MAD
MAD. An abbreviation with several meanings; in this publication, the median absolute deviation of an app's per-meal errors, a robust measure of their spread.
Three letters, three calculations
“MAD” is one of the most overloaded abbreviations in applied statistics. Depending on the field, it can mean:
- Mean absolute deviation of forecast errors. Common in forecasting and operations management, where it is simply the average size of the errors. Arithmetically it is identical to MAE.
- Mean absolute deviation around the mean. The average distance of a set of values from their own mean. A measure of spread.
- Median absolute deviation. The median distance of a set of values from their own median. Also a measure of spread, but a robust one.
When this publication writes MAD, it means the third. Any report that uses the abbreviation without defining it has left out something important.
Why a robust spread measure is useful
Accuracy figures such as MAPE tell you how big an app’s typical miss is. They do not tell you how predictable the misses are. Two apps can share the same average error while one misses by roughly the same amount every time and the other is usually close but occasionally wildly wrong. For someone planning around a calorie budget, the first is easier to live with.
The standard deviation measures spread too, but it squares each distance, so a single large miss dominates it. The median absolute deviation does not: it looks at the middle of the distances, and one wild value barely moves it.
Worked example
Six meals logged in an app, with these misses in kilocalories (app minus truth): −12, −4, +3, +6, +9 and +150. The last is a restaurant curry whose portion the app badly over-estimated.
Median absolute deviation. The median of the six misses is halfway between +3 and +6, which is +4.5. The distances from +4.5 are 16.5, 8.5, 1.5, 1.5, 4.5 and 145.5. Their median is halfway between 4.5 and 8.5: 6.5 kcal.
Standard deviation. The mean miss is about +25.3 kcal, and the standard deviation comes out at about 61.5 kcal, almost entirely because of the curry.
Mean absolute deviation of errors (the forecasting MAD, equal to MAE). The average size of the six misses is 184 ÷ 6, about 30.7 kcal.
Same six numbers, three very different summaries. The median absolute deviation says the app is usually consistent to within a few calories; the standard deviation says it is erratic; the MAE sits between. All three are true descriptions of different things, which is the whole point.
Making it comparable to a standard deviation
For data that follow a normal distribution, the median absolute deviation multiplied by about 1.4826 estimates the standard deviation, which lets you compare the two directly. Here, 6.5 × 1.4826 is about 9.6 kcal: the spread the app would show if the curry had been an ordinary meal. Rousseeuw and Croux discuss this scaling, and more efficient robust alternatives, in their 1993 paper.
How to use it when choosing an app
Read MAD alongside a size measure, never instead of one. A low MAD with a low MAPE describes an app that is both accurate and predictable. A low MAD with a high MAPE describes one that is consistently off, which a correction factor or an adaptive target can partly absorb. A high MAD means misses you cannot plan around. For a side-by-side worked example of all three metrics, see MAPE vs MAE vs MAD.