Why can a plain z-score miss an outlier that a MAD-based modified z-score flags?

## The plain z-score and its self-defeating denominator The standard score of an observation is `z = (x - mean) / s`, where `s` is the sample standard deviation. A rule such as flag anything with `|z| > 3` is the first outlier detector most people learn, and it has a structural flaw: the point under suspicion contributes to both the mean and `s`. A large value drags the mean toward itself, shrinking the numerator, and simultaneously inflates `s`, enlarging the denominator. Both effects push `|z|` down. The outlier is, in effect, allowed to widen the goalposts it must clear. ## Masking and swamping When one extreme value prevents itself or a companion from being flagged, the effect is called masking. It is worst with clusters: two or three extreme values inflate `s` together, so none of them individually looks far from the centre in standard-deviation units. The mirror-image failure is swamping, where the presence of extreme values inflates the spread and shifts the centre enough that perfectly ordinary observations on the opposite side start looking unusual. Both follow from the same cause — a threshold computed from contaminated data. There is also a hard arithmetic ceiling. For a sample of size `n` with `s` computed using the `n - 1` denominator, the largest achievable absolute z-score for any observation is `(n - 1) / sqrt(n)`. For `n = 10` that is about 2.85, so a cutoff of 3 can never fire regardless of how absurd one value is. For `n = 20` the ceiling is about 4.25. Any small-sample z-score rule is partly measuring sample size rather than unusualness. ## The robust replacement The modified z-score swaps in two rank-based quantities: - centre: the median - scale: `MAD = median(|x_i - median|)`, the median of the absolute deviations from the median The statistic is `M = 0.6745 * (x - median) / MAD`, and the conventional cutoff is `|M| > 3.5`. The constant `0.6745` is the standard normal 0.75 quantile; equivalently, `1.4826 * MAD` is a consistent estimator of the standard deviation for normal data, since `1 / 0.6745` is about `1.4826`. Scaling this way makes the modified z-score comparable in magnitude to an ordinary z-score when the data really is normal and clean, which is what makes a threshold near 3.5 meaningful rather than arbitrary. The robustness comes from breakdown points. The sample standard deviation has a breakdown point of 0% — one value moves it without limit. The `MAD` has a breakdown point of 50%, the same as the median, so nearly half the sample would have to be corrupt before the scale estimate itself is compromised. The suspect point no longer sets its own threshold, which is precisely why the modified z-score catches what the plain z-score hides. ## Worked shape of the failure Imagine a small column of nearly identical measurements plus one value orders of magnitude larger. The mean is dragged well above the bulk and `s` is enormous, so the extreme value scores perhaps 2.5 in z units and passes a cutoff of 3. The median sits squarely in the bulk and the `MAD` reflects the bulk's tiny spread, so the same point's modified z-score is huge and is flagged immediately. Nothing about the data changed; only which statistics were allowed to be contaminated. ## Where the MAD breaks The `MAD` is zero whenever more than half the observations share the same value. This happens routinely with coarse counters, rounded readings, zero-inflated columns and low-cardinality integers. A zero denominator makes the modified z-score undefined or infinite for every distinct value, which usually surfaces as an implausible flood of flagged points. When that occurs, switch to a scale estimate built from a wider slice of the distribution — an interquartile-based measure, for instance — or accept that a univariate outlier rule is the wrong tool for that column. Two further caveats apply to both statistics. They are univariate, so a point that is unremarkable on every variable separately but impossible in combination will not be flagged by either. And neither is a hypothesis test: 3 and 3.5 are conventions, not error-rate guarantees, and on genuinely heavy-tailed data a robust rule will still flag real observations by the fistful. ## What interviewers listen for The words masking and breakdown point, the correct definition of `MAD` (median of absolute deviations from the median — not the mean of absolute deviations from the mean), awareness that a scaling constant exists so the cutoff is comparable, and the practical knowledge that a zero `MAD` is a common real-world failure.