimation," the Wikipedia consensus notes. Consequently, models optimized against SMAPE are mathematically incentivized to over-forecast, leading to the exact opposite business problem as MAPE: inflated inventory and over-ordering.[3][6]

Furthermore, SMAPE fails to resolve the binding constraint of zero-value actuals. While it prevents a division by zero when the actual is zero and the forecast is positive, it introduces a new distortion. If the actual observation is zero, any non-zero forecast—whether it is 1 or 1,000—results in a SMAPE of exactly 200%. The metric becomes entirely blind to the magnitude of the error, treating a near-miss and a massive failure as mathematically identical.[1][4]

This property makes SMAPE actively harmful for evaluating intermittent demand. In a March 2026 paper published in the journal Forecasting, researcher Stephan Kolassa demonstrated that for count data, the SMAPE is always minimized in expectation by an integer forecast. More critically, Kolassa proved mathematically that "for sufficiently intermittent data, the sMAPE is minimized by a zero forecast."[1]

Kolassa's analysis of the metric's behavior on low-volume data led to a stark conclusion. "The sMAPE is highly gameable and will usually not reward useful forecasts," he wrote, adding that "in our opinion, the sMAPE is either unnecessary or dangerous; it should not be used." Despite these known mathematical flaws, SMAPE remains widely deployed in retail forecasting systems and machine learning pipelines, often because practitioners assume the "symmetric" naming implies statistical neutrality.[1][6]

The forecasting community has increasingly moved toward alternative metrics that avoid percentage-based scaling entirely. Hyndman and Anne Koehler proposed the Mean Absolute Scaled Error (MASE) in 2006, which scales errors based on the in-sample mean absolute error of a naive benchmark forecast rather than the actual values themselves. This approach bypasses the zero-denominator problem and maintains true symmetry, though it requires historical training data to establish the benchmark scale.[5][6]

The persistence of MAPE and SMAPE highlights a tension in data science between mathematical rigor and business interpretability. Stakeholders intuitively understand percentage errors, making them easier to report in executive summaries than scaled errors or log-ratios. However, the evidence is clear that both metrics distort model behavior. As long as algorithms are rewarded for exploiting these mathematical asymmetries, the resulting forecasts will reflect the biases of the loss function rather than the reality of the data.[6]