The median absolute deviation is the robust cousin of the standard deviation. Instead of averaging squared distances from the mean, it takes the median of the absolute distances from the median. Up to half of the data can be wildly wrong without the result breaking down, which makes it ideal for messy real-world measurements, sensor data and finance.
On its own the MAD is smaller than the standard deviation. Multiplying it by 1.4826 gives the scaled MAD, which equals σ when the data are normal, so it can be used anywhere a standard deviation is needed but outliers are a concern. The modified z-score built from it, 0.6745 × (x − median) ÷ MAD, is a standard outlier test: values with |modified z| above 3.5 deserve a closer look (Iglewicz and Hoaglin).
Enter your data to get the median, the sorted absolute deviations, the MAD and its scaled version, and a table of modified z-scores. The ordinary sample standard deviation is shown alongside so you can see how much any outliers inflate it.
How to use the Median Absolute Deviation Calculator
- Data set
- Scale factor
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How it works
Find the median of the data, compute the absolute deviation of each value from it, and take the median of those deviations. Multiply by 1.4826 for the scaled MAD and use 0.6745 (x − median) ÷ MAD for the modified z-scores.
Worked example
Eight sensor readings: 2.1, 2.4, 2.5, 2.6, 2.8, 3.0, 3.1, 9.7. The median is (2.6 + 2.8)/2 = 2.7. The absolute deviations, sorted, are 0.1, 0.1, 0.2, 0.3, 0.3, 0.4, 0.6 and 7.0, so MAD = (0.3 + 0.3)/2 = 0.3 and the scaled MAD is 0.445. The sample standard deviation is 2.52 – more than five times larger – entirely because of the 9.7. Its modified z-score is 0.6745 × 7 ÷ 0.3 = 15.7, far above 3.5.
Robust versus classical spread
| Statistic | With 9.7 | Without 9.7 |
|---|---|---|
| Median | 2.7 | 2.6 |
| Median absolute deviation | 0.3 | 0.2 |
| Scaled MAD | 0.445 | 0.297 |
| Mean | 3.525 | 2.643 |
| Sample SD | 2.516 | 0.351 |
Formula
MAD = median(|xᵢ − median(x)|) · scaled MAD = 1.4826 × MAD ≈ σ for normal data · modified z = 0.6745 (xᵢ − median) ÷ MAD
Worked example
Inputs — Data set: 2.1, 2.4, 2.5, 2.6, 2.8, 3.0, 3.1, 9.7; Scale factor: 1.4826 – matches σ for normal data.
Results — Median absolute deviation (MAD) = 0.3; Scaled MAD (robust estimate of σ) = 0.4448; Median of the data = 2.7; Sample standard deviation (for comparison) = 2.5161; Number of values = 8; Possible outliers (|modified z| > 3.5) = 9.7.
Key terms
- Median absolute deviation
- The median of |xᵢ − median|.
- Scaled MAD
- 1.4826 × MAD, consistent with σ for normal data.
- Modified z-score
- 0.6745 (x − median) ÷ MAD, a robust standard score.
- Breakdown point
- The share of bad data a statistic can tolerate.
Frequently asked questions
How do I calculate the median absolute deviation?
Find the median, subtract it from each value, take absolute values and find the median of those.
What is the MAD of 2.1, 2.4, 2.5, 2.6, 2.8, 3.0, 3.1, 9.7?
The median is 2.7; the sorted deviations are 0.1, 0.1, 0.2, 0.3, 0.3, 0.4, 0.6, 7, so MAD = 0.3.
Why multiply by 1.4826?
For normal data MAD ≈ 0.6745σ, so 1/0.6745 ≈ 1.4826 turns the MAD into an estimate of the standard deviation.
What is a modified z-score?
0.6745 × (x − median) ÷ MAD. Values with |modified z| > 3.5 are possible outliers. For 9.7 in the example it is 15.7.
Why is the MAD more robust than the standard deviation?
Its breakdown point is 50%: up to half the data can be arbitrarily bad before it is ruined. One bad value can ruin the SD.
Is median absolute deviation the same as mean absolute deviation?
No – both are called MAD, but the mean version averages distances from the mean and is less robust.
How do I compute it in Excel?
=MEDIAN(ABS(range − MEDIAN(range))) entered as an array formula.