MASE Calculator — Mean Absolute Scaled Error

MASE (Mean Absolute Scaled Error), proposed by Rob Hyndman and Anne Koehler in 2006, measures forecast accuracy by dividing the forecast's error by the error a naïve forecast would have made on the training data. Because the scaling constant is fixed per series, MASE is unit-free and comparable across series with wildly different magnitudes — and, unlike MAPE, it is well defined even when actual values are zero. It is computed in three steps:

1. Scaling factor (denominator). Take the in-sample MAE of the m-step naïve forecast on the training series y₁ … yₙ:scale = (1 / (n − m)) · Σₓ₋ₘ₋ₙ | yₜ − yₜ₋ₘ |For non-seasonal data (m = 1) this is simply the mean absolute first-difference of the training series — the error of a “same as last period” forecast.
2. Forecast MAE (numerator).Take the mean absolute error of your forecasts fᵢ against the test actuals aᵢ over the h holdout points:MAE_f = (1 / h) · Σᵢ₋₁ᵣ | aᵢ − fᵢ |
3. Scale it. Divide the two:MASE = MAE_f / scale

The interpretation, following Forecasting: Principles and Practice (FPP3): MASE < 1 means the forecast is on average more accurate than the in-sample one-step (or seasonal) naïve forecast; MASE = 1 matches it; MASE > 1 is worse. For a seasonal series, set m to the season length so the baseline becomes a seasonal naïve forecast (repeat the value from one season ago) rather than a one-step one.

One edge case matters: if the training series is perfectly flat, the naïve baseline makes no error, the scaling denominator is zero, and MASE is a division by zero. This calculator reports that explicitly instead of returning a misleading value. Every MASE it shows is also independently re-derived from the algebraically-equivalent aggregate form (n−m)·Σ|a−f| ÷ (h·Σ|yₜ−yₜ₋ₘ|) and the two routes are compared — when they agree the result is flagged “cross-checked”.

• Hyndman, R.J. & Koehler, A.B. (2006) — Another look at measures of forecast accuracy (Int. J. Forecasting 22(4): 679–688)
• Hyndman, R.J. & Athanasopoulos, G. — Forecasting: Principles and Practice (3rd ed.), “Scaled errors”
• Rob J. Hyndman — MASE publication page and background notes

The formulas and the worked examples on this page were last reconciled against Hyndman & Koehler (2006) and the FPP3 “Scaled errors” section on 2026-07-13. The calculator independently re-derives every MASE through an equivalent aggregate form, so a regression in the math surfaces immediately.