Median absolute deviation:
formula and worked example

What the median absolute deviation measures

Spread describes how widely values are scattered around a centre. The standard deviation is the best known measure, but it squares every gap, so a single extreme value can dominate it. The median absolute deviation, usually written MAD, takes a different route: it finds the typical distance of the data from the median, and does so using medians throughout.

That makes it a robust statistic, one that keeps working when the data contain errors or outliers. It is a common choice for flagging suspicious points in sensor readings, transaction amounts, delivery times and survey responses, where a few extreme values would otherwise distort the picture.

Its units are the same as the data, so a MAD of 2 on prices in rupees means a typical price lies about 2 rupees from the middle one.

The procedure and its inputs

First find the median of the data. Then take the absolute difference of every value from that median, and finally take the median of those differences.

• A to E: the five observations, here 2, 4, 4, 8 and 12.
• Centre: the median of the five values, which is 4, built into the expression.
• Result: the middle value of the sorted absolute deviations, in the same units as the data.

With five values the middle position is the third one after sorting, which the expression picks with its index of 2 counted from zero.

Worked example: five values with a median of 4

Take the observations 2, 4, 4, 8 and 12.

1. 1Sort the data and find the median: 2, 4, 4, 8, 12, so the middle value is 4.
2. 2Find the absolute deviations: |2−4| = 2, |4−4| = 0, |4−4| = 0, |8−4| = 4, |12−4| = 8.
3. 3Sort the deviations: 0, 0, 2, 4, 8.
4. 4Take the middle one: 2.

For comparison, the sample standard deviation of the same data is 4, double the MAD, because the 12 pulls it upward.

Reading MAD next to the median

A MAD of 2 around a median of 4 says half of the values lie within 2 of the middle. That is large relative to the centre, so the data are widely scattered. A MAD of 0.2 around the same median would describe a tight cluster.

Because it is relative to the data scale, compare MAD to the median or to another group's MAD, not to an absolute standard. Two batches of delivery times can be compared by which has the smaller MAD.

Some analysts multiply MAD by 1.4826 so that it estimates the standard deviation when the data follow a normal distribution. That adjusted value is the one commonly used for outlier screening, with the cut-off for flagging chosen by the analyst and varying between fields.

What a small change in one value does

MAD changes only when the middle deviation changes. Moving A from 2 to 2.2 puts it closer to the median of 4, shrinking its deviation from 2 to 1.8, and because that value is the middle of the sorted deviations, the MAD falls to 1.8.

In this example the median stays at 4 after the change, so the centre in the expression remains correct. If an edit shifts the median, the centre must be recomputed before the deviations are taken.

Common slips with MAD

• Confusing median absolute deviation with mean absolute deviation, which averages the deviations from the mean instead.
• Using the mean as the centre rather than the median, which gives a different number and loses the robustness.
• Forgetting to sort before picking the middle value, or picking the wrong position for an even number of values, where the answer is the average of the two middle ones.
• Comparing a raw MAD with a standard deviation directly without the 1.4826 scale factor.

A quick check on delivery times

Five parcels took 2, 4, 4, 8 and 12 days. A new parcel takes 30 days. Does the typical spread change?

1. 1Add the new value: 2, 4, 4, 8, 12, 30. The median is (4 + 8) ÷ 2 = 6.
2. 2Deviations from 6: 4, 2, 2, 2, 6, 24. Sorted: 2, 2, 2, 4, 6, 24.
3. 3MAD is the average of the middle pair: (2 + 4) ÷ 2 = 3.

A late parcel of 30 days barely moves the MAD from 2 to 3, whereas the standard deviation would rise sharply.