Why Opposing Errors Can Cancel

Introduction

A prediction system can overpredict for some cases and underpredict for others. When the signed errors are averaged, those opposite directions can offset one another. As a result, the average can be close to zero even if the system's predictions are substantially wrong case by case.

The mean prediction error describes the average direction of errors under a chosen sign convention. Mean absolute error (MAE), by contrast, describes their average size without allowing positive and negative errors to cancel. Reading the two together helps distinguish little overall signed error from close predictions for individual cases.

Signed Errors and Their Mean

For each case, pair its observed outcome with the prediction made for that same case. Use one consistent convention throughout: prediction error is the observed outcome minus the predicted outcome. A positive error means the prediction was too low; a negative error means it was too high. The errors and both summary measures must use the same outcome units.

These are descriptive summaries of the cases included in the calculation. They require matched observed outcomes and predictions on a common scale, and the formulas above give every case equal weight. No independence assumption is needed just to calculate these summaries. If cases are missing, be clear about which paired cases are included; comparing metrics computed on different sets can be misleading.

With this sign convention, positive MPE indicates average underprediction, while negative MPE indicates average overprediction. MPE is sometimes used as an estimate of systematic bias on the evaluated cases, but it is still an average: it does not show the size or pattern of every case's error. MAE is nonnegative and is expressed in the same units as the outcome.

Why the Signed Average Can Mislead

The reason for cancellation is arithmetic. A signed average combines positive and negative quantities before dividing by the number of cases. If the positive errors add to about the same amount as the magnitudes of the negative errors, the total is small. MAE changes the calculation order: it removes the signs first, so every case contributes a nonnegative amount.

To verify the result, apply the triangle inequality to the sum of errors:

The inequality says that the net signed error cannot have a greater magnitude than the sum of the individual error magnitudes. It does not say that MPE is usually a good substitute for MAE. When errors point in opposing directions, the inequality can be very far from equality. If every error has the same sign, then \(|\mathrm{MPE}|=\mathrm{MAE}\); if signs oppose one another, cancellation can make \(|\mathrm{MPE}|\) much smaller.

“Typical” here describes an average of absolute errors, not a claim that every case misses by exactly the MAE. Some errors can be smaller and some larger. A metric also cannot by itself explain why errors differ; it only summarizes the chosen cases under its own rule.

Using MPE and MAE Together

When reporting both metrics, give the direction and size of the MPE and the outcome units, then report MAE in those same units. For example, “MPE was 0.2 hours and MAE was 8.6 hours” makes two different statements: errors had a small net positive average, and their average absolute size was much larger. Avoid describing a small MPE as evidence that individual predictions are accurate.

The interpretation also depends on which cases were evaluated. A mean near zero on a particular test set summarizes that set; it does not guarantee that errors will cancel in another set or in future use. A few large positive errors can also offset a few large negative errors. Looking at the signed average alongside MAE prevents the direction-of-error summary from being mistaken for an error-size summary.

Common Mistake

• Wrong reasoning: “The mean prediction error is nearly zero, so predictions were nearly correct.” This treats a signed average as though it summarized the size of each error.
• Correction: Positive and negative errors may cancel. Calculate MAE as well: take the absolute value of each error before averaging. Then interpret MPE as the average direction and MAE as the average absolute size, both in context and in outcome units.