Robust measures: MAD, trimmed and winsorized means

Robust summaries such as the median absolute deviation and the trimmed mean change little when a few values are extreme or wrong.

\text{MAD} = \operatorname{median}_i\bigl|x_i - \operatorname{median}(x)\bigr|, \qquad \text{MAD}_{\text{scaled}} = 1.4826 \times \text{MAD}

Median absolute deviation (scaled version estimates σ for normal data).

$\operatorname{median}(x)$: Median of the data.
$1.4826$: Consistency constant $\approx 1/z_{0.75}$ , making the scaled MAD comparable with an SD under normality.

\xbar_{\text{trim}(\gamma)} = \text{mean after removing } \lfloor \gamma n \rfloor \text{ values from each end}

Trimmed mean (R's mean(x, trim = γ)).

$\gamma$: Proportion trimmed from EACH end, e.g. 0.1 or 0.2.

When to use

Data with outliers or heavy tails where you want a summary that represents the bulk of the data.
Winsorized mean: instead of removing the extreme values, replace them by the nearest remaining values, then average.
Breakdown point: the fraction of data that can be arbitrarily wrong before the measure becomes arbitrary — 0% for the mean and SD, 50% for the median and MAD, $\gamma$ for a $\gamma$ -trimmed mean.

How to interpret

Large differences between the mean and trimmed mean, or between the SD and the scaled MAD, signal outliers or skew. Robust measures describe the centre and spread of the typical values, not the totals.

Common mistakes

Reading "20% trimmed" as 20% in total; it is 20% from each end.
Comparing an unscaled MAD with an SD.
Using robust summaries to hide genuinely important extremes (large insurance claims are the point, not noise).

Example

Values 10, 12, 13, 15, 100: mean 30, SD 39.2, but median 13, MAD 2 (scaled 2.97). The 20% trimmed mean drops one value from each end: $(12 + 13 + 15)/3 = 13.33$ ; the 20% winsorized mean is 13.4.