One Bad Reading Multiplies the Standard Deviation by Twenty

The breakdown point is the number that matters

Every summary statistic has a fraction of the data that can be corrupted before it becomes arbitrary, and for the mean and the standard deviation that fraction is zero: one value taken far enough away drags either of them anywhere you like. The median and the median absolute deviation survive up to half the data being wrong, which is the highest any statistic can achieve — beyond half, there is no way to tell the corrupted values from the good ones. That gap of zero against a half is the entire practical difference, and it is why instrument logs, sensor feeds and anything else with a plausible failure mode are summarised by rank-based statistics rather than by averages.

Two different statistics share the same three letters

Median absolute deviation and mean absolute deviation are both abbreviated MAD in ordinary use, and they are not the same quantity: the first takes the median of the distances from the median, the second takes the mean of the distances from the mean. Only the first is robust — the second still has a breakdown point of zero, because the mean appears twice in it. A figure labelled MAD without further qualification is therefore ambiguous, and the difference between the two readings of it can be a factor of twenty on exactly the kind of data where somebody reached for a robust statistic in the first place. Write out which one you mean.

Frequently asked questions

Where does 1.4826 come from?

From making the two statistics agree on well-behaved data. For a normal distribution the median absolute deviation is about 0.6745 standard deviations, so dividing by that — equivalently multiplying by 1.4826 — turns a MAD into an estimate of the standard deviation that matches when nothing has gone wrong. The factor is specific to the normal distribution and to no other, which is worth remembering before quoting a scaled MAD for data that is visibly skewed. Unscaled, the MAD is still a perfectly good measure of spread; the factor exists only so the number can be compared with standard deviations quoted elsewhere.

Should I always prefer the robust version?

No, and the cost of robustness is efficiency. On clean, well-behaved data the standard deviation extracts more information from the same number of observations than the MAD does — it uses every value's actual magnitude rather than its rank, and that is a real advantage when no value is wrong. The choice is therefore about what you expect: robust statistics for measurements that can fail, classical ones for data you trust. Running both is often the most informative option of all, because a large gap between them is itself the finding — it says something in the data is behaving unlike the rest.