Median Absolute Deviation¶
A robust measure of univariate dispersion defined as the median of the absolute distances from the sample median, resistant to a minority of extreme observations.
Core Idea¶
The median absolute deviation (MAD) is a robust scale statistic for one-dimensional quantitative data. First find the sample median, take every absolute distance from that median, and then take the median of those distances. For the data 1, 1, 2, 2, 4, 6, 9, the center is 2 and the absolute deviations are 1, 1, 0, 0, 2, 4, 7, whose median is 1. For the data 1, 1, 2, 2, 4, 6, 9, the center is 2 and the absolute deviations are 1, 1, 0, 0, 2, 4, 7, whose median is 1.
Scope of Application¶
Use MAD with data units, missing-value treatment, sample/population meaning, raw or scaled convention, tie handling, and intended distribution stated. Use MAD with data units, missing-value treatment, sample/population meaning, raw or scaled convention, tie handling, and intended distribution stated.
- Robust statistics. Estimates scale under contamination.
- Outlier detection. Builds robust standardized distances.
- Quality control. Monitors skewed or heavy-tailed data.
- Signal processing. Summarizes noise amplitude.
- Exploratory analysis. Compares spread with medians.
Clarity¶
MAD can be zero when at least half the observations equal the median, even though other values vary. The closest near miss sets the boundary: Mean absolute deviation from the median is closest: it uses the same distances but averages them, losing the defining median aggregation and some robustness.
Manages Complexity¶
Robustness does not mean universal efficiency or adequacy. Discrete, asymmetric, multimodal, censored, or very small samples can need different scale descriptions and interval methods. The central outlier resistance–efficiency tradeoff is this: Robustness protects against extremes but can sacrifice precision in ideal light-tailed models. A second simple formula–software convention tension matters because Scaled and raw outputs can share the same name.
Abstract Reasoning¶
Use three linked moves: confirm one quantitative scale and clean missing values; compute the sample median; form absolute deviations from that median. As a collapse test, the case exits when deviations are centered elsewhere or summarized by something other than their median. A fourth check is to take their median and retain units.
Knowledge Transfer¶
Median-centered robust distance transfers to residual analysis, but univariate data and the exact two-stage median operation delimit MAD. The nearest stopping boundary is explicit: Mean absolute deviation from the median is closest: it uses the same distances but averages them, losing the defining median aggregation and some robustness. The inclusion test remains: A statistic is MAD when it takes the median of absolute deviations from the sample median under a declared scaling convention. The structure no longer applies when the case exits when deviations are centered elsewhere or summarized by something other than their median. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. It supplies both center and aggregation.
Neighborhood in Abstraction Space¶
Median Absolute Deviation sits in a moderately populated region (45th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Empirical Measurement & Statistical Inference Methods (50 abstractions)
Nearest neighbors
- Shapiro–Wilk Test — 0.89
- D'Agostino's K-squared test — 0.89
- M-Estimator — 0.88
- Statistical regularity — 0.86
- Bootstrapping populations — 0.85
Computed from structural-signature embeddings · 2026-10-08